Unified View of Matrix Completion under General Structural Constraints
Matrix completion problems have been widely studied under special low dimensional structures such as low rank or structure induced by decomposable norms. In this paper, we present a unified analysis of matrix completion under general low-dimensional structural constraints induced by any norm regularization. We consider two estimators for the general problem of structured matrix completion, and provide unified upper bounds on the sample complexity and the estimation error. Our analysis relies on generic chaining, and we establish two intermediate results of independent interest: (a) in characterizing the size or complexity of low dimensional subsets in high dimensional ambient space, a certain partial complexity measure encountered in the analysis of matrix completion problems is characterized in terms of a well understood complexity measure of Gaussian widths, and (b) it is shown that a form of restricted strong convexity holds for matrix completion problems under general norm regularization. Further, we provide several non-trivial examples of structures included in our framework, notably including the recently proposed spectral k-support norm.

This paper introduces a general analysis of matrix completion with norm regularization. The analysis setting can be thought of as a generalization of the setting of Negahban and Wainwright, in which we assume that the target matrix has small L-\infty norm, and attempt to prove that natural optimization formulations stably recover the target matrix, up to an error which is proportional to the maximum of the noise level and a floor (which is a consequence of only assuming the target is not spiky). The paper motivates this problem through several examples, including spectral-k-support norms, and norms for cut matrices. It states results showing that the estimation error and number of samples required is controlled by the squared Gaussian width of the descent cone of the regularizer at the target, times a logarithmic oversampling factor. This has a nice interpretation - namely, that "in general" the number of measurements in the standard basis only needs to be a logarithmic factor larger than the number of Gaussian measurements. While it is not clear how mainstream some of the examples are, it is certainly welcome to have general tools which show that accurate recovery from elementwise subsamples is not a phenomenon which is restricted to the nuclear norm.  Smaller issuesThe main theorem becomes vacuous when the number of sampled entries approaches w_G(E_R)^4. This seems to mostly be a technical issue, but it should be corrected. For general norms w_G could be quite small, and taking w_G^4 samples could represent an interesting scenario. There are imprecisions in the mathematics. The calculations in the supplement seem to move back and forth between a Bernoulli sampling model (which is implicitly used in A.1.1, (18), (a) - otherwise || PO[ X ] ||_F^2 is not a sum of independent random variables, due to collisions) and the uniform sampling model, which is used elsewhere. Proposition 4: \alpha_sp( \hat{\Delta} ) \in A(c_0, E_R ). Should just be \hat{\Delta} \in A(c_0, E_R). While the paper analyzes matrix completion in any norm (and hence with any prior information that can be encoded in a norm), there are some norms that are not compatible with the spikiness condition. For example, the paper gives robust PCA as an example of an additive decomposition that could be analyzed using the tools here. There exist analyses of robust PCA with missing entries, but all that can be shown is that observed portion of the sparse component is recovered.  Unless I'm mistaken, it should be possible to prove a better bound than (12) on the quantity w_{\Omega,W}(S), using a fairly simple argument. For S of interest here, w_G(S) > 1, and so the bound (12) is larger than w_G(S) (often substantially larger). One can write (abbreviating \Omega as O and P_\Omega as PO):      w_{O,W}(S) = E_OE_W sup_{X \in S-S} < X, PO(W) >   (11)                         <= 2 E_O E_W sup_{X \in S} < X, PO(W) >                         =2 E_O w_G( PO(S) ),(*)Consider the natural metric space on S, with d(s,t) = ( E | < s - t, w > |^2 )^(1/2); projection onto \Omega is a contraction, with respect to this distance and so it does not increase the \gamma_2 function, hence,         w_G(PO(S)) <= K w_G(S), givingw_{O,W}(S) <= K' w_G(S) via (*) and the fact that \gamma_2 gives a tight bound on the expected supremum of a Gaussian process. Am I missing something? The authors should be careful about anonymizing their supplementary material for conference submissions.


Synaptic Sampling: A Bayesian Approach to Neural Network Plasticity and Rewiring
We reexamine in this article the conceptual and mathematical framework for understanding the organization of plasticity in spiking neural networks. We propose that inherent stochasticity enables synaptic plasticity to carry out probabilistic inference by sampling from a posterior distribution of synaptic parameters. This view provides a viable alternative to existing models that propose convergence of synaptic weights to maximum likelihood parameters. It explains how priors on weight distributions and connection probabilities can be merged optimally with learned experience. In simulations we show that our model for synaptic plasticity allows spiking neural networks to compensate continuously for unforeseen disturbances. Furthermore it provides a normative mathematical framework to better understand the permanent variability and rewiring observed in brain networks.

The paper proposes a mechanism for explaining Bayesian inferenceand network plasticity in the brain using an algorithm very similar to Stochastic Gradient Langevin Dynamics. Clarity: The paper is well written. Even though my background is machine learning and not neuroscience, I was able to follow most of the paper. Originality: The mechanism itself is well studied in the machine learning literature where it is called Stochastic Gradient Langevin Dynamics (SGLD) (see Ref[1] and analysis in Ref[2]). This is also well known in physics where it is usually called the Langevin equation with noisy force (see e.g. section 5.3.'Noisy inexact algorithms' of Ref[3]and references therein). The difference seems to be that the step size (b(\theta) in Eqn. 2) is usually constant in existing work. Also, SGLD has been used only for posterior inference over weights, not for learning the network structure and rewiring.Significance: Although the proposed mechanism is well known in ML and physics, the paper could be very interesting to the neuroscience community. Both network plasticity and whether the brain does Bayesian inference are important questions, and this seems like a plausible explanation worth investigating more. I would suggest to rewrite the paper so that the mechanism itself is presented as existing work in ML, and focus more on how this is realized in biological neural networks.Quality: The paper is convincing and shows experimentally how Bayesian inference using the proposed mechanism can help plasticity. Not having much of a neuroscience background, I wasn't able to tell whether this is actually biologically plausible or not, e.g. a) how are priors represented in biological neural networks? b) Also a problem with SGLD is that it is difficult to mix between modes of the posterior distribution. Is this not a problem in biological neural nets?The paper should cite these (and/or related) references:[1] Bayesian Learning via Stochastic Gradient Langevin Dynamics, Welling and Teh, ICML 2011[2] Approximation Analysis of Stochastic Gradient Langevin Dynamics by using Fokker-Planck Equation and Ito Process, Sato and Nakagawa, ICML 2014[3]The Hybrid Monte Carlo algorithm on parallel computers, A.D. Kennedy, Parallel Computing 1999


Stop Wasting My Gradients: Practical SVRG
We present and analyze several strategies for improving the performance of stochastic variance-reduced gradient (SVRG) methods. We first show that the convergence rate of these methods can be preserved under a decreasing sequence of errors in the control variate, and use this to derive variants of SVRG that use growing-batch strategies to reduce the number of gradient calculations required in the early iterations. We further (i) show how to exploit support vectors to reduce the number of gradient computations in the later iterations, (ii) prove that the commonly-used regularized SVRG iteration is justified and improves the convergence rate, (iii) consider alternate mini-batch selection strategies, and (iv) consider the generalization error of the method.

Comments:- Tricks like non-uniform sampling and then correcting by importance sampling have been around for much longer, see e.g. [1] which claims a speed-up of 3x. Those older approaches chose the probability based on error instead of L_i -- could you discuss/compare such an approach, at least in practice? I'd suspect this may be useful in practice where the L_i may not be known.- Channeling Leon Bottou: please discuss the limitations, and the cases where your method may *not* be applicable.- I would really appreciate an open-source code release accompanying the paper, which would increase its impact.- The experimental section and figures are clearly the weakest part of the paper, I'd recommend a serious second pass. For example, all plots should have error bars, state over how many runs they were averaged, labeled more clearly (SVRG -> full batch in top plots), the test-error plot does not have the resolution to be readable (maybe log-scale is better even if it won't go to zero). The same holds for the figures in the appendix. I do not understand why there is no difference at all between uniform sampling and Lipschitz sampling (in experiment 3)? Of course, a more challenging domain than logistic regression would be appreciated, but that can rightfully be shown in another paper. [1] Geoff Hinton, "To recognize objects, first learn to generate images", 2007.Typos:L19 a variant L53 increases over L77 expectation missing L96, L132 expectation symbol misformatted L164 B^s L289 can be used L316 superscript instead of subscript L325 sometimes L416 sometimes often?


The Poisson Gamma Belief Network
To infer a multilayer representation of high-dimensional count vectors, we propose the Poisson gamma belief network (PGBN) that factorizes each of its layers into the product of a connection weight matrix and the nonnegative real hidden units of the next layer. The PGBN's hidden layers are jointly trained with an upward-downward Gibbs sampler, each iteration of which upward samples Dirichlet distributed connection weight vectors starting from the first layer (bottom data layer), and then downward samples gamma distributed hidden units starting from the top hidden layer. The gamma-negative binomial process combined with a layer-wise training strategy allows the PGBN to infer the width of each layer given a fixed budget on the width of the first layer. The PGBN with a single hidden layer reduces to Poisson factor analysis. Example results on text analysis illustrate interesting relationships between the width of the first layer and the inferred network structure, and demonstrate that the PGBN, whose hidden units are imposed with correlated gamma priors, can add more layers to increase its performance gains over Poisson factor analysis, given the same limit on the width of the first layer.

Acknowledge the author's rebuttal.I overall maintain my general sentiment about the paper and look forward to more discussion of this work's relationship to traditional neural / deep networks.---In this paper the authors present a deep belief network in which the intermediate hidden layers are represented by nonnegative weights.They apply this model to text documents as a "deep belief topic model" (my own phrasing).They derive an inference / update step, in which dirichlet vectors are propagated up the network and gamma weights are propagated back down it, and perform an empirical evaluation of the model.Overall this paper was written fairly clearly.I thought the paper was mostly fine, but I believe that the empirical analysis could be improved (more on that shortly).I also would be shocked if nobody has tried to implement a neural network with weights outside the range [0, 1] before; it would be good to see some background material discussed even if it wasn't used to motivate this model.I also had a few other minor comments: - It's not clear whether a fixed budget is appropriate for fixing the size of the various layers (and, in the absence of more detail about the method, I am skeptical).It still seems that these layers could easily be too large or too small despite the fixed budget approach.I suspect that some held-out subset of data could be used to decide whether to grow or shrink the model. - It's not clear whether adding any layers beyond the second layer help.The performance in the figures indicates that they're marginally better, but it's difficult to be confident given how much variance there is in the plots (with the possible exception of Figure 3).Further, the topic descriptions from sample words (line 417) aren't substantially different; it's very possible that the authors' observation that the topics are more specific at the top layer are simply because that layer is largest. - The authors describe a process to jointly train all layers in contrast to training in a greedy fashion (line 264).In this respect it resembles the way a traditional neural network (from 20 years ago) was fit, i.e., forward/back propagation, except that it is with samples. - It seems that this paper could introduce the idea of the Gamma-Poisson belief network without needing to use an application which uses the CRT (i.e., the application itself is complicated, and it's not clear that this paper necessitates a complicated application to evaluate a neural network architecture.Regarding the experimental validation, I believe it could be improved.First, it would be useful to see more comparisons with baselines.For the classification task, for example, it would be good to see an SVM or ridge regression using word counts as covariates, which I have seen outcompete topic models for classification. (N.B. I don't think the model needs to beat these baselines).It would be good to see some comparison with e.g. a typical neural network, in which weights are in [0,1], perhaps with the bottom layer adapted to suit the Poisson/Multinomial observations.Particularly, claims like that on 145 ("..clearly shows that the gamma distributed nonnegative hidden units could carry richer information than the binary hidden units and model more complex nonlinear functions") could an should be backed up with a comparison with a model using binary hidden units.It's not clear that a binary network couldn't express similar nonlinear functions with the right architecture.Is it necessary to remove stopwords?Topic models like LDA can handle them fine if you're okay with a stopword topic (and it's not clear why that would hurt an evaluation).Nit picks:  - The authors discuss classification in the experiment section (line 349) before the actual task is introduced in 353, which was a bit confusing. - Line 56: "budge" -> "budget"


Local Smoothness in Variance Reduced Optimization
We propose a family of non-uniform sampling strategies to provably speed up a class of stochastic optimization algorithms with linear convergence including Stochastic Variance Reduced Gradient (SVRG) and Stochastic Dual Coordinate Ascent (SDCA). For a large family of penalized empirical risk minimization problems, our methods exploit data dependent local smoothness of the loss functions near the optimum, while maintaining convergence guarantees. Our bounds are the first to quantify the advantage gained from local smoothness which are significant for some problems significantly better. Empirically, we provide thorough numerical results to back up our theory. Additionally we present algorithms exploiting local smoothness in more aggressive ways, which perform even better in practice.

All variance reducing stochasting algorithms use a sampling distribution over the training points. Though the first works used a uniform distribution, it has quickly been extended to distributions based on the Lipschitz constants of the gradients of each individual function. These constants, however, were computed once for the entirety of the optimization.Orthogonally, the optimal stepsize for these methods relies on the maximum of these constants. Schmidt et al proposed a line-search for the SAG algorithm which adapted this stepsize throughout the optimization.The current paper merges these two techniques in some way by adapting the sampling distribution along the optimization. Though obvious in retrospect, I do not know of any work implementing this idea and making it work.I think Algorithms 1 and 2 are of very limited interest since they apply in very contrived cases. In particular, I think the authors should make clearer than, in most cases, these algorithms will not bring anything over the classical ones (a brief mention is made at the bottom of page 5).I am however impressed by the results obtained by empirical SVRG and this enough makes it worth publicizing. It is unfortunate that these results end up in the appendix.Overall, my main comment would be to rewrite this paper as a practical implementation with promising results and expand the experimental section, while shrinking the more theoretical part which is obvious.Other comments: - The beginning of section 2 with the SGD (typo) and SVRG updates is quite confusing. Please reword it. - I did not see mentioned the cost of drawing from a non-uniform distribution. It seems to be that, to be efficient, you need to build a tree of the probabilities. Do you need to rebuild the tree everytime a local L changes?In conclusion, I'm leaning for acceptance because of the impressive results but I think it should definitely be recast as an empirical paper.


Discriminative Robust Transformation Learning
This paper proposes a framework for learning features that are robust to data variation, which is particularly important when only a limited number of training samples are available. The framework makes it possible to tradeoff the discriminative value of learned features against the generalization error of the learning algorithm. Robustness is achieved by encouraging the transform that maps data to features to be a local isometry. This geometric property is shown to improve (K, )-robustness, thereby providing theoretical justification for reductions in generalization error observed in experiments. The proposed optimization framework is used to train standard learning algorithms such as deep neural networks. Experimental results obtained on benchmark datasets, such as labeled faces in the wild, demonstrate the value of being able to balance discrimination and robustness.

This paper presents a method to learn a transformation using a loss function in which it is possible to control the trade-off between robustness and class discrimination. The loss function is a linear combination of a discriminative element, which tries to maximise the between-class separation and a robustness element, which tries to preserve isometry between source feature space and target (classification) mapping. The method is theoretically justified based on the metric learning method of [1], thought the focus here is on learning a mapping function, rather than a metric. The proposed loss function is differentiable and can be used in a Gradient descent framework. It was applied as part of a Neural Network, enabling non-linearity by having multiple layers.The experiments prove the concept, for instance, they show that when a reduced training set is available, one benefits from using smaller values of lambda (the discrimination ratio), which is quite intuitive.However, I felt that the experiments are a bit limited. It would have been interesting to see more practical applications of the proposed method, showing when it is relevant to be able to tune between robustness and discrimination power. Perhaps this method could be applied in domain adaptation datasets. Minor style issues: - sentences starting with "And..." - missing spaces between words and brackets, e.g. "decreases(Fig. 3b)" - "attentions have" -> "attention has" Note: I'm pleased to see that clarity has improved in the final version of the paper and that more details and evaluations are presented in the experimental results section.


Structured Estimation with Atomic Norms: General Bounds and Applications
For structured estimation problems with atomic norms, recent advances in the literature express sample complexity and estimation error bounds in terms of certain geometric measures, in particular Gaussian width of the unit norm ball, Gaussian width of a spherical cap induced by a tangent cone, and a restricted norm compatibility constant. However, given an atomic norm, bounding these geometric measures can be difficult. In this paper, we present general upper bounds for such geometric measures, which only require simple information of the atomic norm under consideration, and we establish tightness of these bounds by providing the corresponding lower bounds. We show applications of our analysis to certain atomic norms, especially k-support norm, for which existing result is incomplete.

=== Summary ===In this paper, the authors propose to bound geometric quantities that appears in statistical error bounds of structured estimation problems with atomic norms.It was previously shown that the estimation error of Dantzig-like estimator, with random Gaussian design, could be bounded by using three key quantities: the Gaussian width of the unit ball of the regularizer, the Gaussian width of the spherical cap of the tangent cone and the restricted norm compatibility constant. The goal of this article is to provide easy to compute bounds for these three quantities, in the case of sign-invariant atomic norms.In section 3, the authors introduce upper bounds on the three geometric quantities, which are the main contribution of the paper. The Gaussian width of the unit ball can be bounded by decomposing the set of atoms as the union of simpler sets, for which the Gaussian width are easy to compute. This result already appeared in [14]. The Gaussian width of the spherical cap of the tangent cone can be bounded using a single subgradient of the regularizer, taken at the true parameter vector w^*. Finally, the restricted norm compatibility constant can be bounded using two other "norm compatibility constants" which are simpler to compute (because the set on which they are defined are simpler). The authors show on two examples, the L1 and OWL norms, how these results can easily be applied.In section 4, the authors provide lower bounds for the two last quantities. The lower bounds does not match the upper bounds, but show that the upper bound are relatively tight, and thus usefull.In section 5, the bounds obtained in this paper are applied to the k-support norm, improving previously known results.=== Significance ===I believe this paper is addressing an important problem, providing interesting and useful new tools for studying sparse estimation problems with atomic norms. The paper is clearly written and well organized. The use of examples helps to understand the results and how to apply those in practice. Finally, the bounds derived in the paper allowed to obtain new results for the k-support norm, showing the relevance of the results.


A Nonconvex Optimization Framework for Low Rank Matrix Estimation ⇤
We study the estimation of low rank matrices via nonconvex optimization. Compared with convex relaxation, nonconvex optimization exhibits superior empirical performance for large scale instances of low rank matrix estimation. However, the understanding of its theoretical guarantees are limited. In this paper, we define the notion of projected oracle divergence based on which we establish sufficient conditions for the success of nonconvex optimization. We illustrate the consequences of this general framework for matrix sensing. In particular, we prove that a broad class of nonconvex optimization algorithms, including alternating minimization and gradient-type methods, geometrically converge to the global optimum and exactly recover the true low rank matrices under standard conditions.

The paper addresses convergence of alternating algorithms in non-convex formulations of matrix completion and matrix sensing. Global convergence with geometric rate is established under certain conditions.The paper is very ambitious. It addresses 3 algorithms (alternating exact minimisation, alternating gradient descent, gradient descent) for 2 problems (completion and sensing). Because it is impossible to address the 6 all possible cases, many parts are either sketched or relegated to supplementary materials. As the results, the paper feels a bit scattered and doesn't read very well.I would prefer that the authors address one special case rigorously and then explain how the results apply to other cases with minor modifications.The experiments are trivial - they should be either discarded or greatly improved (preferred).** General comments - the role of initialisation in the proposed convergence results is not clear. For example, it doesn't seem to intervene in the proposed Theorems (though this is announced in line 38). I suppose that initialisation is critical to obtain the global convergence ?- after reading Section 3.1, it feels like the theoretical study is done, that Eq. on line 223 provides the necessary results. What is it that Theorems 3.4 and 3.6 specifically address in the context of the discussion of Section 3.1 ? - more generally, the paper doesn't sufficiently emphasises what are the difficulties that are being solved and doesn't provide enough intuitions about the solutions. For example, Section 4 is not very inspiring and does not sufficiently explains how the proof works at the general level. A key contribution seems to be the use of the projected Oracle divergence as defined in Eq. (3.5) and I wish it was more motivated. - how realistic/stringent are the conditions of Theorems 3.4 and 3.6 in practice ? - please explain the difference between alternating gradient descent and gradient descent - examining Algorithm 1 doesn't tell much. - the issue of identifiability and the chosen forms of U and V could be more discussed, as well as the relation between the respective convergences of U, V and M=UV.** Specific comments - I don't see why the decomposition of M* in line 309 should depend on t ? - what's the meaning of Assumption 3.5 ? I don't see how it prevents sparsity. The use of star indices is confusing (clashes with the use of star to denote ground truth). - what's Omega in line 241, Eq. 3.10, etc ? - the sentence in lines 122-123 seems useless. - what's the renormalisation issue in line 188 ? - weird phrasing in lines 228-229. - caption of Fig.1 essentially repeats the text of Section 5.** typos - entires - board - takeS line 75 - existS line 276 - noting that line 125 - that our line 253


Learning Structured Output Representation using Deep Conditional Generative Models
Supervised deep learning has been successfully applied to many recognition problems. Although it can approximate a complex many-to-one function well when a large amount of training data is provided, it is still challenging to model complex structured output representations that effectively perform probabilistic inference and make diverse predictions. In this work, we develop a deep conditional generative model for structured output prediction using Gaussian latent variables. The model is trained efficiently in the framework of stochastic gradient variational Bayes, and allows for fast prediction using stochastic feed-forward inference. In addition, we provide novel strategies to build robust structured prediction algorithms, such as input noise-injection and multi-scale prediction objective at training. In experiments, we demonstrate the effectiveness of our proposed algorithm in comparison to the deterministic deep neural network counterparts in generating diverse but realistic structured output predictions using stochastic inference. Furthermore, the proposed training methods are complimentary, which leads to strong pixel-level object segmentation and semantic labeling performance on Caltech-UCSD Birds 200 and the subset of Labeled Faces in the Wild dataset.

Summary: A framework for learning complex structured output representations is presented. To this end variational auto-encoders (VAE) are extended to `conditional VAEs,' i.e., conditioned on the input data x.Quality - The paper is mostly well written, could however be improved occasionally. Clarity - The idea is clearly presented but some details are missing. Originality - Conditional VAEs seem to be a straightforward extension of standard VAEs, but certainly worth a discussion Significance - The significance could be improved by a more extensive evaluation showing results for various modificationsComments: - I think the term generative is typically used when learning distributions that also involve the input data. This seems different from the proposed conditional variational auto-encoder technique that is not capable of generating data via p(x). Hence readers could get confused.- Three different methods are possible for inference, i.e., estimation of y given x. Two of them are evaluated using toy data. But which of the ones discussed in Sec. 3.1. do the authors recommend?- The authors mention a gap (l. 148) measurable by the regularization term during/after learning, i.e., the proposal q(z|x,y) and the prior p(z|x) don't exactly agree. How big is this gap?- The proposed solution to close the gap is to introduce a Gaussian stochastic neural network which models the reconstruction term directly, i.e., no regularization using a KL divergence. A weighted combination of the conditional VAE (computes regularization+reconstruction) and the newly introduced network (computes reconstruction) is then proposed as the desirable cost function. This however seems counter-intuitive to me. I would have expected the regularization gap between both distributions to be closer to zero if more weight is placed on the regularization rather than on the reconstruction term. Could the authors provide intuition? How did the authors set/cross-validate \alpha and what was the resulting value?- There is a significant amount of semantic image segmentation techniques using CNNs. For fairness the authors might consider citing a few more recent ones.- More details regarding the noise injection into data x is useful for a reader. What exactly did the authors do?- Moreover as a reader I'm very interested in quantitative results regarding the modifications described in Sec. 3.3.2, i.e., how important is the `latent-to-output' pathway really? In addition I'm curious as to how much performance improved by using the direct output prediction as the input for the conditional prior network? Further I'd appreciate if the authors could clarify whether the conditional prior network as trained using only the direct output prediction as its input or whether both data x and prediction \tilde{y} were used? Although some of the modifications are investigated in Tab. 3, a more careful ablation analysis would seems very useful to a reader.- Since the conditional VAE involves multiple networks I'm wondering whether its improved performance is a result of the larger amount of trainable parameters. Can the authors comment?- Since efficiency is claimed in the abstract I'm wondering about the time for training and inference. Can the authors comment?------------- As pointed out by fellow reviewers some citations could be added for completeness.


Fast Two-Sample Testing with Analytic Representations of Probability Measures
We propose a class of nonparametric two-sample tests with a cost linear in the sample size. Two tests are given, both based on an ensemble of distances between analytic functions representing each of the distributions. The first test uses smoothed empirical characteristic functions to represent the distributions, the second uses distribution embeddings in a reproducing kernel Hilbert space. Analyticity implies that differences in the distributions may be detected almost surely at a finite number of randomly chosen locations/frequencies. The new tests are consistent against a larger class of alternatives than the previous linear-time tests based on the (non-smoothed) empirical characteristic functions, while being much faster than the current state-of-the-art quadratic-time kernel-based or energy distancebased tests. Experiments on artificial benchmarks and on challenging real-world testing problems demonstrate that our tests give a better power/time tradeoff than competing approaches, and in some cases, better outright power than even the most expensive quadratic-time tests. This performance advantage is retained even in high dimensions, and in cases where the difference in distributions is not observable with low order statistics.

Summary: This paper proposes two fast methods for nonparametric two sample testing. These are based on the notion of random metrics on distributions. The authors propose two approaches to make random metrics: one based on smooth characteristic functions, and one based of kernel mean embeddings with analytic kernels. Both the approaches essentially make use of representations of probabilities as analytic functions (in an RKHS). Analytic functions are different from each other almost surely w.r.t. any absolutely continuous distribution. Therefore almost surely unique representations of probabilities can be defined, by evaluating of the analytic representations of those probabilities on a random sample from an absolutely continuous distribution. By doing such evaluations on J random samples, the probabilities can be a.s. uniquely represented as J dimensional vectors. The random metrics are defined as distances between these J dimensional vectors. The resulting tests are fast to compute, and have high statistical powers.I think this paper is nice and should be accepted. It is clever to represent distributions by finite dimensional vectors with analytic functions. I believe this approach can also speed up other methods based on kernel mean embeddings, and therefore would have significant impacts. Quality: Both the theoretical analysis and experiments of the paper are well conducted. The following are some comments.As J goes to infinity, Eq. (10) tends to || \mu_P - \mu_Q ||_{L2(\nu)}, where \nu is the absolutely continuous distribution of T_j. This further equals to || T^{1/2} (\mu_P - \mu_Q) ||_H, where T denotes the integral operator defined by the kernel and the distribution \nu. This can be seen as a smoothed version of MMD, as T^{1/2} has an effect of smoothing. Therefore Eq. (10) may be seen as an estimator of the smoothed MMD. This might explain why the test based on mean embedding performs well empirically for high-dimensional settings. Is this understanding correct? And if the space of the rebuttal permits, please explain how the performance change when J is increased. Clarity: This is paper is well organized and easy to understand.Originality: I believe that clever idea of using analytic functions to represent distributions is original. Significance: The paper proposes approaches for representing distributions a.s. uniquely as finite dimensional vectors. I believe these are beneficial not only to two sample testing, but also to other problems of nonparametric statics. Examples include independence testing, Bayesian inference with graphical models, and reinforcement learning. These have been effectively tackled with kernel mean embeddings, but they are known to be slow. The approaches proposed in this paper can be used to speed up those kernel methods. Therefore I believe this paper would have significant impacts. Minor points: Line 164: if P \neq Q then d(P,Q) > 0 a.s. ?


Regret Lower Bound and Optimal Algorithm in Finite Stochastic Partial Monitoring
Partial monitoring is a general model for sequential learning with limited feedback formalized as a game between two players. In this game, the learner chooses an action and at the same time the opponent chooses an outcome, then the learner suffers a loss and receives a feedback signal. The goal of the learner is to minimize the total loss. In this paper, we study partial monitoring with finite actions and stochastic outcomes. We derive a logarithmic distribution-dependent regret lower bound that defines the hardness of the problem. Inspired by the DMED algorithm (Honda and Takemura, 2010) for the multi-armed bandit problem, we propose PM-DMED, an algorithm that minimizes the distribution-dependent regret. PM-DMED significantly outperforms state-of-the-art algorithms in numerical experiments. To show the optimality of PM-DMED with respect to the regret bound, we slightly modify the algorithm by introducing a hinge function (PM-DMED-Hinge). Then, we derive an asymptotically optimal regret upper bound of PM-DMED-Hinge that matches the lower bound.

This paper studies the stochastic partial monitoring problem. Asymptotic lower bound is derived and an asymptotically optimal algorithm is proposed.Overall Comments:- The lower bound might be easier to be derived from the results in the following paper: Graves, Todd L., and Tze Leung Lai. "Asymptotically efficient adaptive choice of control laws incontrolled markov chains." SIAM journal on control and optimization 35.3 (1997): 715-743.- I did not check the proof of the optimality for the algorithm but I think the result is reasonable: The lower bound requires a solution to an optimization problem which represents how many times each action needs to be played to eliminate "confusing environments". The algorithm tries to approximate these number of plays by the empirical estimate of p. A uniform exploration rate (\sqrt(log t)) is used the guarantee the approximation quality of the solution to the optimization problem.    Quality, clarity, originality and significance:The paper is well written and the contribution of asymptotic analysis is interesting. The existing CBP algorithm for stochastic partial monitoring is designed for achieving the T^2/3 minimax bound and does not exploit the observation structure efficiently and thus is not asymptotically optimal. It could be more interesting to consider designing an algorithm which is both asymptotically and minimax optimal but this might be harder and beyond the scope of this paper. I am not very sure about the significance. Although the ideas of deriving the lower bound and the algorithm are not completely novel fitting these ideas into the partial monitoring framework still require some efforts.


SGD Algorithms based on Incomplete U -statistics: Large-Scale Minimization of Empirical Risk
In many learning problems, ranging from clustering to ranking through metric learning, empirical estimates of the risk functional consist of an average over tuples (e.g., pairs or triplets) of observations, rather than over individual observations. In this paper, we focus on how to best implement a stochastic approximation approach to solve such risk minimization problems. We argue that in the largescale setting, gradient estimates should be obtained by sampling tuples of data points with replacement (incomplete U -statistics) instead of sampling data points without replacement (complete U -statistics based on subsamples). We develop a theoretical framework accounting for the substantial impact of this strategy on the generalization ability of the prediction model returned by the Stochastic Gradient Descent (SGD) algorithm. It reveals that the method we promote achieves a much better trade-off between statistical accuracy and computational cost. Beyond the rate bound analysis, experiments on AUC maximization and metric learning provide strong empirical evidence of the superiority of the proposed approach.

-- Review after rebuttal -- I thank the authors for their responses. I notched up my recommendation a bit, but still I cannot offer a clear accept for this interesting paper. My quibble with Prop. 1 was not fully resolved. The authors said "the variances \sigma are w.r.t. the gradient of the *true* loss L(\theta)". This is a bit unclear but I take it to mean that\theta should be \theta^* in Prop. 1?That would be an important typo that was not acknowledged. Furthermore, in later search through the citation the authors provide and other works, I found an inconsistency in the results, which basically boils down to the \sum n'_k being a productin Prop. 1. Unfortunately, I cannot disclose other works to maintain author anonymity. Together with the way the authors originally presented this result, this makes me skepticalabout Prop. 1. Regarding my comments on excess mathematical clutter, these were generally acknowledged by the authors but it is unclear how much can be done in such limited time.Still, I decided to increase my rating for this paper, so if other reviewers are happy with therebuttal the paper could get through.--  In this paper the authors investigate ways to perform stochastic approximation that use U-statistics in a way that is computationally and statistically efficient. The quite surprising result is that using "incomplete U-statistics" is significantlymore efficient than "complete U-statistics", in both efficiency aspects. This is surprising because complete U-statistics use averages over all"data tuples", albeit reducing the sample size.I am skeptical about this result, which begins in Proposition 1. Specifically the variance for the complete U-statistics is proportional to 1/\sum n_k' whereas the variance for the incomplete U-statistic is 1/\prod {\choose..}. (see Eq. in Prop. 1) This difference is huge and the mathematical formulation is counter-intuitive. If we set n_k'=n_kshouldn't the complete U-statistic be efficient like theory suggests? It seems that efficiency should depend onterms like n_k' / n_k. This would not follow from Prop. 1. Unfortunately there is no proof for Prop. 1, but a citation that is not very helpful.I strongly suggest that the authors provide a simple proof for that fact because it is alsoimportant for Theorem 2, and for the paper in general.The other major problem is mathematical clutter. For example, Def. 1 is a quarter page long and overwhelming. A more intuitive explanation of U-statistics is possible since this is a well-developed subject in statistics. I would recommend to build up from primitives, e.g., introduce U-statistics when K=1, and thengeneralize. I also think that Proposition 2 and Theorem 1 are standard theoryin stochastic approximation, and should be in the Appendix. This sectionconfused me the most. How is normality used?Is the purpose of this section to show that the conditional variance is importantfor convergence of SGD? If it is, then the section is unnecessary because it replicates standard theoryresults. If not, the authors should try to be more explicit and precise of what this section isabout and what are the new results.The experimental results are limited and they don't seem to supportthe claim "the incomplete U-estimator significantly outperforms its complete variant". For example, the results in Figure 1 show that the difference between the two methodsis not statistically significant, unless plotted standard deviations do not reflectstandard errors; in such case the authors should make this explicit.Finally, there are several parts in the paper that I was confused or struggled through excesswording e.g.,1) In the abstract: "In this paper, we ague that in the large-scale settingthe inductive principle of SA for risk minimization should be implemented in a very specific manner. ...". This sentence spans 3 lines and gives no information.2) In Lines #48-49 a description of SGD is given that is not typical. In plain SGD there is no averaging.3) Overall the use of "tuples" is very confusing (starting from the abstract.) "Tuple" can mean many different things. Because this term is heavily usedand only becomes more clear in Page 3, I would suggest a very early definition.4) Indexes in Line #106 of Def. 1 are non-overlapping?5) In Line #173 it says that the complete U-statistic is "far from optimal". I think this isevident by definition?6) Remark 1 is confusing and could be omitted, or added at a discussion section, orin conclusion.7) The "conditional performance analysis" is confusing as well. In Line #207 it says that the analysis is "conditional on the observed data samples". But from what follows it seems that we simply condition on the number of samples, not the data samples themselves, is that correct?8) In Line #215 it says "We point out that the present analysis can be extended to more generalcontexts" and then the authors cite the work of Moulines and Bach on non-asymptoticanalysis of stochastic approximations. This is a very confusing statement.What extension do the authors mean?Minor: * Line #184 D_B should have "mathcal" * Constant C in Line #295 is used earlier,


Learning with Symmetric Label Noise: The Importance of Being Unhinged
Convex potential minimisation is the de facto approach to binary classification. However, Long and Servedio [2010]  proved that under symmetric label noise (SLN), minimisation of any convex potential over a linear function class can result in classification performance equivalent to random guessing. This ostensibly shows that convex losses are not SLN-robust. In this paper, we propose a convex, classification-calibrated loss and prove that it is SLN-robust. The loss avoids the Long and Servedio [2010]  result by virtue of being negatively unbounded. The loss is a modification of the hinge loss, where one does not clamp at zero; hence, we call it the unhinged loss. We show that the optimal unhinged solution is equivalent to that of a strongly regularised SVM, and is the limiting solution for any convex potential; this implies that strong 2 regularisation makes most standard learners SLN-robust. Experiments confirm the unhinged loss' SLN-robustness is borne out in practice. So, with apologies to Wilde [1895], while the truth is rarely pure, it can be simple.

SUMMARYThe paper presents a solution to binary classification with symmetric label noise (SLN). They show that, in order to obtain consistency (w.r.t. to the 0-1 loss in the "noiseless" case) while using a convex surrogate, one must use the loss $\ell(v,y) = 1 - vy$ -- the "unhinged loss" -- , which is shown to enjoy some useful properties, including robustness to SLN. In a more restricted sense of robustness, it is the only such loss, but in any case it overcomes the limitations of other convex losses for the same problem.Different implications of using the unhinged loss are discussed; the problem of classification with SLN with the unhinged loss and "linear" classifiers is investigated and solved analytically. The authors also present an empirical evaluation to motivate that their theoretical considerations have practical impact. CONTRIBUTIONSThe main contribution is an SLN-robust loss (consistent in spite of label noise), and a demonstration that this function enjoys certain properties that make it suitable for classification with SLN, including with linear classifiers.The contributions are relevant for the problem of classification with SLN; they have limited scope outside the problem, which in this reviewer's opinion somewhat specific, but at the same time they address the it effectively and describe a clear-cut solution for the problem. SUPPORTThe support for the contributions is adequate. The results are clearly stated and the proofs are correct easy to follow. The illustrative empirical support of the theoretical results is also suitable, and may shed light on the finite-sample behavior of minimizers of SLN robust losses. TECHNICAL QUALITYThe paper is technically sound and well-executed.A discussion of the limitations of strong SLN robustness (as opposed to SLN robustness) would be informative, as would a discussion about the limitations of the SLN problem as a model for other problems would (though somewhat non-central).More considerations about the finite-sample behavior of minimizers of the unhinged loss would strengthen the discussion of the empirical results. ORIGINALITYThe work is well-placed in the literature, the results well-connected to existing ones, and contributions, novel. CLARITYThe paper is well written and easy to read. The different facts about the unhinged loss are brought together cohesively.Considering binary classification problems in general, the problem of classification with SLN is presented as more realistic than the general case, but this reviewer would appreciate some more motivation as to why one believes so (see detailed comments). (Appendix B.1 seems to briefly touch on this matter.) Perhaps SLN robustness is a step towards robustness to some types of outliers? FOR THE REBUTTALI would appreciate if the authors could give a brief motivation on SLN problems, and their importance in a more general binary classification context. DETAILED COMMENTSI am a bit confused as to the motivation behind SLN. On the one hand, in two dimensions it would be odd to just flip classes uniformly over the space. On the other hand, maybe this behavior in higher dimensions is not too strange and makes sense to be considered. Are there any practical issues motivating the interest in SLN (applications, robustness, etc.), or if the interest is of theoretical nature?Although the loss is unhinged, unboundedness is dealt with by bounding the scores. This essentially turns the unhinged loss into a hinge loss again. I would suggest looking into [1,2], from which we can infer that with $\mathcal{F} = \mathbb{R}^{[-1,1]}$ the hinge and unhinged losses are equivalent and, by a scaling argument, that so are the unhinged and the "scaled" hinge $f(v) = (1 + v/B)_+$ with $\mathcal{F} = \mathbb{R}^{[-B,B]}$. In terms of regret/excess risk (i.e., ignoring constants added to the losses) one can also obtain equivalence with $f(v) = \frac{1}{B}(1 + v)_+$. These results can be used to even obtain tight risk bounds with calibration functions [3].Regarding Conjecture 1 in Appendix B.2, I would point out that linear scorers and (some) proper losses do not go very well together: even in the absence of noise, minimizers of these losses fail to be consistent w.r.t. the best linear classifier in the class [4]. Perhaps the proof techniques used in [4] can provide some directions to develop proofs for Conjecture 1. I would also imagine that Conjecture 1 with linear scorers with $\|w\| \leq C$ for constants $C$ would be quite interesting to investigate.Based on your experimental results, and considering that the theoretical results are asymptotic in nature, can you make any statements about SLN robustness in the finite-sample setting based on your data? REFERENCES[1] Zou, Zhu & Hastie, 2006: "The margin vector, admissible loss and multi-class margin based classifiers" [2] Avila Pires, Szepesvari & Ghavamzadeh, 2013: "Cost-sensitive multiclass classification risk bounds" [3] Steinwart, 2007: "How to compare different loss functions and their risks" [4] Long & Servedio, 2013: "Consistency versus Realizable $H$-Consistency for Multiclass Classification" SUGGESTED FIXES[l:75] "argmin" [l:120-121] "w.r.t." [l:144-145] Your appendix references seem to be broken. [l:234-237]This intuition is compatible with a "graphical" view of the SLN setup, where we would lay out points in space and uniformly flip their labels with a given probability. This also begs the question as to whether this setup is realistic at all, of course, but it makes sense that a classifier should be able to "ignore" large errors introduced by the noise in order to succeed. [l:245] Interestingly, Proposition 5 is also the case for the hinge loss with scaled scores, which is equivalent to the unhinged loss with bounded scores. Note that you instantly get even risk bounds in that case. [l:270] In Proposition 6, you can have R_{01}(s) = B \cdot R_{\ell}(s) (by looking at the minimizers directly). [l:308-311] Because you already have the expectations, you do not need to regularize in the Ridge sense -- I was going to suggest that you can regularize by biasing the centroid estimates, but you already allude to that in the next paragraph. [l:341-347] The statement here seems like a natural consequence of over-regularization. The conclusions you use for SLN robustness are interesting, nonetheless. [l:344-345] $\limsup_{\lambda \rightarrow \infty}$? [l:362] The appendix references are broken here too. [l:445] "$\ell$-risk w.r.t. some convex" [l:480] "any $\rho < \frac{1}{2}$" [l:494] This notation where one integrates out the subscript can be confusing... I would suggest using conditional expectation. [l:514] I believe you need $\alpha \geq 0$. [l:535] $(1 - 2\eta)v + 1$POST-REBUTTAL REMARKSThis paper has the potential to reach a larger audience on account of its proposed algorithm, which the authors should tap. Although SLN has attracted interest from the theoretical and practical points of view, I have the impression that it is too harsh a noise model. Nevertheless, this work is a step in the direction of understanding how to have robustness to noise in classification.The lack of comparison between SLN robustness and its strong counterpart seems to have been a concern in other reviews. While strong SLN robustness ensures that a loss is suitable for classification with SLN, it also restricts the number of such suitable losses. It would be interesting to understand whether how limiting this restriction is.The Appendix carries a relatively large portion of work that may not have been peer-reviwed, but this reviewer verified the proofs (Appendix A).


On the Limitation of Spectral Methods: From the Gaussian Hidden Clique Problem to Rank-One Perturbations of Gaussian Tensors
We consider the following detection problem: given a realization of a symmetric matrix X of dimension n, distinguish between the hypothesis that all upper triangular variables are i.i.d. Gaussians variables with mean 0 and variance 1 and the hypothesis that there is a planted principal submatrix B of dimension L for which all upper triangular variables are i.i.d. Gaussians with mean 1 and variance 1, whereas all other upper triangular elements of X not in B are i.i.d. Gaussians variables with mean 0 and variance 1. We refer to this as the 'Gaussian hidden clique problem'. When L = (1 + ) √ n ( > 0), it is possible to solve this detection problem with probability 1 − o n (1) by computing the spectrum of X and considering the largest eigenvalue of X. We prove that when L < (1 − ) √ n no algorithm that examines only the eigenvalues of X can detect the existence of a hidden Gaussian clique, with error probability vanishing as n → ∞. The result above is an immediate consequence of a more general result on rank-one perturbations of k-dimensional Gaussian tensors. In this context we establish a lower bound on the critical signal-to-noise ratio below which a rank-one signal cannot be detected.

This paper is about hypothesis testing of Gaussian Hidden Clique problem. It shows that, for any \epsilon, the case where there exists a planted clique of size L = (1-\epsilon) \sqrt{n} cannot be distinguished against the case where there is no hidden clique, using spectral methods. This complements the positive result of [12], where it is possible to distinguish the two cases using a spectral method, when L > (1+\epsilon) \sqrt{n}. The main technical contribution is a general analysis based on rank-1 perturbation of order-k symmetric Gaussian tensors, which may be of independent interest.Quality: The result is technically sound as far as I have checked. Clarity: The paper is well-written overall; I am confused about Equation (24) -- why does it follow directly from Lemma 5?Originality: Although the proof technique based on Lemma 2 is quite interesting, to be honest, I am not able to evaluate the novelty since I am not familiar with the relevant literature of proving such lower bound.Significance: the paper establishes an interesting result, which nicely complements the previous positive result on spectral methods. But the downside is that this is only a lower bound for ``spectral methods", which makes the usage rather restrictive. It seems that extending this lower bound to any efficient detection algorithm is not easy, since the technique heavily relies on the fact that the eigenvalues are conjugation-invariant.


Statistical Topological Data Analysis - A Kernel Perspective
We consider the problem of statistical computations with persistence diagrams, a summary representation of topological features in data. These diagrams encode persistent homology, a widely used invariant in topological data analysis. While several avenues towards a statistical treatment of the diagrams have been explored recently, we follow an alternative route that is motivated by the success of methods based on the embedding of probability measures into reproducing kernel Hilbert spaces. In fact, a positive definite kernel on persistence diagrams has recently been proposed, connecting persistent homology to popular kernel-based learning techniques such as support vector machines. However, important properties of that kernel enabling a principled use in the context of probability measure embeddings remain to be explored. Our contribution is to close this gap by proving universality of a variant of the original kernel, and to demonstrate its effective use in twosample hypothesis testing on synthetic as well as real-world data.

Summary: topological data analysis is a field which studies the geometric "shape" of a data set, using methods from algebraic topology. A tool that has been proven useful is the so-called persistent homology together the persistence diagrams. One is particularly interested in computing these persistence diagrams as they give the "topological summary" of the data shape. However a statistical treatment of these persistent diagrams is rather computationally expensive and difficult. A way to circumnavigate this difficulty goes by embedding probability measure on the space of persistence diagrams into a a reproducing kernel Hilbert space, since Hilbert spaces are highly regular and have an abundance of available well-studied methods that largely simplify calculations. Such an embedding can be achieved if the corresponding kernel is "universal", via the so-called "mean map". The paper proves that a slight modification of the recently introduced "PSS kernel", called the "u-PSS kernel", has the property of being "universal" (with respect to the 1-Wasserstein metric) [prop 2]. For example, one can then test whether two different samples exhibit significantly different homological features, while avoiding the direct computation of persistence diagrams and reducing it to the evaluation of the corresponding kernel. Major negative points- The paper is generally hard to read for people not so familiar with algebraic topology or functional analysis. Despite the background chapter, several rather more advanced notions or results in functional analysis are assumed familiar and not explained in further detail. - Many relevant definitions, argumentations from other papers and calculations are rather sketchy or even omitted.- The core argumentation of the paper is not written down in a concise chronological "start-to-finish argument", rather the different pieces of the argument are spread out throughout the paper, randomly mixed in with different information of secondary importance which makes the reading at times rather confusing.- The actual contribution is the proof of proposition 2, which in turn just boils down to proving that (\mathcal{S},d_{W,1}) is a compact metric, due to Theorem 1. In my opinion, the actual contribution is not that significant. The author mainly put together several already-published definitions and results, to achieve his result without introducing anything novel.   Positive points- the field of topological data analysis is a rather new and exciting area with many interesting applications, and the result seems rather exciting. As algebraic topology is a rather powerful tool, however not frequently used due to its computational complexity, a result leading to implementations of reduced complexity can lead to interesting and very diverse implications. - the experiments /implementations illustrate well the generality of theoretical result, and the diversity of its applications.Final conclusion: 5/10 (weak rejection)


Local Expectation Gradients for Black Box Variational Inference
We introduce local expectation gradients which is a general purpose stochastic variational inference algorithm for constructing stochastic gradients by sampling from the variational distribution. This algorithm divides the problem of estimating the stochastic gradients over multiple variational parameters into smaller sub-tasks so that each sub-task explores intelligently the most relevant part of the variational distribution. This is achieved by performing an exact expectation over the single random variable that most correlates with the variational parameter of interest resulting in a Rao-Blackwellized estimate that has low variance. Our method works efficiently for both continuous and discrete random variables. Furthermore, the proposed algorithm has interesting similarities with Gibbs sampling but at the same time, unlike Gibbs sampling, can be trivially parallelized.

SUMMARY This paper presents an idea that would be very interesting to NIPS readers. It presents a preliminary analysis of an estimator of a quantity that every variational inference researcher would want to know about. The mathematical exposition is good: the authors promise more technical and theoretical studies down the road. The experimental study is the weakest point of the paper, where they miss the opportunity to explore practical aspects of their estimator (specifically, other integration techniques beyond Gaussian quadrature). Overall it is a good paper.QUALITY The paper presents a new method in a careful and well-paced manner. The mathematical development is good. However, the experimental section could use a bit more work. My main comment is that I would have rather had the authors study alternative ways of computing (or approximating) the internal expectation in Equation (9). Instead, they study two uninteresting toy models in Sections 4.1 and 4.2, where it is not clear what insight they expect the reader to extract. The experimental study in Section 4.3 is good, because it addresses a case where ReGrad is not applicable. This is important because there appears to be little reason to use LeGrad for differentiable probability models.CLARITY The paper is well written and accessible.I am not particularly keen on how Section 3.1 is structured. It's not clear to me why the authors have bundled a time-complexity study with the (in my opinion) much more interesting connection to Gibbs sampling. I would rather prefer that they minimize the time-complexity paragraph and push it to the experimental section; then, they could devote this space in the narrative to the connection to Gibbs (and maybe even expand on it some more).Figure 2: caption describes (a) (a) (c). should be (a) (b) (c).Figure 3 also has some caption issues.ORIGINALITY The authors propose a new idea for estimating the gradient of an expectation that pops up in many of the state-of-the-art variational inference techniques proposed in the last few years. This paper should have no trouble on the originality front.SPECIFICS line 151: extra comma in equation, needs reformatting.line 196: ... that IT has with ...line 250: ... we consider --a-- sigmoid belief networks ...


A Structural Smoothing Framework For Robust Graph-Comparison
In this paper, we propose a general smoothing framework for graph kernels by taking structural similarity into account, and apply it to derive smoothed variants of popular graph kernels. Our framework is inspired by state-of-the-art smoothing techniques used in natural language processing (NLP). However, unlike NLP applications that primarily deal with strings, we show how one can apply smoothing to a richer class of inter-dependent sub-structures that naturally arise in graphs. Moreover, we discuss extensions of the Pitman-Yor process that can be adapted to smooth structured objects, thereby leading to novel graph kernels. Our kernels are able to tackle the diagonal dominance problem while respecting the structural similarity between features. Experimental evaluation shows that not only our kernels achieve statistically significant improvements over the unsmoothed variants, but also outperform several other graph kernels in the literature. Our kernels are competitive in terms of runtime, and offer a viable option for practitioners.

This paper proposes graph kernels that take smoothed matching into account when computing a structural similarity, i.e., generated by graphlet kernels, Weisfeiler-Lehman subtree kernels, and shortest-path kernels.Moreover, the paper proposes an extension of the Pitman-Yor process for smoothing subgraph distributions, which originates novel graph kernels.The empirical evaluation shows that the proposed kernels outperform their non smoothed counterpart. Additionally, they also outperform other graph kernels and are competitive in terms of runtime.The idea of using NLP smoothing for graph matching is interesting and reasonable. However, the model proposed by the authors is a subcase of a not-so-recent smoothed tree kernels. For example, see:Danilo Croce, Alessandro Moschitti, and Roberto Basili. Structured lexical similarity via convolution kernels on dependency trees. In Proceedings of the 2011 Conference on Empirical Methods in Natural Language Processing, pages 1034-1046, Edinburgh, Scotland, UK., July 2011. Association for Computational Linguistics.which proposed a smoothing function in the partial tree kernel by Moschitti, ECML 2006. Such function is more general than the smoothing function proposed in the submitted paper since it can be any valid kernel similarity.Note that the smoothed partial tree kernel can be straightforwardly applied to DAG, see:Aliaksei Severyn, Alessandro Moschitti: Fast Support Vector Machines for Convolution Tree Kernels. Data Mining Knowledge Discovery 25: 325-357, 2012.Finally, there has been older work using tree kernels with similarities:Stephan Bloehdorn and Alessandro Moschitti, Exploiting Structure and Semantics for Expressive Text Kernels, Proceeding of the Conference on Information Knowledge and Management, Lisbon, Portugal, 2007


Color Constancy by Learning to Predict Chromaticity from Luminance
Color constancy is the recovery of true surface color from observed color, and requires estimating the chromaticity of scene illumination to correct for the bias it induces. In this paper, we show that the per-pixel color statistics of natural scenes-without any spatial or semantic context-can by themselves be a powerful cue for color constancy. Specifically, we describe an illuminant estimation method that is built around a "classifier" for identifying the true chromaticity of a pixel given its luminance (absolute brightness across color channels). During inference, each pixel's observed color restricts its true chromaticity to those values that can be explained by one of a candidate set of illuminants, and applying the classifier over these values yields a distribution over the corresponding illuminants. A global estimate for the scene illuminant is computed through a simple aggregation of these distributions across all pixels. We begin by simply defining the luminance-to-chromaticity classifier by computing empirical histograms over discretized chromaticity and luminance values from a training set of natural images. These histograms reflect a preference for hues corresponding to smooth reflectance functions, and for achromatic colors in brighter pixels. Despite its simplicity, the resulting estimation algorithm outperforms current state-of-the-art color constancy methods. Next, we propose a method to learn the luminanceto-chromaticity classifier "end-to-end". Using stochastic gradient descent, we set chromaticity-luminance likelihoods to minimize errors in the final scene illuminant estimates on a training set. This leads to further improvements in accuracy, most significantly in the tail of the error distribution.

The algorithm presented here is simple and interesting. Pixel luminance, chrominance, and illumination chrominance are all histogrammed, and then evaluation is simply each pixel's luminance voting on each pixel's true chrominance for each of the "memorized" illuminations. The model can be trained generative by simply counting pixels in the training set, or can be trained end-to-end for a slight performance boost. This algorithm's simplicity and speed are appealing, and additionally it seems like it may be a useful building block for a more sophisticated spatially-varying illumination model.I found the paper reasonable easy to read, though much of the math could use a revision to make things more clear and hopefully reduce the (unusually large) number of symbols which are defined for what seems to be a fairly simple algorithm. For example, why define x-hat and g()? is v(n) == v_n? Is { g(v(n), m-hat_i}_i really the easiest way to refer to that set --- and is that math even correct, given that i is used as a subscript twice? Perhaps some figures which correspond to the math would be useful, as the only variable which appears in a figure is L[].The evaluation shows that this technique produces state-of-the-art results on the standard color checker dataset. The evaluation contains one component which I take issue with. When comparing the authors results to the results published in [19], we see that the authors very slightly outperform [19]. However, in this paper the authors have presented a modified set of error metrics, which in the supplement they explain are derived from the metrics in [7,19] subject to a correction, with the claim that this correction undoes an unfair advantage that the authors of [7,19] have achieved by using incorrectly black-leveled images. It may be true that [7,19] are evaluated incorrectly, but such an allegation should be taken very seriously, and should not be dealt with in this way. This number (as I understand it) assumes that the models of [7,19] were trained on the incorrect dataset and then evaluated on the correct dataset, which will obviously increase error metrics. But this does not mean that these numbers are indicative of how [7,19] would perform if trained *and* tested on the correct dataset. Indeed, given that incorrect black-level correction can completely ruin a color constancy algorithm, I would not be surprised in [7,19] actually performs better if trained and tested correctly. So I think the correction the authors present here is not valid.Though it is unfortunate that [7,19] contains incorrect numbers which may be cited, we should not combat such innacuracies with other innacuracies. If published, the number presented here may be taken as truthful by future work, when in fact it is even more speculative and unreliable as the numbers presented in [7,19]. So I think the authors should not present their modified error numbers. The error metrics presented here should be the same as in [7,19], but perhaps indicated with an asterix or with a grayed-out font color that they are likely not trustworthy, with an explanation of why in the supplement. Then the onus will be on [7,19] to revise their results, and we wont have two contradictory (and equally incorrect) numbers for the performance of [7,19] circulating in publications. I don't suggest this to diminish the author's work, as even if [7,19] are taken on face value the work presented here outperforms it. This is simply a matter of scientific honesty and ettiquette that we should handle as carefully as possible.


Online Rank Elicitation for Plackett-Luce: A Dueling Bandits Approach
We study the problem of online rank elicitation, assuming that rankings of a set of alternatives obey the Plackett-Luce distribution. Following the setting of the dueling bandits problem, the learner is allowed to query pairwise comparisons between alternatives, i.e., to sample pairwise marginals of the distribution in an online fashion. Using this information, the learner seeks to reliably predict the most probable ranking (or top-alternative). Our approach is based on constructing a surrogate probability distribution over rankings based on a sorting procedure, for which the pairwise marginals provably coincide with the marginals of the PlackettLuce distribution. In addition to a formal performance and complexity analysis, we present first experimental studies.

In the setting of dueling bandits and under the PL model assumption, the authors consider two tasks: finding an approximate best arm and finding an approximate ranking of the arms. Both tasks relates to the estimations of the pairwise marginals of the underlying PL model, which is done with an early-stopped version of the QuickSort. This Budgeted QuickSort allows to build algorithms which sample complexity is O(M log^2 M). Experiments compare the sample complexity of the proposed algorithms against state-of-art methods.The problem is not well-motivated, which is even more visible by the lack of experiments based on real-world dataset. This been said, I like the algorithms for their simplicity and the theoretical study that support them is correct. I didn't check proofs in appendix, but the results seems realistic. I would have appreciated a little discussion about the choice of the method to build confidence intervals in PLPAC -- e.g. is the algorithm sensitive to the method ?The paper is correctly organized and reasonably easy to follow. I still find a bit curious to put an algorithm (AMPR-PLPAC) in supplementary material.As I was saying, the problem is not well-motivated but the gain in terms of sample conplexity over existing algorithms is significant.


Online F-Measure Optimization
The F-measure is an important and commonly used performance metric for binary prediction tasks. By combining precision and recall into a single score, it avoids disadvantages of simple metrics like the error rate, especially in cases of imbalanced class distributions. The problem of optimizing the F-measure, that is, of developing learning algorithms that perform optimally in the sense of this measure, has recently been tackled by several authors. In this paper, we study the problem of F-measure maximization in the setting of online learning. We propose an efficient online algorithm and provide a formal analysis of its convergence properties. Moreover, first experimental results are presented, showing that our method performs well in practice.

The problem of learning binary classifiers optimizing non-decomposable performance metrics has recently gained a lot of traction; this paper considers the popular F-measure specifically, and proposes an algorithm for optimizing F-measure in the online learning setting. The authors give a formal consistency analysis for the online algorithm. Experimental results on fairly large real-world datasets are presented.The main idea in the paper -- and the online algorithm itself -- is neat and simple. It is easy to understand and more or less easy to implement (except for the caveat of estimating a model for the class-conditional probabilities in each round, which can be quite tricky). The consistency analysis is also fairly novel. Though, I'm not quite sure if the way the guarantee is presented in Theorem 3 (and 5) is precise. Perhaps it would be clearer to make the regret, which in this case is F((y_1, ... , y_t), (\hat{y}_1,...,\hat{y}_t)) - F(\tau^*)), explicit in the statement (which the authors do mention later in Section 6).Recently, [9] introduced a framework for optimizing & analyzing non-decomposable measures in the online setting. It is not quite convincing from the discussion in Section 6, how exactly the analysis of OFO is stronger than that in [9]. In fact, the framework in [9] based on a certain per-round non-decomposable loss is quite intuitive. In particular, [9] also includes convergence rates for regret. It is also surprising that the method in [9] is not compared to in experiments (at least a justification is needed otherwise).The other concern is with respect to the organization of the paper itself. The main idea/contribution which is Algorithm 1 and its analysis appear much later in the paper. Sections 2 and 3 are redundant for the most part. In particular, I fail to see the point of Section 3 in the context of this paper and its key contribution --- at best, it can be summarized in a Background section that combines the key ideas in Sections 2 and 3. This way, the readers can get to Section 4 (& 5), which is the focus of this paper, without distractions (and possibly confusions). The authors even had to move some useful results on Synthetic data to the Appendix -- I suggest including this in the main paper.Responses are satisfactory. Raising my score to Accept.


Algorithmic Stability and Uniform Generalization
One of the central questions in statistical learning theory is to determine the conditions under which agents can learn from experience. This includes the necessary and sufficient conditions for generalization from a given finite training set to new observations. In this paper, we prove that algorithmic stability in the inference process is equivalent to uniform generalization across all parametric loss functions. We provide various interpretations of this result. For instance, a relationship is proved between stability and data processing, which reveals that algorithmic stability can be improved by post-processing the inferred hypothesis or by augmenting training examples with artificial noise prior to learning. In addition, we establish a relationship between algorithmic stability and the size of the observation space, which provides a formal justification for dimensionality reduction methods. Finally, we connect algorithmic stability to the size of the hypothesis space, which recovers the classical PAC result that the size (complexity) of the hypothesis space should be controlled in order to improve algorithmic stability and improve generalization.

SUMMARYThis paper defines a notion of algorithmic stability that is shown to be equivalent to uniform generalization. Different consequences for generalization are explored in connection to techniques used to control algorithmic stability directly, richness of the hypothesis space and techniques used to control it, and also a notion of size of the input space. CONTRIBUTIONS* A notion of algorithmic stability that corresponds to uniform generalization. * Theoretical justifications for different popular learning strategies used to improve algorithmic stability. * Strengthening of the understanding of generalization and: the input space, the hypothesis space, the learning algorithm.These contributions strengthen the understanding of generalization, and can lead to interesting future work that deepens the understanding of different techniques used to increase generalization (hold-out set validation, regularization, dimensionality reduction, etc.). Proposing a notion of stability that is equivalent to (uniform) generalization is challenging, and I think this makes the contributions of this paper strong. SUPPORTThe proofs are correct, simple and easy to understand. The claims that provide a justification for techniques that improve generalization are well-founded in the theoretical results. The proof sketches are informative as well, while maintaining the pace of the exposition. TECHNICAL QUALITYThe paper is technically sound, and the consequences of the central results we well-explored. The authors have defined stability on expectation, not with high-probability (in contrast to sample complexity bounds, which hold with high-probability). The expectation allows us to obtain low-probability bounds with Markov's inequality, but it is not clear whether high-probability bounds can be obtained without fixing a specific algorithm. ORIGINALITYThe paper seems well-inserted in the literature, but my knowledge of related work in the area of stability is shallow and I cannot make a credible assessment of originality. CLARITYThe paper is well-written, with clear explanations and clear proofs. This theoretically-oriented work is well executed and it is an accessible read (in this reviwer's opinion). Based on the appreciation of other reviewers, though, it seems that the clarity of the paper can be improved.FOR THE REBUTTALPlease comment on the issue of expectation bounds vs. high-probability bounds. Can high-probability bounds be obtained without fixing an algorithm, or are the expectation bounds alone a significant contribution?It would be good if the authors could comment on the relationship between their definition of stability and existing definitions, this has been raised by other reviewers too.DETAILED COMMENTS The definition of stability proposed is interesting; it is sensible, as it can be interpreted as "the output of the algorithm does not depend too much on any single example", but I was wondering how it relates to existing definitions (e.g., the ones where one observation in the sample is dropped, or where it is swapped for another one). On the other hand perhaps the point is not so much to compare them, in case the endeavour in previous works has been to characterize generalization with some notion of stability.Considering Section 5.3, do you think we should be caring about Shannon Entropy of the output hypothesis (when designing algorithms), rather than the size of the hypothesis space? I am asking this because of the issues with bad ERM learners in multiclass classification [1], in which case we have to do something besides "just ERM" to be successful.Example 2 made me think of hold-out set validation (where a validation set is used to increase the "stability" of the algorithm) and also of minimization using Bregman projections (minimize a convex loss, then do a Bregman projection on a set) -- it also seems similar to post-processing.I see that the concepts in Definition 8 match the losses you use to show that stability is necessary for uniform generalization. Do you think it allows you to conclude that classification (with the 0-1 loss) is a hardest problem from the point of view of stability/generalization? REFERENCES:[1] Daniely, Sabato, Ben-David & Shalev-Shwartz (2013). "Multiclass learnability and the ERM principle."FIXES[l:60] "nature of the observation" [l:127] I am not a fan of this notation, why not use conditional expectation? (Using the subscript of the expectation only to indicate the distribution of some r.v. is helpful, though.) [l:152] The symbolic convention somewhat breaks when you use $\mathcal{L}$ for the algorithm, so why not use simply $A$? [l:173] I would suggest removing the last sentence from this paragraph. [l:190-191] "variables are independent of each" [l:265-266] "H_2 contains less information" [l:317] Here and in the appendix you used $\mathcal{V}$ to denote the total variation, while you defined it with a $\mathcal{T}$. [l:357-358] "always holds:" [l:383] You inverted the arguments of the KL divergence here. [l:400] $\mathbb{P}(Z_{\mathrm{trn}} = z)$ (the parentheses need to be fixed). [l:420] I would suggest using verbs in the past here "In this paper, we showed" [l:421-422] "always vanishes at"(Appendix) [l:185] (double ellipsis) [l:228] Perhaps you can add the reference to H. Robbins (1955) and Shalev-Swartz & Ben-David's book at the end of the appendix, or expand the one to H. Robbins (1955) as a footnote.POST-REBUTTAL REMARKSThere seems to be little understanding of the connections between stability by conditioning vs. stability by observation elimination vs. stability by observation switching. It would seem that the definition arose as a tool that worked for the desired results, and the authors should be upfront about the limited understanding of the aforementioned connections.The authors suggest that a strength of their main result is its algorithm-independence. However this algorithm-independence is the case only for expectation guarantees on stability, not high-probability guarantees, which are often what one is interested in. On the other hand, it is known how to convert expectation guarantees for a base algorithm into high-probability guarantees for the same algorithm combined with a hold-out set validation scheme (cf. Shalev-Shwartz & Ben-David, "Understanding Machine Learning: From Theory to Algorithms"). Therefore, the authors should mention this limitation of their results. The contributions are nevertheless solid and interesting -- of course, not as strong as algorithm-independent high-probability guarantees, which could be pursued as future work.


Efficient Exact Gradient Update for training Deep Networks with Very Large Sparse Targets
An important class of problems involves training deep neural networks with sparse prediction targets of very high dimension D. These occur naturally in e.g. neural language models or the learning of word-embeddings, often posed as predicting the probability of next words among a vocabulary of size D (e.g. 200 000). Computing the equally large, but typically non-sparse D-dimensional output vector from a last hidden layer of reasonable dimension d (e.g. 500) incurs a prohibitive O(Dd) computational cost for each example, as does updating the D × d output weight matrix and computing the gradient needed for backpropagation to previous layers. While efficient handling of large sparse network inputs is trivial, the case of large sparse targets is not, and has thus so far been sidestepped with approximate alternatives such as hierarchical softmax or sampling-based approximations during training. In this work we develop an original algorithmic approach which, for a family of loss functions that includes squared error and spherical softmax, can compute the exact loss, gradient update for the output weights, and gradient for backpropagation, all in O(d 2 ) per example instead of O(Dd), remarkably without ever computing the D-dimensional output. The proposed algorithm yields a speedup of D 4d , i.e. two orders of magnitude for typical sizes, for that critical part of the computations that often dominates the training time in this kind of network architecture.

Summary:This paper gives a novel technique for gradient-based learning in neural networks with a large but sparse target vector. The technique calculates the exact, correct gradient direction, while never computing the full large-dimensional output, albeit for a restricted class of loss functions. The paper shows, theoretically and empirically, that this drastically improves the computational complexity of gradient descent learning. Experiments document the speedup, but show that the loss function restrictions may negatively impact performance.Major comments:This paper presents a promising technique for speeding up neural language models with large, sparse output vectors. Remarkably, the paper shows how to perform exact gradient updates (for a restricted set of loss functions) in an efficient way that scales in the sparsity of the targets, rather than in the size of the output vector. This paper is not a tweak of previous approaches to dealing with large sparse targets, but a wholly new direction which may inspire a variety of related algorithms and improvements. It will be no surprise to the authors that the paper would be strengthened greatly if it used the proposed method to attain state of the art performance on some task. The experiments in the paper show that the method indeed attains in practice the speedup promised, but this speedup is not leveraged to improve performance. It may be that the loss function restrictions are severe enough to stop this method from being practically useful. Most concerning is the comparison between hierarchical softmax and the proposed method. While H-softmax is only a heuristic, it is about equally fast and performs better than the proposed LST models on the example dataset in the paper. However the theoretical contributions of the paper are already substantial, and even if it takes some time to find a compelling application, the insights given here seem likely to be useful in some future application.The experimental results are also valuable in showing that there are no numerical stability issues (other than the ill-conditioning of U which is identified and addressed).The paper is very clearly written, with the algorithm spelled out step-by-step in useful tables. Additionally, code will be released upon publication, which should also aid clarity and adoption of the method in practice. The supplementary material adds the practically important detail of how to perform mini batch updates. The clarity of the algorithm could be further improved by explicitly including pseudocode for the rank-1 V update used to control numerical stability.Minor comments:pg 2 ln 097 "workFinally" ->"work. Finally" Fig 1 caption: "inO(Kd)"->"in O(Kd)" pg 4 ln 172: "non-lienarity"->"non-linearity" pg 5 ln 258: "theK"->"the K" pg 5 ln 259: "sited" -> "instead" pg 5 ln 261: "theW^Ty"->"the W^Ty" pg 7 ln 329: "appropriate do a rank-1" -> "appropriate rank-1" pg 7 ln 363: I don't believe the LST acronym has been introduced yet at this point in the text. pg 8 fig 2: the right hand plot does not appear to contain all points in the lefthand plot (the red curve never goes above the blue curve, for instance). Citation formats are inconsistent (some have whole name; some abbreviate first names; some say NIPS, some say Advances in Neural...) Citation 4 has a typo


Weakly-supervised Disentangling with Recurrent Transformations for 3D View Synthesis
An important problem for both graphics and vision is to synthesize novel views of a 3D object from a single image. This is in particular challenging due to the partial observability inherent in projecting a 3D object onto the image space, and the ill-posedness of inferring object shape and pose. However, we can train a neural network to address the problem if we restrict our attention to specific object classes (in our case faces and chairs) for which we can gather ample training data. In this paper, we propose a novel recurrent convolutional encoder-decoder network that is trained end-to-end on the task of rendering rotated objects starting from a single image. The recurrent structure allows our model to capture longterm dependencies along a sequence of transformations, and we demonstrate the quality of its predictions for human faces on the Multi-PIE dataset and for a dataset of 3D chair models, and also show its ability of disentangling latent data factors without using object class labels.

In this paper the authors propose a novel recurrent convolutional encoder-decoder network for learning to apply out-of-plane rotations to 3d objects such as human faces and 3d chair models. The proposed network starts from a basic model, where its encoder network disentangles the input image into identity units and pose units, then with the action units applied on pose units to control the rotation direction, its decoder network which consists of convolution and unsampling decode the identity and pose into an image of rotated object and the corresponding object mask. To support longer rotation trajectories, the proposed network is then extended to have the recurrent architecture where the encoded identity unit of input image is fixed and the pose unit is changed by a sequence of action units, and finally both identity and pose units are fed into decoder to generate the result image.        One of main contribution of this paper is learning to disentangle the representations for identity/appearance and pose factors, where the identity units are shown to be a discriminative view-invariant features in the cross-view object recognition task. In addition, this disentangling properties will benefit more and predict better rendering while using the longer rotation trajectories in the curriculum training stages for training the proposed recurrent convolutional encoder-decoder network.        The paper is well-written, easy to follow, and the motivation for different parts of proposed method is all clearly described. Also the qualitative results for predicted rendering of rotated images and quantitative evaluation on cross-view object recognition task provide good support for the method, especially the disentangled representations for pose and identity factors.        Some minor weakness are listed as follows and hopefully the authors can address them in the rebuttal period:       - The proposed network can only support discrete rotation angles, depending on the set of rotation angles shown in the training data. Do the authors have any initial idea how to extend the proposed method to support continuous rotation angles?       - The proposed recurrent convolutional encoder-decoder network is trained with fixed-length, which is actually contradictory to general recurrent neural networks.


Learning Structured Densities via Infinite Dimensional Exponential Families
Learning the structure of a probabilistic graphical models is a well studied problem in the machine learning community due to its importance in many applications. Current approaches are mainly focused on learning the structure under restrictive parametric assumptions, which limits the applicability of these methods. In this paper, we study the problem of estimating the structure of a probabilistic graphical model without assuming a particular parametric model. We consider probabilities that are members of an infinite dimensional exponential family [4], which is parametrized by a reproducing kernel Hilbert space (RKHS) H and its kernel k. One difficulty in learning nonparametric densities is the evaluation of the normalizing constant. In order to avoid this issue, our procedure minimizes the penalized score matching objective [10,11]. We show how to efficiently minimize the proposed objective using existing group lasso solvers. Furthermore, we prove that our procedure recovers the graph structure with high-probability under mild conditions. Simulation studies illustrate ability of our procedure to recover the true graph structure without the knowledge of the data generating process.

I have read the authors' rebuttal and it did not change my opinion of the paper. -------- 1 Summary of the paper:This paper proposes to learn the structure of a kernel-based graphical model encoding a joint probability distribution (probabilities belonging to an infinite dimensional exponential family[1]). By using score-matching as the objective of the learning procedure, the normalisation constant does not need to be estimated, which is scalable and sufficient to learn the structure. Moreover, because of the kernel-based parameterisation, the optimisation problem reduces to an instance of the group lasso. The structure recovered by the algorithm is asymptotically consistent. The method proposed is compared to the graphical lasso to estimate a gaussian distribution and to both glasso and an algorithm knowing the model family on a paranormal distribution[1] S. Canu and A. Smola. Kernel methods and the exponential family. Neurocomputing, 69(7):714-720, 2006.2 Summary of the review: While I did not verify the theoretical analysis provided in the paper and the experiments could be explained better, I find the paper well organized, the contributions reasonably clear, the idea very interesting and potentially impactful. 3 Qualitative evaluation Quality: medium + extensive theoretical analysis (not checked fully, seems sound) +- small scale empirical evaluation - no discussion or empirical evaluation of how to choose the kernel parameters. Is a grid-search + cross-validation feasible and sufficient? How sensible is the result to the parameters chosen? - no code providedClarity: low + well organised paper, clear language +- lots of mathematical developments that are not easy to follow, especially with notations not necessarily introduced in the main paper.- the parameters of the second model learned in the experiments are not provided.Originality: good + I am not familiar with kernel-based methods or kernel-based graphical models, but I could not find really similar works beyond the ones cited.+ significant development with respect to previously published cited work - still, other works using kernel with PGMs exists. It could be interesting to relate the paper under review to them, for the informed reader like myself. For example, a quick search on google yielded the following papers: "Nonparametric Tree Graphical Models via Kernel Embeddings", "Kernel Conditional Random Fields: Representation and Clique Selection", "Learning Graphical Models with Mercer Kernels". Significance: good + structure learning for non-gaussian variable is an important open problem + I find the proposed idea very interestingImpact: 24 Other Comments:Details: . I understand you focus on the structure estimation problem, but can the normalisation constant be estimated easily enough when the model has been learned if one also wants the density? Are there some restricted families of kernel that make it possible? . I think section headers should not be all capitalized. . I am not familiar with RKHS. So I could not easily go from l204 to l207. I would suggest deriving it in the supplementary material. . a table of all notations in the supplementary material would make the paper a bit easier to follow. . l 319: why is it Omega(f)^2? (was Omega(f) in equation 8) . experiment 2: what is the dimensionality of the problem? What are the parameter values? . it would be nice to have the expression for D, E and F in theorem 3.2, for example in the appendix.typos: . l25: evaluation of --> the evaluation of . l53: "the" structure (lots of similar missing "the" in the paper - though I'm not a native speaker.) . l 76: integrable . l86: H_2?. l233: S^c not defined . l323: Omega*_{S^c} not defined