A Bio-inspired Redundant Sensing Architecture
Sensing is the process of deriving signals from the environment that allows artificial systems to interact with the physical world. The Shannon theorem specifies the maximum rate at which information can be acquired [1]. However, this upper bound is hard to achieve in many man-made systems. The biological visual systems, on the other hand, have highly efficient signal representation and processing mechanisms that allow precise sensing. In this work, we argue that redundancy is one of the critical characteristics for such superior performance. We show architectural advantages by utilizing redundant sensing, including correction of mismatch error and significant precision enhancement. For a proof-of-concept demonstration, we have designed a heuristic-based analog-to-digital converter -a zero-dimensional quantizer. Through Monte Carlo simulation with the error probabilistic distribution as a priori, the performance approaching the Shannon limit is feasible. In actual measurements without knowing the error distribution, we observe at least 2-bit extra precision. The results may also help explain biological processes including the dominance of binocular vision, the functional roles of the fixational eye movements, and the structural mechanisms allowing hyperacuity.

The authors introduce an architecture for introducing redundancy in the reference value sets used by various ADC techniques (e.g. SAR). These degrees of freedom potentially allow one to optimize the combinations of references used to generate a threshold/comparison-point so as to minimize the error from component mismatch. An ADC circuit has been fabricated to implement such techniques, and they are indeed found to reduce mismatch error.
Update in response to author's rebuttal: Many thanks to the authors for taking the time to consider my points and to respond to them. In general, I would say that I look forward to seeing a revised version of the paper that makes the adjustments the authors promise, but the promised changes are sufficiently significant that (sight unseen) I only feel comfortable bumping up the technical quality by a point. My detailed responses to the responses, in order: -I do understand it is difficult to discuss too much prior work in the redundancy-for-mismatch-reduction space given the 9 page limitation. Nonetheless there is a complete absence of this at present in the paper, which may even count as a fatal flaw. (As an aside, note that McNeill's (JSSC 2005) work does *not* increase either power consumption or chip area (therein lies the innovation), although the authors are absolutely correct in noting that off-chip calibration power/space is unaccounted for. This is the kind of "Related work" discussion I would have appreciated seeing in the paper.) -While it is reassuring that the authors are planning to introduce more implementation details, my interest is less in the detailed implementation details and more in better motivation for the algorithmic choices that are made. With that said, if the paper were to spend more time discussing their (significant) contributions on the hardware implementation end, what I see as flaws in the algorithmic presentation would be far more forgivable. -The language may be simple, but it is surprisingly difficult to extract the core algorithmic ideas from a reading of the paper. This could very well be a reading comprehension issue on my end, but I doubt it. -Understood and agreed on the purpose of the simulation/implementation, and glad to hear that the authors are considering contrasting to stronger baselines. -I appreciate the clarification re: eq (5). It does make more sense to me now. -The author's listing of their assumptions in their rebuttal is much appreciated, but it is very difficult to parse from the paper. A similarly rational motivation for the structure of their design (N0/N1/etc.), as they promise, would also help a great deal. -Scalar quantization is 1-dimensional quantization. I remain unclear on why the authors are referring to it as zero-dimensional quantization (a term I neither understand nor have heard before --- and a simple google search seems to confirm its lack of usage in the literature).  Positives:  -The fundamental idea explored by the authors --- the use of redundancy to reduce mismatch errors --- is a compelling one. I would add that they are far from the first to explore this fertile area; see, for instance, McNeill's work (JSSC 2005, “Split ADC” Architecture for Deterministic Digital Background Calibration of a 16-bit 1-MS/s ADC). Their approach in adjusting which components to utilize represents a significant improvement over post-processing calibration. As the authors note, there are numerous situations where information is simply beyond recovering in the latter case. -While I am generally suspicious of "neural" analogies, the authors do an admirable job of drawing inspiration from the human visual system while optimizing a clear practical objective. -The authors implement their proposed solution not only in software and through simulations, but by fabricating an actual circuit. This adds tremendously to the credibility of their results. With that said, the authors do not go into details about their hardware implementation or its validation, and the focus of my review is on their algorithmic work (the authors themselves note that the hardware implementation is to be described elsewhere).  (bigger) areas for improvement: -Fundamentally, I do not feel that the justification and explanation of the authors' algorithmic contributions is up to the NIPS standard. More on this below. -A lesser concern is the virtual lack of comparison to alternatives/nontrivial-baselines, either in terms of mismatch error vs quantization error, or in terms of the power/area tradeoff the authors frequently make reference to. This makes it very difficult to objectively evaluate the authors' contributions. -The model for binary coding introduced in Sec 2.2 is extremely difficult to follow ---one has to read through multiple times just to understand the terminology. What is a unit cell, versus a reference, versus a component? I was only able to piece this together through a combination of domain knowledge and guesswork from the expressions.  -Furthermore, there is a lack of motivation for the binary coding approach beyond its use in "artificial systems" (it's relevant for SAR etc., but the authors provide no references or examples for this). This is critical, as the entire space of optimizations the authors are concerned with exist within this paradigm. -The model for component noise (eq. (5)) is supplied without any justification, either in words or in citations. Why, for instance, does the variance scale with the mean? Why not the standard deviation? -The description of the scheme in Section 3 is extraordinarily painful to decode. I would feel genuinely uncomfortable seeing this manuscript published the way this is currently written. One has to read the section twice to even understand the basic architecture that is being used, and terminology is used without any introduction/definition (e.g. the use of \hat{c} versus c --- there may even be a typo on line 269 as the \hat is used in the argument but not the subscript, but I would not be able to swear by this, as the \hat significance is never provided.). -Related to the above, the form of the proposed solution (eq. (8)) is not sufficiently motivated or justified. A red flag: not only do the authors never explain the purpose/intuition behind parameters N_0/N_1/s_1, but they later turn to simulations (lines 310-313) just to explain their impact on the solution. If there are simple  insights that lead one to the system designed by the authors, this manuscript certainly does not make them clear. -Results reported for DNL/INL make for fantastic confirmations that the techniques proposed work in practice. With that said, confirmation of functionality is insufficient in a space as crowded as this one (mismatch calibration/correction), and performance needs to be compared against nontrivial and informative baselines.  (smaller) areas/points for improvement: -The authors frequently refer to scalar quantization as zero-dimensional. Unless I am missing something big, I believe they mean one-dimensional. -Eq. (6): should NOT index c_i by i, since that index is being used with a different interpretation by X_i. -line 213: should be fig. 3(a).  -What is fig. 3a showing? Expected absolute error over many samples, or a single sample? -line 241: \in [0,1] should be \in \{0,1\}, as the former refers to the unit interval.


Catching heuristics are optimal control policies
Two seemingly contradictory theories attempt to explain how humans move to intercept an airborne ball. One theory posits that humans predict the ball trajectory to optimally plan future actions; the other claims that, instead of performing such complicated computations, humans employ heuristics to reactively choose appropriate actions based on immediate visual feedback. In this paper, we show that interception strategies appearing to be heuristics can be understood as computational solutions to the optimal control problem faced by a ball-catching agent acting under uncertainty. Modeling catching as a continuous partially observable Markov decision process and employing stochastic optimal control theory, we discover that the four main heuristics described in the literature are optimal solutions if the catcher has sufficient time to continuously visually track the ball. Specifically, by varying model parameters such as noise, time to ground contact, and perceptual latency, we show that different strategies arise under different circumstances. The catcher's policy switches between generating reactive and predictive behavior based on the ratio of system to observation noise and the ratio between reaction time and task duration. Thus, we provide a rational account of human ball-catching behavior and a unifying explanation for seemingly contradictory theories of target interception on the basis of stochastic optimal control.

This paper attempts to show that previously researched ball catching heuristics can be understood as optimal control policy strategies by modelling catching as a POMDP, and making several assumptions.   This paper also explores how varying parameters can affect the optimal strategy in a simulated catching task by varying the task length and the reaction time as well as the noise in the partially observable states.  The authors spend a good amount of effort developing a computational model of a human catching a ball with observational uncertainty. This drives their point that heuristics researched in the past may not be sufficient to model scenarios where the reaction time is a significant fraction of task time, or in scenarios where there is a substantial amount of noise.
This is an interesting paper which explores the catching policies of reaction and planning. While the background is present, it is not presented in a way that fully justifies the experimental simulations. There is some discussion of outfielder vs. catcher catcher scenarios, but those are not fully defined or simulated. A more thorough investigation of these tasks would help to illustrate the simulated model of reaction and planning.   While the paper provides generous background and model development the experimental configuration is not fully described, and the results demand additional investigation. This is a cursory introduction to the justification of a model which includes reaction and planning, but additional experimentation would allow for this relationship to be fully elucidated.   The statistics are insufficient to show significance and the paper suffers from a lack of proof-reading and editing.   Please follow up with these references to further develop the modelling and dynamics as well as the tradeoff between planning and reaction and set ups involving robotic systems:  Kim, Seungsu, Ashwini Shukla, and Aude Billard. "Catching objects in flight." IEEE Transactions on Robotics 30.5 (2014): 1049-1065. Zago, Myrka, et al. "Visuo-motor coordination and internal models for object interception." Experimental Brain Research 192.4 (2009): 571-604. Cesqui, Benedetta, et al. "Catching a ball at the right time and place: individual factors matter." PLoS one 7.2 (2012): e31770. Kim, Seungsu, and Aude Billard. "Estimating the non-linear dynamics of free-flying objects." Robotics and Autonomous Systems 60.9 (2012): 1108-1122.


Bayesian optimization under mixed constraints with a slack-variable augmented Lagrangian
An augmented Lagrangian (AL) can convert a constrained optimization problem into a sequence of simpler (e.g., unconstrained) problems, which are then usually solved with local solvers. Recently, surrogate-based Bayesian optimization (BO) sub-solvers have been successfully deployed in the AL framework for a more global search in the presence of inequality constraints; however, a drawback was that expected improvement (EI) evaluations relied on Monte Carlo. Here we introduce an alternative slack variable AL, and show that in this formulation the EI may be evaluated with library routines. The slack variables furthermore facilitate equality as well as inequality constraints, and mixtures thereof. We show our new slack "ALBO" compares favorably to the original. Its superiority over conventional alternatives is reinforced on several mixed constraint examples.

This manuscript proposes a Bayesian optimization approach to problems with mixed inequality and equality constraints. The main tool used in the approach is a slack-variable version of the augmented Lagrangian, which allows the incorporation of mixed constraints and has some advantages over the use of the augmented Lagrangian without slack variables. A detailed description of the proposed method is given, including guidance on how to efficiently solve various subproblems that arise via standard library routines (in this case in R). A set of experiments on various difficult problems demonstrate that the proposed method works better than several alternatives, including one proposed only earlier this year.
This manuscript is very interesting, extremely well written, and solves an important problem, namely Bayesian optimization with mixed constraints. The main contribution of the paper is an extension of the recently-published ALBO technique for incorporating inequality constraints into Bayesian optimization problems to handle equality constraints as well. The approach is to incorporate the equality constraints via the addition of slack variables into the augmented Lagrangian. Remarkably, the addition of these variables leads to some computational niceties even in the case that there are no equality constraints at all. Specifically, the calculation of expected improvement in this setting is simplified and may be resolved via quadrature rather than via Monte Carlo sampling. This is a nice result and discussed at length, including specific recommendations for implementation. Further, the authors provide guidance on how to solve for the slack variables, which removes the only real barrier to adopting the proposed slack-variable formulation wholesale. A set of well-designed and insightful experiments establish that the proposed method is superior to (in some cases markedly so) to alternatives, in terms of the performance as a function of the number of function evaluations.  The paper is well-written and easy to read, with only a few minor issues, which I list below. The paper is also remarkably complete, providing a nice level of detail in all parts of the discussion, and helpful, more-detailed pointers in the supplementary material. I only have a few suggestions for how the paper could be strengthened. One is that I think some brief discussion regarding the wall-clock time spent computing the EI would be of interest. The authors do a good job convincing me that this is readily implemented and presumably quite fast, although I don't have a good intuition for how slow the "library routines" are. It would also be helpful to put this time in perspective with the time required by the Monte Carlo approximation used in ALBO. Again, I think this comparison would only be in the favor of the authors. Also regarding wall-clock time, it would be interesting to see how this compares end-to-end across the tested methods, perhaps on only one of the experiments.  I also have a minor complaint about the computation of EI in (2). The EI in that expression is being computed against f_min, defined to be min {y_1 ... y_n}. This is not really correct due to the noise in the observations. Rather, f_min, if I am to interpret its meaning from the notation, is a random variable with an admittedly annoying distribution. In fact, I would hazard to guess that y_min is usually an optimistic estimate of f_min in Bayesian optimization settings (and things get even more troublesome if the noise is heteroskedastic!). The result is that the computation in (2) is measuring the value of points against an inflated benchmark, which I believe would lead to over-enthusiastic exploration. This is a well-known fact that is almost always swept under the rug, and indeed the formulation in (2) is widespread. Nonetheless, I feel compelled to complain on philosophical grounds.  Minor corrections:  - Line 45: I find "Here we dub that method ALBO" somewhat awkward. I might recast as "We will refer to this method as ALBO." - Lime 137: I find "re-develop" a bit awkward as well. Perhaps "rewrite?" - Under line 153: Please typeset (13) using the amsmath align environment rather than eqnarray, which is giving you unsightly spacing around the ='s. - I might suggest adding a little negative space to the _f subscripts used on Y and \sigma; they're typeset with too much space to my eyes (inspect 6\rho\sigma_f(x) on line 166 for an example). - Line 170: "how \lambda and and \rho updated" -> "how \lambda and \rho are updated" - The sans serif font should probably be scaled down a bit. I believe this is a problem with the NIPS style file and will email the proceedings chair.


Learning brain regions via large-scale online structured sparse dictionary-learning
We propose a multivariate online dictionary-learning method for obtaining decompositions of brain images with structured and sparse components (aka atoms). Sparsity is to be understood in the usual sense: the dictionary atoms are constrained to contain mostly zeros. This is imposed via an 1 -norm constraint. By "structured", we mean that the atoms are piece-wise smooth and compact, thus making up blobs, as opposed to scattered patterns of activation. We propose to use a Sobolev (Laplacian) penalty to impose this type of structure. Combining the two penalties, we obtain decompositions that properly delineate brain structures from functional images. This non-trivially extends the online dictionary-learning work of Mairal et al. (2010), at the price of only a factor of 2 or 3 on the overall running time. Just like the Mairal et al. (2010) reference method, the online nature of our proposed algorithm allows it to scale to arbitrarily sized datasets. Preliminary xperiments on brain data show that our proposed method extracts structured and denoised dictionaries that are more intepretable and better capture inter-subject variability in small medium, and large-scale regimes alike, compared to state-of-the-art models.

This paper provides an online algorithm of learning structured dictionary to characterize neuroimaging data across individuals, by extending a well-established framework in [14] with a Sobolev prior to encourage spatial contiguity of dictionary atoms. The authors applied their method on empirical data, reported the effect of varying tuning parameters and compared the performance with two alternative methods in the field.
Technical quality: I would rate marginally below 3 if given the option. The model definition was clear. The algorithms followed a well established framework [14], and appeared solid. However, maybe a bit undermined by its presentation, the paper did not seem to clearly demonstrate the empirical advantage of introducing the Sobolev prior in the results section. Especially, in Figure 2, it was not clear in what aspect the SSMF method was better than the two alternatives.  Was the spatial boundary more smooth quantitatively? Were there certain regions, which were known to be connected, only identified by SSMF? etc.  Besides (vaguely) saying results by SSMF were “more structured”, the authors could explicitly describe some criteria justified by neuroscience literature, and give quantitative measurements of the improvement according to these criteria.   Added after reading the rebuttal: The results provided in the rebuttal are very informative. I changed my rating on technical quality from 2 to 3. I hope the authors include these new results in their revision, and in addition, demonstrate that the atoms learned by SSMF are more structured (with quantitative measurements) and more meaningful (with reference to neuroscience literature).  Novelty/originality: There is notable original contribution in this paper, which is the extension of [14] that applied Sobolev priors to encourage spatially structured dictionary atoms.   Potential impact or usefulness:  Dictionary learning is a useful approach for explaining neuroimaging data across a large group of subjects, and introducing Sobolev prior or similar spatial priors should improve the results in theory. This paper provides an efficient online algorithm of doing so, and could add a good tool in the toolbox of the neuroscience/medical imaging community. The authors could also explore other applications of this method such as image segmentation and regional sensing.   Presentation: The general organization of the paper is okay. However, the paper appeared not to be a well-edited manuscript; there were confusing descriptions and typos that made it harder to understand the contents. I list below the confusing parts. Some of them seem to be easy-to-fix typos, but some may require more detailed explanations.   --In the matrix decomposition equation between Line 67 and 68,  should the transpose symbol on the group of $V^j$ (j = 1,..., r) be on each individual $V^j$’s? --Line 73, the reference of HCP dataset was missing.  --Line 83 “The Graph-Net penalty can imposes ...” --Line 146: bold sentence might be a co-author’s comment which was not fixed. --Last line of Equation (6) and Equation (8) did not agree on the sign $ V^j +A_t[j,j]^{-1}(VA_t^j -B_t^j)$  versus $A_t[j,j] V^j -(VA_t^j - B_t^j)$. Also in the last line of equation (6), I assume $V$ refers the dictionary in the last iteration, (i.e. the jth column is not the variable $v$ to optimize).  -- Line 157, “a single, a single” -- Line 169, “ We did an extensive study of th how” -- In Section 3.3, the paragraph near 147 just below Algorithm 2,  it would be nice if the authors explicitly gave the mathematical/algorithmic solution to Equation (6), even in the appendix.  -- Figure 2, the results by different methods were presented at different horizontal and sagittal slices, which made it hard to directly compare the results.  Based on the figure, I did not follow the description in Line 222 that claimed the results by SSMF model were more “structured”.  -- Figure 3, it was unclear what was predicted. Was it regressing brain data on behavioral metrics, if so, was it base on $U_i$’s? The legends were not explained in the main text nor the figure caption. The figure did not seem to show the PCA baseline mentioned in the caption.  I did not follow the description in Line 224-227, which claimed that R^2 greater than 0.1, (which curves should I look at?).


Total Variation Classes Beyond 1d: Minimax Rates, and the Limitations of Linear Smoothers
We consider the problem of estimating a function defined over n locations on a d-dimensional grid (having all side lengths equal to n 1/d ). When the function is constrained to have discrete total variation bounded by C n , we derive the minimax optimal (squared) 2 estimation error rate, parametrized by n, C n . Total variation denoising, also known as the fused lasso, is seen to be rate optimal. Several simpler estimators exist, such as Laplacian smoothing and Laplacian eigenmaps. A natural question is: can these simpler estimators perform just as well? We prove that these estimators, and more broadly all estimators given by linear transformations of the input data, are suboptimal over the class of functions with bounded variation. This extends fundamental findings of Donoho and Johnstone [12] on 1-dimensional total variation spaces to higher dimensions. The implication is that the computationally simpler methods cannot be used for such sophisticated denoising tasks, without sacrificing statistical accuracy. We also derive minimax rates for discrete Sobolev spaces over d-dimensional grids, which are, in some sense, smaller than the total variation function spaces. Indeed, these are small enough spaces that linear estimators can be optimal-and a few well-known ones are, such as Laplacian smoothing and Laplacian eigenmaps, as we show. Lastly, we investigate the adaptivity of the total variation denoiser to these smaller Sobolev function spaces.

The authors answer some open questions about the statistical performance of TV denoising in higher dimensions: They provide new lower bounds for classes of bounded total variation in higher dimensions, as well as lower and matching upper bounds for linear estimators over the same class. Their results show that the performance of linear estimators can be significantly worse than TV denoising if one picks a suitable n-dependent scaling for the total variation, which can be motivated by asking for a natural embedding of the TV class into other regularity classes to hold. Moreover, they give lower bounds for Sobolev classes in higher dimensions and show that contrary to the TV case, these are attained by linear estimators. Finally, they pose the question of whether TV denoising manages to achieve the minimax risk for Sobolev classes in dimension d >= 3 the same way it does for d <= 2.
Novelty  The authors did a fine job of completing the picture of statistical minimax rates for total variation classes. Since TV denoising is heavily used in 2D, any results pertaining to that regime are important. Maybe the most interesting contribution is their establishing the “right” scaling (which is not constant) to be used for the variation term to allow comparison with other derivative based regularity classes. After this has been done, most of the results are interesting, but can be shown with standard techniques together with new estimates of spectral statistics of the grid graph.  For example, the minimax rate for the TV class simply follows from the observation that the l_1 ball is embedded in it, so it can be immediately reduced to that case which is already available in the literature.  I also agree with the authors on the conjectured minimax performance of TV denoising for Sobolev classes in higher dimensions being intriguing.  Quality  The proofs are sound and are executed with sufficient level of detail to be understood by someone with limited background in that field. It would have been even nicer to provide a bit of background on some of the concepts involved in the proofs, e.g. (linear) minimax rates for quadratically convex sets, but since this is restricted to the proofs in the supplementary material, this choice for the sake of brevity might be appropriate.  The comments and discussion are helpful in order to appreciate the formulation of the results.  Impact  The paper offers some nice insight into theoretical performance bounds for a widely used denoising algorithm, which will be of interest to the imaging community as well as theoretical statistics. On the other hand, it does not offer new algorithmic insights nor fundamentally new proof techniques. However, it is notable that most of their lower bound results hold for arbitrary graphs. Also, the authors offer some venues for further exploration such as a possible generalization to trend filtering and investigating the performance of TV denoising over Sobolev classes in higher dimensions.  Clarity  The introduction and motivation sections are very well written and succeed in motivating the results that follow. The presentation of the results is clear and the comments helpful in putting them into context.  Minor comments/corrections:  Main paper:  Line 4: parametrized _by_?  Supplement:  Line 8: “and in the second”: That seems to be dangling here.  Display following line  39: “C_n” n missing  Display following line 49, third to last line: Parentheses around the summands would be helpful  Display following line 63, first line: Is it actually an equality here?  Line 64: Donoho’s paper could already be referenced with a number here.  Second display following 68: Variable clash for i_j, used twice in different contexts  Display following line 82: The paper uses k instead of \tau. Also, it should be \theta instead of \beta.  Third line of display following line 82: Here, something about the nullspace is being used to guarantee that we can write I \simeq D D^\dag. This should be explained in slightly more detail.  Display following line 90: Again, the nullspace has to be well-behaved to be able to write this.


Variational Bayes on Monte Carlo Steroids
Variational approaches are often used to approximate intractable posteriors or normalization constants in hierarchical latent variable models. While often effective in practice, it is known that the approximation error can be arbitrarily large. We propose a new class of bounds on the marginal log-likelihood of directed latent variable models. Our approach relies on random projections to simplify the posterior. In contrast to standard variational methods, our bounds are guaranteed to be tight with high probability. We provide a new approach for learning latent variable models based on optimizing our new bounds on the log-likelihood. We demonstrate empirical improvements on benchmark datasets in vision and language for sigmoid belief networks, where a neural network is used to approximate the posterior.

This work extends hash-based inferences techniques to the learning of latent models. It bridges the gap between existent the theory for hash-based inference for discrete latent variable models and the existent practical inference techniques (sampling and variational methods). The authors provide a theoretical bound on the marginal likelihood of the latent variable models when using hash-based inference then apply it to learning sigmoid belief networks where they claim an improvement as compared to existing benchmarks.
This paper provides theoretical bounds that are tighter than existing variational bounds for the problem of learning latent variable models. The authors extend applied existing theory of hash-based learning and amortized inference to design a black-box learning algorithm. They later applied it to learning a Sigmoid Belief Network. The main advantage to this approach seems to be the partitioning of the search space for posterior distributions into buckets/subsets that are faster to search than with a typical sampling method. The proposed inference scheme then leverages mean-field inference (used heavily in the context of variational inference) within each subset. One of the main technical contributions is the tighter bound on the likelihood using two aggregate estimators which was an extension of an existing work (specific to undirected graphical models) to the directed models setting. However, it was not intuitive how this was more advantageous to stochastic (called mini-batch in the paper) or batch variational inference schemes. While the general variational bounds on the marginal likelihood are provenly tighter, we would have appreciated a less terse explanation (in comparison to mini-batch + mean-field). Additionally, the empirical analysis seemed lacking as there weren't enough details regarding  recent developments and heuristics in the domain were discarded as complementary rather than competing. However, it is clear that the goal of the paper was to prove the superiority of this novel hash-based inference scheme to the common sampling or variational (they used stochastic variational inference for learning the ELBO of the basic SBN unless I misunderstood?) and for that their analysis might be adequate. Overall, despite a few typographical errors, this paper was written eloquently and the methodology was presented adequately and the analysis of the results was quite appropriate.


Learnable Visual Markers
We propose a new approach to designing visual markers (analogous to QR-codes, markers for augmented reality, and robotic fiducial tags) based on the advances in deep generative networks. In our approach, the markers are obtained as color images synthesized by a deep network from input bit strings, whereas another deep network is trained to recover the bit strings back from the photos of these markers. The two networks are trained simultaneously in a joint backpropagation process that takes characteristic photometric and geometric distortions associated with marker fabrication and marker scanning into account. Additionally, a stylization loss based on statistics of activations in a pretrained classification network can be inserted into the learning in order to shift the marker appearance towards some texture prototype. In the experiments, we demonstrate that the markers obtained using our approach are capable of retaining bit strings that are long enough to be practical. The ability to automatically adapt markers according to the usage scenario and the desired capacity as well as the ability to combine information encoding with artistic stylization are the unique properties of our approach. As a byproduct, our approach provides an insight on the structure of patterns that are most suitable for recognition by ConvNets and on their ability to distinguish composite patterns.

In this paper, the authors proposed a way to generate markers (images which encode information, like a QR code) by training an end-to-end network to maximize the recognizability in some marker-aid scenarios for robots. The network consists of a generator which encodes information and generates a marker, a renderer which simulates how the markers are placed in a real environment, and a recognizer responsible for information extraction. Qualitative experiments were detailed, and results under various scenarios were compared.
Briefly, the overall idea of serializing the whole encoding--environment simulation--recognition into an end-to-end network is interesting and worth exploration. The presentation is clear and comprehensive. To me, however, there is still large room to fulfill the idea and its application possibility. Specifically, more experiments should have been done to support and demonstrate the idea.  About the idea. The end-to-end learning is intuitive and proved effective, at least qualitatively, but the model structure binds the synthesizer and the recognizer together, assuming that we already know the recognizer in the first place. In real situations, the two parts are mostly fully decoupled, and this structure limits its applications. For example, different robots may vary in different computational capabilities, but they all need to keep the same recognizer network inside its memory. In other words, unlike a barcode or a QR code which includes only encoding and decoding rules of sequence/image in its protocol, this method also takes the recognizer into its "protocol". Another concern is on the rendering network. The purpose of the rendering network is to simulate the real environment as much as possible, thus directly determining the performance of the method. To ensure the differentiability of the end-to-end network, the rendering network may not perform some of the transformations, such as bending, as mentioned, and some other non-linear transformations, though these can be solved in other ways: limit the marker transformation types in the real world. Also, the rendering in the real world is far more complicated than 2D rendering(example: shadows).         One concern on texture loss. The purpose of texture loss is to incorporate aesthetics into the markers. However, the loss enables but the similarity to the original image, no patterns learned, as is shown in Figure 4. What we expected are some patterns which resemble a QR code: differs in content but follows the same style. Regarding this, the texture loss may harm the recognition accuracy, which limits its use in the real world.          About the experiments. Since there seems no universal criterion for the performance of a protocol, it is acceptable to have only qualitative results and no quantitative comparisons. However, results of existing protocols should be reported and compared, such as of a QR code and of a bar code. For example, the encoder can be replaced with a QR code encoder, and the recognizer a QR decoder (an off-the-shelf one or a trained network exactly same as the used structure in the experiments). Question: in Figure 5, what are the black boxes?          For the renderer, it would be more interesting to investigate that how to encode information can resist certain transformations, such as affine transformations only, lighting variations only, motion blurs only, optical blurs only, or even no transformations at all, where we assume that a robot can adjust its pose to locate the marker and take an image almost same as the originally generated one. This can provide perspective on how to design new protocols to reduce the effects of certain transformations.


Finite-Sample Analysis of Fixed-k Nearest Neighbor Density Functional Estimators
We provide finite-sample analysis of a general framework for using k-nearest neighbor statistics to estimate functionals of a nonparametric continuous probability density, including entropies and divergences. Rather than plugging a consistent density estimate (which requires k → ∞ as the sample size n → ∞) into the functional of interest, the estimators we consider fix k and perform a bias correction. This is more efficient computationally, and, as we show in certain cases, statistically, leading to faster convergence rates. Our framework unifies several previous estimators, for most of which ours are the first finite sample guarantees.

The paper gives a finite sample analysis (bias and variance bounds) for a framework of nonparametric methods, estimating functionals of densities (such as divergences and entropies) by plugging in a k-nearest neighbor density estimate and taking a sample average. The interesting property of the methods being analyzed is that instead of increasing k with the sample size-- which leads to consistent density estimates-- k is fixed. The implication of considering fixed k is that the variance of the density estimate does not go to zero asymptotically.   The non vanishing variance in the density estimate manifests as bias in the estimate of the functional. The methods being considered do some correction to obtain an asymptotically unbiased estimator. The variance of the functional estimate is asymptotically 0, in spite of the non vanishing variance of the density estimate, because of a sample averaging.   The paper derives finite sample bounds for the bias and the variance of such methods under certain weak assumptions on the underlying density.  The paper argues that the bounds are superior to most of the state of art approaches and those approaches that might have better convergence are computationally too demanding, concluding that the fixed-k approach with bias correction gives competitive convergence with significant computational advantage.
Congratulations to the authors for their excellent work.   Quality: This is a high quality paper with significant theory and sophisticated math. The claims of the paper are well supported by the derived error bounds, though, I have a some concerns about the math at a couple of places (see under Concerns). The authors have clearly stated the assumptions under which the bounds apply and have given an objective comparison of their work with previously known results.  Clarity: Overall the paper is very well written and reads easily. The authors have provided intuition which makes it easy to follow the math. The assumptions are very well motivated and discussed. Though there are some formatting issues, typos and notation reuse.   Originality: The error bounds proved in the paper are novel. The proofs are non trivial and interesting. The authors have done an exceptional job of discussing the related work.   Significance: The problem of estimating functionals of densities is very important in several machine learning and statistics problems. The competitive error bounds derived in the paper and the computational advantage of fixed-k nearest neighbor based methods  justify using them for estimation.   Main concern: As I understand, the argument used to apply lemma 4 to derive bounds on $E[f(\epsilon_k(x))]$, is that $f(\epsilon_k(x))$ can be expressed as $f(h(z))$ where $z=\frac{p_*(x)}{\hat{p}_k(x)}$ and h(z)=(\frac{kz}{cnp_*(x)})^{1/D}, and $f o h$ is an increasing function.  However, the definition of $h$ is not independent of z; its definition changes with its argument $z$ because it contains $p_*(x)$ as well in the denominator. In this light, it is not straightforward how lemma 4 can be applied.   The appendix version of lemma 4 seems to derive the bounds on $E[f(\epsilon_k(x))]$ directly. The proof starts with defining $g$, but never uses it. Also, it might be better to have the same result in the appendix and the main paper or lemma with a detailed proof should be added to derive one from the other.  (Line 236) Expression for C_f when f(x)=log(x) is given. It seems like the M_{f,p}(x) is evaluated to be 1. However, I do not see how it can be a constant. Also the expectation should be with respect to p, but the integral is with respect to the borel measure.  In lemma 4, line 218 says that the inequality 7 is a lower bound, however, it is actually an upper bound. Also upper bound for E(f)  =E(f_+)-E(f_-) cannot be derived with <= in 7.  From the expression for the variance bound (line 254), it seems that the upper bound increases with k, instead of decreasing as one would expect.    Minor comments: There are some typos below Theorem 5: line 233 M_{f,o}, Equation below line 233 the second term should be E w.r.t q, not p.  Line 232 gives the impression that q and p should have the same L, it might be better to have L_1, L_2.   Before line 235, it would be good to have expression for the bias bounds for functionals of p and q and perhaps a lemma in the appendix.   Notation reuse: r is used as the exponent in the norm as well as the radius of the ball.  It would be good to include proofs that  all the functionals in table 1 satisfy the assumptions of the theorems.   The intuition provided in line 208 about asymmetry and bias is not is not obvious.  Line 274 about testing problems is difficult to parse.  Overall, I recommend accepting the paper if they address my concerns.  After author's response: The authors have responded to my concerns satisfactorily. Though, I would like to suggest them to add few technical comments to Lemma 4 and Theorem 5 for more clarity.


On Valid Optimal Assignment Kernels and Applications to Graph Classification
The success of kernel methods has initiated the design of novel positive semidefinite functions, in particular for structured data. A leading design paradigm for this is the convolution kernel, which decomposes structured objects into their parts and sums over all pairs of parts. Assignment kernels, in contrast, are obtained from an optimal bijection between parts, which can provide a more valid notion of similarity. In general however, optimal assignments yield indefinite functions, which complicates their use in kernel methods. We characterize a class of base kernels used to compare parts that guarantees positive semidefinite optimal assignment kernels. These base kernels give rise to hierarchies from which the optimal assignment kernels are computed in linear time by histogram intersection. We apply these results by developing the Weisfeiler-Lehman optimal assignment kernel for graphs. It provides high classification accuracy on widely-used benchmark data sets improving over the original Weisfeiler-Lehman kernel.

The paper describes an approach to construct valid (symmetric and p.s.d.) optimal assignment kernels that are built on strong base kernels. Therefore they introduce the properties of strong kernels and show, that they are induced by a hierarchy on a set of elements with increasing weights (from root to leaves) and, vice versa, every strong kernel induces such a hierarchy. Thus, on every set, by building a hierarchy, e.g. on the set of vertices of a graph, a strong kernel is induced. Using this hierarchy they derive a feature mapping and show that the scalar product of this feature mapping equals the strong kernel which therefore proofs that the strong kernel is a valid kernel. Subsequently they show that the optimal assignment can be calculated by a histogram intersection kernel where the histograms are calculated using the earlier introduced feature mapping. This allows for efficient calculation of the optimal assignment kernel once the hierarchy is built.  They experimentally show that by integrating their optimal assignment kernel into existing methods one can reach higher accuracies for graph classification than state-of-the-art methods.
The novelty and clarity of the work seem limited.   ------- Weisfeiler Lehman-OA -------  The least convincing part of the paper is the description of the Weisfeiler-Lehman OA kernel:  - the second argument in equation (2) should be v not u. - the authors miss to cite Shervashidze et al. (JMLR 2011) which already presents variants of the Weisfeiler Lehman kernel, that outperform the classic one. It would have been appropriate to compare to the best of these variants. - If the proposed algorithm just counts all matching labels in the refinement sequence of all pairs of nodes, then it is completely unclear whether it actually differs from the classic Weisfeiler-Lehman kernel (which takes the linear product between the refinement sequence histograms, which is the sum over the dirac kernels on the refinement sequence of all pairs of nodes). The authors have to make this difference much clearer, define all functions/variables used (k_delta) and give an intuition will this - at best minor modification - of the WL kernel should lead to better results.     ------------------ Proof Theorem 1 ------------------ The proof of Theorem 1 is not very clear. It is only shown that the histogram intersection kernel is equal to W(B) but it is not explicitly shown that this bijection B is necessarily the one maximising W(B). From my understanding this follows from the construction of the hierarchy which assures that the kernel between to nodes is maximised. However, this must be shown explicitly in the proof of Theorem 1 in order to proof that the histogram intersection kernel is equal to the optimal assignment kernel. Otherwise the proof is not complete. This missing link makes the paper also very hard to follow and requires to first understand lemma 2 in order to follow the reasoning.  ------------------ Proof Lemma 2 ------------------ Fig. 1 is supposed to illustrate the proof of lemma 2, but it contains labels that are not mentioned in the text of the main article (e.g. b_1, b_2, p',c). I suggest to either modify Fig. 1 such that it fits the sketch of the proof outlined in the main article or to move the figure to the supplement. Since the construction of the hierarchy is essential for the proof of the main theorem 1, I would strongly suggest to add the complete proof to the main paper. Furthermore, there are two typos in the proof: 1. in line 128 main article, change X_{i} to \chi_{i} 2. in line 24 supplement, I suppose it should say 'since z is above LCA(B)'  ------------------ Figure 3 ------------------ The caption of figure 3 is unclear to me. It would be helpful to mention, that the elements in X and Y can be represented by the hierarchy in Fig. 2.   I am not sure what is meant by 'there are three vertices in X that are children of a in the hierarchy, but not represented explicitly'. As far as I understand the method, \chi = {a,b,c}, and one can build a tree T on \chi as in Fig. 2. Together with the weights w, this results in the  hierarchy in Fig. 2. Now, using that X = {a,a,a,b,c} and Y = {a,b,b,c,c} and applying the definition of the histogram kernel, one can arrive at the histograms in Fig. 3b. I don't see the link between the above sentence in the figure caption. Furthermore, in Fig. 3a, I guess the edges between elements in X and Y are supposed to represent the bijections. Still, their link to the histogram kernels remains unclear to me, and it would be good to explain this detail in the caption as well.   ------------------ Missing references ------------------ 1. In the paper, the Hungarian method is mentioned twice. Please add a citation or a definition. 2. The chapters 2 and 3 in the supplement are not referred to in the main article. 3. Outline difference between optimal assignment kernels and convolution kernels in more detail, or add reference


A state-space model of cross-region dynamic connectivity in MEG/EEG
Cross-region dynamic connectivity, which describes the spatio-temporal dependence of neural activity among multiple brain regions of interest (ROIs), can provide important information for understanding cognition. For estimating such connectivity, magnetoencephalography (MEG) and electroencephalography (EEG) are well-suited tools because of their millisecond temporal resolution. However, localizing source activity in the brain requires solving an under-determined linear problem. In typical two-step approaches, researchers first solve the linear problem with generic priors assuming independence across ROIs, and secondly quantify cross-region connectivity. In this work, we propose a one-step state-space model to improve estimation of dynamic connectivity. The model treats the mean activity in individual ROIs as the state variable and describes non-stationary dynamic dependence across ROIs using time-varying auto-regression. Compared with a two-step method, which first obtains the commonly used minimum-norm estimates of source activity, and then fits the auto-regressive model, our state-space model yielded smaller estimation errors on simulated data where the model assumptions held. When applied on empirical MEG data from one participant in a scene-processing experiment, our state-space model also demonstrated intriguing preliminary results, indicating leading and lagged linear dependence between the early visual cortex and a higher-level scene-sensitive region, which could reflect feedforward and feedback information flow within the visual cortex during scene processing.

The paper elegantly presents an approach to source localization that is applicable to both MEG and EEG data (and other modalities), highly sensitive to temporal variations and allows direct extraction of statistical dependance and time lags, from a set of mutually correlated signals (linearly generated from sources). One of the main attributes advocated in this work is that separation into statistically separable sources need not be done independently of causality, or that of time-lag analysis of sources. Rather, the authors present a single model that decomposes the data into sources and models their time dependence, based on expectation-maximization approach. An application to MEG analysis is presented.
The Introduction is generally very good (with minor exceptions described below).   Comparison to other models is required. Only one alternative approach is compared to the suggested method and another one-step model (DCM) is not lawfully described.    I suggest the authors discuss other applications beside EEG/MEG as many of the alternative approaches were shown to be useful to many modalities.   Please introduce consistent spacing before citations (in many cases the space doe not exist at all).   Specific comments:  1. 46. Perhaps the authors can also point to the ill-poshness of the problem due to SNR and destructive local activities: even if we had the number of observation roughly equal to the number of sources still unravelling source activity from scalp dynamics would have been a major challenge. Please elaborate if this is not the case.  2. 54. Please expand “based on different assumptions.” or if those assumptions are already mentioned before, say it.  3. 60. Perhaps the authors can add a reference to the claim? 4. 61. Is this claim addressed later? If yes, say so, if not, expand this claim, or give proper reference, or omit this claim altogether.  5. 77. describe the model “in detail”.  6. 69-71. The text seems to undermine Ref 1. Please explain why DCM is irrelevant to the claim. If it’s relevant please expand on this issue.  7. 85. “includes”. 8. 94. In what neighborhood, of the sensor? Not clear: please revise the entire sentence, including the improper use of  “according to the physics” (92), as the discussed notions combine physical (and biophysical) attributes with anatomical ones.  9. 100 “Gaussian distribution”. 10. 108. “The state variable u_t…” In general  I suggest sentences won’t start with symbols.  11. The definition of A_0 should be fixed. It doesn’t make sense to say $l$ and $\forall l$ — Please make it more formal.  12. 122. “Equations…” 13. 123. The way the sentence is written should be “is a 0/1…”. Later the authors can omit “in the lth row and ith column”.  14. 123. The way Q_J is described is not a proper definition (not clear what is the exact value of Q_J for the case corresponding to equation 4). In fact the “only if” could be a wrong assertion, if I understand correctly  what is meant by the authors.   15. I don’t see how the authors “integrated J_t out.” Isn’t it just a substitution?  16. 132-134 is clear but informal. Please improve the text by making the definitions more formal/correct.  17. 134. Do you need the square when designating the Forbenuis norm? Isn’t the norm given by the notation || ||_F ? 18. 142. “and” is repeating twice.  19. 190. Typo: “pericalcarine suci.”  20. 194: Typo: “entries was.” 21. 198-199. It’s not so clear what is meant by “were different.”  22. 200. Do you mean “nanoamper”? 23. 215. “which underlines recognition of what we see” is not entirely clear and also a too bold description. 24. 244-247. The statistical analysis is not presented in a clear manner. It is also not clear how the p-values are derived given that the T^2 observations are statistically dependent. Please make this part clearer and make sure the assumptions of the statistical test are met.  25. 289. The reference to “cognitive processes” as an exceptional case is not clear, whereas the authors suggest a simple dynamical behavior for other neural processes. The DCM model should be extended also here (and earlier in the paper).  26. 290. The reasoning for not comparing against DCM is weak and should be improved (“Part of of the reason…” — the comparison is required). The authors should make efforts to directly compare their results to DCM.     Ref 1. David et al. “Dynamic causal modeling of evoked responses in EEG and MEG”


Minimax Estimation of Maximum Mean Discrepancy with Radial Kernels
Maximum Mean Discrepancy (MMD) is a distance on the space of probability measures which has found numerous applications in machine learning and nonparametric testing. This distance is based on the notion of embedding probabilities in a reproducing kernel Hilbert space. In this paper, we present the first known lower bounds for the estimation of MMD based on finite samples. Our lower bounds hold for any radial universal kernel on R d and match the existing upper bounds up to constants that depend only on the properties of the kernel. Using these lower bounds, we establish the minimax rate optimality of the empirical estimator and its U -statistic variant, which are usually employed in applications.

The paper deals with the estimation of Maximal Mean Discrepancy (MMD) with radial kernels from the minimax point of view.  It gives new lower bounds for the minimax probability, which coincide with the known upper bounds, up to multiplicative constants (depending on the properties of the kernel).  A first lower bound for Gaussian kernels is straightforwardly obtained from Le Cam's method, following the same arguments as Tolstikhin, Sriperumbudur and Muandet (2016), but this lower bound depend on the dimension of the data. A second lower bound is developed, not depending on this dimensionality parameter and generalized to radial kernels, based on classical arguments from the minimax theory for functionals (the two fuzzy hypotheses method, as named by Tsybakov (2008)).  This new lower bound enables to establish a classical minimax parametric rate of estimation of the MMD, and to prove that the empirical estimator and its $U$-statistic variant, used in the MMD two-sample tests for instance, are optimal in the presented minimax sense.
The paper is very interesting as it provides the first known lower bounds for the minimax rate of estimation of the MMD distance between two distributions based on finite random samples, considering general radial kernels, which match with the known upper bounds up to constants. Theses bounds lead to a classical parametric rate of estimation, not depending on the dimension of the data. The text could however be more fluent, and easier to read for nonspecialists of the minimax estimation theory. For instance, the results are all expressed with respect to the minimax probabilities. Could the link with the more classical minimax risk be clearly written ? Moreover, to my mind the only main drawback of this work is that it does not tackle the real target issue of minimax testing, which is usually closely related to the minimax estimation of functionals. The obtained parametric rate of estimation is indeed known to coincide with the parametric rates of testing when one considers weak distances (such as the Bounded Lipschitz or the MMD distances) between distributions. Finally, I noticed a few mistakes in the text (in (1) a absolute value bar is lacking, and $r_{n,m,k}(\mathcal{P})$ should be replaced by  $r_{n,m,k}(\mathcal{P})^{-1}$ in (2).


Learning Transferrable Representations for Unsupervised Domain Adaptation Brain of Things
Supervised learning with large scale labelled datasets and deep layered models has caused a paradigm shift in diverse areas in learning and recognition. However, this approach still suffers from generalization issues under the presence of a domain shift between the training and the test data distribution. Since unsupervised domain adaptation algorithms directly address this domain shift problem between a labelled source dataset and an unlabelled target dataset, recent papers [11,33] have shown promising results by fine-tuning the networks with domain adaptation loss functions which try to align the mismatch between the training and testing data distributions. Nevertheless, these recent deep learning based domain adaptation approaches still suffer from issues such as high sensitivity to the gradient reversal hyperparameters [11] and overfitting during the fine-tuning stage. In this paper, we propose a unified deep learning framework where the representation, cross domain transformation, and target label inference are all jointly optimized in an end-to-end fashion for unsupervised domain adaptation. Our experiments show that the proposed method significantly outperforms state-of-the-art algorithms in both object recognition and digit classification experiments by a large margin.

In this paper, the authors proposed a new approach based on deep learning by incorporating an idea of transductive learning for transfer learning. Experiments are conducted on a fully transductive setup. Though the experimental experimental results look promising, there are some major concerns on the proposed model as well as the experimental setup. Please refer to my detailed comments.
Regarding the proposed model, below are the major concerns:  1. In Abstract and Introduction, the authors highlighted several times that transfer learning or domain adaptation aims to align the mismatch between the training and testing data distributions, such that good generalization can be obtained across domains or tasks. In the problem setup, the authors further explicitly state that \hat{x}_i and x_i follow different distributions p_s and p_t, respectively. However, different from some existing methods, like [19] and [Pan etal., Domain adaptation via transfer component analysis, IEEE TNN, 2011], the proposed model indeed does not explicitly minimize the distance or align the mismatch between the training and testing distributions. There is no guarantee that based on the new representation, the mismatch issue between distributions can be addressed. Therefore, I would suggest the authors the rephrase those claims carefully. The new high-level representation learned by deep learning may be able to reduce the difference between domains, but is not able to align distributions explicitly.  2. Compared to other transfer learning methods, the new idea here is to introduce a tranductive step to make use of predicted labels on the target domain unlabeled data. This looks new to transfer learning. However, this idea is indeed borrowed from the co-training technique in semi-supervised learning (transductive setting). The adaptation step, which learns common and task-specific parameters, is quite standard in deep learning based transfer learning methods. Thus, technically, I would consider the proposed model is a combination of existing techniques.  3. The solution to labeling target domain unlabeled data, i.e., (2), is quite heuristic. The basic idea of the Reject Opinion is to introduce a kind-of confidence on the predicted labels. However, it would be more solid if some theoretical analysis is provided.  4. Discussions on convergence of the alternating optimization procedure are missing.  Regarding experiments,  1. I guess the performance of the proposed method is quite sensitive to the value of k in k-nn on different datasets. Sensitivity analysis on k is missing. In addition, in practice, how to tune the value of k is lack of discussions.   2. I guess the performance of the proposed method is also sensitive to the value of \gamma in the proposed Rejection Option. Sensitivity analysis on k is missing. Similarly, in practice, how to tune the value of \gamma is lack of discussions.  3. Experiments are only conducted on a fully transductive setup, which is not practical. Actually, after the labels of the target domain unlabled data are estimated by the proposed model, a k-nn can be applied to make predictions on out-of-sample target domain test data. It would be more interesting to show results on an out-of-sample test dataset.


Scan Order in Gibbs Sampling: Models in Which it Matters and Bounds on How Much
Gibbs sampling is a Markov Chain Monte Carlo sampling technique that iteratively samples variables from their conditional distributions. There are two common scan orders for the variables: random scan and systematic scan. Due to the benefits of locality in hardware, systematic scan is commonly used, even though most statistical guarantees are only for random scan. While it has been conjectured that the mixing times of random scan and systematic scan do not differ by more than a logarithmic factor, we show by counterexample that this is not the case, and we prove that that the mixing times do not differ by more than a polynomial factor under mild conditions. To prove these relative bounds, we introduce a method of augmenting the state space to study systematic scan using conductance.

This paper is interested in comparing the mixing rates of Gibbs sampling using either systematic scan or random updates. The basic contributions are two: First, in Section 2, a set of cases where 1) systematic scan is polynomially faster than random updates. Together with a previously known case where it can be slower this contradicts a conjecture that the speeds of systematic and random updates are similar. Secondly, (In Theorem 1) a set of mild conditions under which the mixing times of systematic scan and random updates are not "too" different (roughly within squares of each other).
I think this paper addresses an important issue and makes valuable contributions, and thus should be published. I have a few concerns, hence my lower rating for the last question above (which I think could be addressed relatively easily, however).       - The proofs are sketched out in quite an informal manner. I think this is fundamentally *OK* and even perhaps a positive thing. However, I think a bit more discussion needs to be given to how the arguments might be made more formal. For example, in Section 2.1, I think the proof is intended to hold only in the limit of M going to infinity. Please give a stament of what should hold in what limit-- this wasn't clear to me. Another example is the assumption that the bridge state has low mass in Section 2.2. This seems like a dubious assumption. Again this seems to be intended to hold only in the limit of a large model but exactly how this should work is not sketched out. (Also, if you are going to make this assumption, why not include it in Eq. 3?) Finally, the assumption that each of the islands mix rapidly within themselves again seems to weaken the strength of Section 2.2 All these assumptions are probably fine-- and improve the readability of the paper-- but I think more guidance needs to be given how one would make things more formal if desired.       - I don't understand the experiments. Specifically, how are the mixing times calculated? This isn't discussed at all. I do not understand what was done to only consider states next to the bridge.       - The argument at the end of Section 2.2 seems sufficient to compare f(n) to n*f(n) but it isn't clear why f(n) should itself be n.       - What is |x| in P(x) on p. 4? Is this simply the length of x?       - All the results seem to have the nature of adding/removing a factor of n, or squaring the mixing times. Is there anything fundamental about this? The abstract seems to lose something in only referring to this as "polynomial".


Estimating the class prior and posterior from noisy positives and unlabeled data
We develop a classification algorithm for estimating posterior distributions from positive-unlabeled data, that is robust to noise in the positive labels and effective for high-dimensional data. In recent years, several algorithms have been proposed to learn from positive-unlabeled data; however, many of these contributions remain theoretical, performing poorly on real high-dimensional data that is typically contaminated with noise. We build on this previous work to develop two practical classification algorithms that explicitly model the noise in the positive labels and utilize univariate transforms built on discriminative classifiers. We prove that these univariate transforms preserve the class prior, enabling estimation in the univariate space and avoiding kernel density estimation for high-dimensional data. The theoretical development and parametric and nonparametric algorithms proposed here constitute an important step towards wide-spread use of robust classification algorithms for positive-unlabeled data.

The paper proposes a method for estimating the class prior given on labelled sample, as well as noisy positive samples. This builds upon an existing approach to estimate the prior when the positive samples are clean. Experiments show good performance compared to some existing methods.
UPDATE AFTER REBUTTAL:  Regarding the main result highlighted in the rebuttal: it's indeed nice to relate the observed P(y|x) to the true one, but again this was already done in [1] using similar ideas, just without noise in the positives. The result here thus didn't really surprise me. As mentioned in my review, it's certainly valuable to have all this written down, but to me falls below the novelty threshold.   Regarding the related work on asymmetric noise: my basic point is that from (Scott et al., 2013) Lemma 1 + Proposition 3 (part 2), one has a means of relating the output of a "mixture proportion estimator" (MPE) to the mixing weights for generic asymmetric positive and negative label noise. As I understand, that means that the estimator of Sanderson & Scott that I mentioned could be used in the asymmetric noise scenario. The authors are correct that Sanderson & Scott focussed on an application to (multiclass) positive + unlabelled learning, without noise in the positives; but the point remains that their estimator is conceivably applicable in the scenario of noisy positives as well. I believe this is similarly true of Ramaswamy et al.'s estimator.  ==============  Learning from positive and unlabelled data is a fairly well studied problem, and the paper makes a reasonable case for attention to be made in the case where the positives are subject to noise. Existing methods in the PU literature for estimating the class prior are not directly applicable here, and so it is of interest to design methods for the scenario. The paper thus is well motivated.  On the technical side, the paper has a fairly careful theoretical analysis of the identifiably issues in this problem, which is appreciated. This is used to determine a canonical form in which estimation is performed, leading to a practical algorithm for the estimation. The basic idea is to build upon the AlphaMax framework of (Jain et al., 2016), and cast the estimation problem as determining the mixing weight for a certain mixture model. This estimation is performed on the outputs of any probabilistically-consistent scorer on the inputs (i.e. it does not suffer from the curse of dimensionality.) Thus, the basic technical content seems sound.  I do however have two concerns about the novelty of the work.   (1) The analysis in section 3 appears to closely follow that of (Jain et al., 2016), with some natural extensions to account for the fact that one is in the noisy positive setting. Unless I missed something, there don't seem to be any fundamentally new insights or techniques used in the proofs of the section: the two basic messages of (Jain et al., 2016) are translated, namely that working with all possible distributions leads to non-identifiability, while disallowing distributions that are themselves mixtures alleviates the problem. One has two calls to AlphaMax for the estimation, but the basic idea of mixture modelling is as per the original paper.   The idea of then using univariate transformations in section 4 appears particularly similar to what is done in (Jain et al., 2016). The paper claims that the latter was restricted to the case where the transformation results in calibrated probability, but doesn't Thm 9 of the latter also allow for arbitrary transformations?   Certainly it is valuable for all this to be written down in the noisy PU setting, but this leads to my next concern.   (2) The precise distinction of the scenario considered here to the general asymmetric label noise setting, seems worthy of a little more discussion. This is first to make clear why one does not just conduct all analysis in the general case, rather than assuming that one of the observed  distributions is in fact the underlying marginal (i.e. why one does not work in the general mutually contaminated framework of Scott et al., 2013). Is there some reason analogues of Theorem 1 would not exist here? (As noted, at least some of the results have precedent in Scott et al., 2013, albeit in a different language.)   The second reason is there are methods for the latter which I think could be applied for the scenario considered here, in which case the paper should compare against them. In particular, I refer to the practically viable estimators of  Sanderson and Scott. Class proportion estimation with application to multiclass anomaly rejection. AISTATS 2014.  Liu and Tao. Classification with Noisy Labels by Importance Reweighting. PAMI 2016. (arXiV draft since 2014)  Ramaswamy et al. Mixture Proportion Estimation via Kernel Embedding of Distributions. ICML 2016. (arXiV draft since early 2016, so reasonable for authors to be unaware of)  All three of the above seem like they would be relevant at the end of the third paragraph in the introduction, as a contrast to the cited work of Scott et al., 2013 that relies on density estimation. I note that the first method in particular also does not require the use of calibrated probabilities, but rather a probabilistically consistent ranker (as the estimate is based on a derivative of the RHS of the ROC curve).  Overall, I think the paper's contribution is reasonable, being an extension of (Jain et al., 2016) to the noisy PU case, but I think the novelty is a weak point.  Other comments:  - section 3 is said to include "a few missing results needed for our approach" -- explicitly identifying which these are seems prudent.  - I found the use of a subscript and superscript in a^lambda_mu a bit confusing. Consider just a(lambda, mu)?  - the introduction of \Pi^res could perhaps be deferred till after Lemma 1.  - consider swapping statements 4 and 5 for lemma 1, since the proof seems to do the same.  - in the proof of lemma 1, consider not having all the equations inline to improve readability.  - typo: "Theorem 1 (identifiablity)"  - I felt that section 4 could be more explicit about the suggested algorithm, i.e. that one simply perform the mixture modelling on the classifier outputs on the positive and unlabelled samples.  - it might be worth explaining in more detail what exactly the probabilistic model summarised in Figure 1 corresponds to. That is, is it meant to represent an assumed generative view of the data, which must be satisfied for the data generation process? Or a means by which one processes the available data one has? Put another way, suppose one is able to draw from the unlabelled distribution, and then separately from the noisy positive distribution; what relevance does Fig 1 have?  - is (12) related to the result about the label probability function (Proposition 1) in  Ward et al. Presence-only data and the EM algorithm. Biometrics, 2009.


Understanding the Effective Receptive Field in Deep Convolutional Neural Networks
We study characteristics of receptive fields of units in deep convolutional networks. The receptive field size is a crucial issue in many visual tasks, as the output must respond to large enough areas in the image to capture information about large objects. We introduce the notion of an effective receptive field, and show that it both has a Gaussian distribution and only occupies a fraction of the full theoretical receptive field. We analyze the effective receptive field in several architecture designs, and the effect of nonlinear activations, dropout, sub-sampling and skip connections on it. This leads to suggestions for ways to address its tendency to be too small.

The paper attempts to “debug” deep convolutional neural networks (CNNs) by analyzing their receptive field in the visual tasks. It provides theoretical evidence that under certain assumptions the effective receptive field (ERF) is asymptotically distributed as a Gaussian and, therefore, smaller than the theoretical receptive field (i.e., smaller than expected). The authors discuss how different blocks of CNNs affect the ERF. They suggest that performance of CNNs can be improved if the ERF is increased or grows faster width depth. Most of their theoretical findings are supported in the experimental part on image datasets (mainly CIFAR-10).
Technical quality. - The quality of experiments is overall good, but still could be improved. For instance, it is not informative to provide absolute values of classification accuracy (e.g., accuracy of 89% on CIFAR-10 is neither good for today standards, nor meaningful). Instead, a comparison should be provided in which contribution of authors ideas are clear.  Since there is such comparison (although, only in the rebuttal) for training speed clearly showing improvements, and questions of other reviewers regarding technical quality are addressed, I am able to give higher scores. The author should check if they use correlation or convolution in eq. (7)-(9) and (14). It should not be important for the distribution, but still seems that you use correlation while say that it's convolution.  Novelty/originality. The idea to analyze the effective receptive field sounds novel. It would be interesting to see analysis of more nonlinearities inherent to networks.   Potential impact or usefulness. This work potentially can be really useful and employed to improve initialization or architectures of CNNs. The authors provide some empirical evidence that making the effective receptive field larger by a special initialization method speeds up training of CNNs, which is quite valuable. The theoretical part can also be picked up and further extended by others.  Clarity and presentation. - Additional proofreading would be nice. The paper has several rather minor errors/typos. Next, all headings should be lower case (e.g., Section 4.2) according to the NIPS style.  All figures should have numbers and captions, otherwise, it can be hard to refer to them. Sentences containing equations should follow punctuation rules (e.g., equations through lines 78-82 and other instances). In several cases, additional text should be provided along with notations to ease reading of the paper. For instance, “...the receptive field size as n grows.” (line 240). So, what is ‘n’? Even though somewhere in the paper ‘n’ was denoted as something, the reader will have to search for it. - Also, few minor comments regarding references. 1) Stay consistent with proceeding titles, e.g., either use full titles as  “Advances in neural information processing systems” or abbreviated as “NIPS” for all references. 2) If the paper from arXiv you refer to was published in another (peer reviewed) place, it’s better to cite that place. For example, “Deep Residual Learning for Image Recognition” was published at CVPR.


Efficient Second Order Online Learning by Sketching
We propose Sketched Online Newton (SON), an online second order learning algorithm that enjoys substantially improved regret guarantees for ill-conditioned data. SON is an enhanced version of the Online Newton Step, which, via sketching techniques enjoys a running time linear in the dimension and sketch size. We further develop sparse forms of the sketching methods (such as Oja's rule), making the computation linear in the sparsity of features. Together, the algorithm eliminates all computational obstacles in previous second order online learning approaches.

The authors strive to make second-order online learning efficient by approximating the scaling matrix A_t by a low-rank sketched version based on S_t^T S_t. They prove scale-invariant regret guarantees for this approach when the desired matrix A_t is well approximated in this way, and show that the algorithm can be implemented efficiently for sparse data.
The quadratic update time and space requirements of second-order online methods like Online Newton Step make such algorithms unsuitable for most practical problems. The present work takes a significant step in addressing this. The primary contribution of the paper are variations of Online Newton Step that remove this drawback using a sketching approximation to the scaling matrix and a clever implementation of sparse updates.  The primary theoretical contributions are the analysis of the RP and FD versions of the algorithm. For RP they show a regret bound which holds when the matrix G_T (the matrix of observed gradients) is actually low-rank. Given the structure of the loss functions assumed,  f_t(w) = \ell(< w, x_t >), gradients will always be in the direction of the examples x_t, and so I think this theorem only holds when the data is actually low-rank. But if that was the case you could always simply project the data onto a basis and run ONS on the lower dimensional space. Hence, this result feels somewhat limited. The FD result Thm 3 is thus stronger (and is the one presented in detail in the body of the paper) since it depends instead on the spectral decay of G_T^T G_T. This point should be clarified in the paper.  The authors emphasize the fact their results are scale-invariant, but this also comes at some cost. Most results in online convex optimization apply to arbitrary convex functions, possibly satisfying additional conditions like strong convexity or smoothness. This work assumes a much more restrictive class of functions where f_t(w) = \ell(< w, x_t >), essentially a generalized linear model. This appears necessary to introduce the concept of "invariant learning", but the importance of this approach in practice isn't clear to me. One can choose the fixed norm bound on the feasible set after scaling the features, and in practice one can often avoid projecting onto a bounded feasible set at all as long as the learning rate is set reasonably. Further, while the author's claim the bound in Theorem 1 is "qualitatively similar" to the standard setting, the explicit dependence of the dimension d is a significant difference in my mind.  The experimental section presents results on both synthetic and real-world datasets. The synthetic data nicely highlights the potential for improvements from this approach; only some of this is demonstrated on real-world datasets. The author's only present experiments on the Oja-SON variant, which is unfortunate since it lacks theoretical guarantees. The importance of the theory is somewhat called into question by this, since it implies the assumptions on A_t necessary for RP-SON and FD-SON to work well may not actually hold in practice (as discussed above, this seems quite likely for RP-SON). Further, in order to outperform AdaGrad (which is 10x faster and substantially easier to implement), a significant trick was needed (lines 254-259). Given that this approach was necessary to get good performance in practice, a more thorough discussion is warranted.


Achieving the KS threshold in the general stochastic block model with linearized acyclic belief propagation
The stochastic block model (SBM) has long been studied in machine learning and network science as a canonical model for clustering and community detection. In the recent years, new developments have demonstrated the presence of threshold phenomena for this model, which have set new challenges for algorithms. For the detection problem in symmetric SBMs, Decelle et al. conjectured that the so-called Kesten-Stigum (KS) threshold can be achieved efficiently. This was proved for two communities, but remained open for three and more communities. We prove this conjecture here, obtaining a general result that applies to arbitrary SBMs with linear size communities. The developed algorithm is a linearized acyclic belief propagation (ABP) algorithm, which mitigates the effects of cycles while provably achieving the KS threshold in O(n ln n) time. This extends prior methods by achieving universally the KS threshold while reducing or preserving the computational complexity. ABP is also connected to a power iteration method on a generalized nonbacktracking operator, formalizing the spectral-message passing interplay described in Krzakala et al., and extending results from Bordenave et al.

This work is about the community detection threshold in stochastic block models (SBM). It deals with arbitrary SBM with $k$ communities (i.e. the communities need not to have equal proportion, the interactions between groups is not restricted to within-groups and inter-groups values and $k$ may be larger than 2).    The paper establishes that a linearized acyclic belief propagation algorithm achieves the Kesten-Stigum (KS) threshold for group detection. Here, detection is defined in a weaker sense than the previous definition from Decelle et al. In the specific case of $k=2$ and equal group proportions it boils down to Decelle et al.'s definition but is weaker in general.
Major comments:  - As mentioned by the authors in the introduction, the SBM is widely used throughout many areas, and not only for detecting communities. The authors introduce no condition on the parameter matrix Q so that their setup includes (in principle) cases as diverse as communities, anti-communities (graphs of a bipartite type) and any other type of connectivity structure. However, I believe that their algorithm (based on belief propagation and thus on the community structure) will reveal only communities. If I am right, they should specify where this underlying assumption comes into play. Is it only that definition 3 becomes useless when the structure is not the one of communities?  - Most of the arguments given here are not understandable to someone not familiar to the field. For e.g. on line 186-187: ‘For any small t, we would expect that \bar w. W^t \bar w is almost equal to \lambda_i^t \|\bar w \|_2^2’. Why is this the case? Maybe you could refer to some textbooks to ease the understanding.    Minor comments:  - The bibliography needs to be corrected. I believe Ref [12] appeared in Probab. Theory Relat. Fields (2015, vol 162:431–461) with a slightly different title. Authors ordering is wrong for ref [5]. Ref [15] also appeared in PTRF in 2015. Ref [16] appeared in STOC'16. Ref [19] has appeared in FOCS 2015, etc.  - please avoid usage of undefined acronyms (e.g. SDP, SAW,…) - End of section 2.1 uses notation introduced at the beginning of section 2.2 (matrix P and eigenvalue \lambda_1). It would be more convenient to move this paragraph.  - Theorem 1: notation ABP(G,m(n),r,c,\lambda_1,\dots,\lambda_h)) only appears in the next section.   - Remark 1: ‘one should use variable that are accurate to within a factor of…’: to which variable are you referring here? Also the term ln(\lambda_1^2/\lambda_2^2) could be simplified into 2ln(\lambda_1/\lambda_2)  - Algorithm description: as the graph G is not directed, the algorithm description should be cautious when referring to edge (v,v’) \in E(G) as here these are directed edges. This remark is valid for the vanilla version as well as the full version (and also the supplementary material).  - line 296: ‘whenever there is no repeated vertex within r of a nonzero c_i’. What do you mean?    Typos: - line 81-82: ’SBM model’ is redundant.  - line 286: do you mean a ‘r-nonbacktracking walk’?  - line 288: the sum is over v_0, \dots v_t but v_t has no meaning here. Should be v_m?  - line 304: second factor on the right-hand side: is this \lambda_s or \lambda_j? Same applies on line 306.   In Supplementary material:  - Before Theorem 1: ’SBP algorithm’ should be ABP.  - Corollary 1: Notation ‘SBM(n,k,a,b)’ -while easily understandable- has never been defined.


Sparse Support Recovery with Non-smooth Loss Functions
In this paper, we study the support recovery guarantees of underdetermined sparse regression using the 1 -norm as a regularizer and a non-smooth loss function for data fidelity. More precisely, we focus in detail on the cases of 1 and ∞ losses, and contrast them with the usual 2 loss. While these losses are routinely used to account for either sparse ( 1 loss) or uniform ( ∞ loss) noise models, a theoretical analysis of their performance is still lacking. In this article, we extend the existing theory from the smooth 2 case to these non-smooth cases. We derive a sharp condition which ensures that the support of the vector to recover is stable to small additive noise in the observations, as long as the loss constraint size is tuned proportionally to the noise level. A distinctive feature of our theory is that it also explains what happens when the support is unstable. While the support is not stable anymore, we identify an "extended support" and show that this extended support is stable to small additive noise. To exemplify the usefulness of our theory, we give a detailed numerical analysis of the support stability/instability of compressed sensing recovery with these different losses. This highlights different parameter regimes, ranging from total support stability to progressively increasing support instability.

The authors present sharp criteria for l_1-sparse support recovery under small noise in the context of non-smooth regression losses l_\infty and l_1. Traditionally, the problem studied in support recovery left the loss shape fixed at MSE (l2^2) loss and explored different non-smooth penalties, starting from the work of Fuchs on l1 penalty up until the recent, possibly most general partly smooth penalties of Vaiter et al. In the present contribution, the authors keep the simple l1 penalty but explore the effect of changing the loss function. Concretely, two loss functions, l_\infty and l_1 are studied. Both share the characteristic of being polyhedral.  A proof for both cases is given, along with some confirmatory experimental results.
This paper presents a new angle of generalization of the l_1 support recovery criteria to other types of loss functions, which are generally useful in regression: l_\infty corresponds to a uniform noise assumption, whereas l_1 corresponds to a sparse noise assumption. In performing this generalization, it exhibits the functionality of some quantities also obtained in the classic MSE + l1 setting, but which at the time were too specific to be given names. We thus have a non-trivial step forward in the understanding of the mechanisms involved. The proofs in the main paper are sound.  However, my main comments concern the readability of this contribution. This paper took me by far the longest to review, because while mathematically perfectly rigorous, it is as if the authors a) underestimate the fact that their contribution is some non-trivial convex analysis, that they b) do not seem to be interested in making any concessions to the reader by providing their own intuitions or figures elucidating them and c) that this may lead to reviews that completely miss the point of the paper.  Specifically, my points are  * It would be great to have a small diagram explaining the regression problem in 1D or 2D, especially since l_1 and l_\infty losses are equal, resp. equivalent in these two settings. There is sufficient space to add this without deleting anything.  * up until a certain point in the paper, the paper works with general dual pairs \alpha, \beta whose harmonic mean is 1. At some point this specializes to the polyhedral losses and later in the paper specializes to the l_\infy loss (l1 in supp mat). These transitions have to be made more clear. It would be helpful also to state what intuitions are needed to keep the general (\alpha, \beta) (i.e. \alpha \in (1, \infty) is not polyhedral so the proof is different, but here is how you would go about it, but here is why we don't do this in this paper).  * In general, the first reading of this paper is difficult, because many notations are introduced a posteriori. One arrives at table 1 and spends some time looking for definitions of \tilde\Phi and S. These are introduced in the lines below. Then one assumes |S| = |J|, which seems surprising, and is referred forward to Lemma 2 for discussion, instead of giving a quick intuition that this is not only "for simplicity" but even "almost surely holds" in a probabilistic setting and is a reasonable assumption, before referring to the details mentioned in lemma 2. There are also several formulae which introduce a symbol and state in the next lines ", where [symbol] means [meaning]". Could these be put before the formula?  * l120 concerning the situation where I != J, there is also [Tibshirani, The Lasso problem and uniqueness]. While the paper seems to state something that is implicit in many previous contributions, it would be worth being a little more explicit in how these relate  * It took me 3 readings to see the bold r in regularization in l139 to make the link to FO_r. Please introduce explicitly FO_r and FO_\beta. Additionally, it would be helpful to make it a bit clearer, by adding an intermediate step, what "corresponds" means in l139. Not all readers are versed enough in convex duality to read this easily. In general, since there still remains half a page of space, a way of obtaining the Fenchel dual (e.g. by introducing the lagrangian, splitting primal variables and minimizing it wrt both of them) and first order conditions could also be given.  * It would be great to have a bit more intuition on the Lagrange multipliers v_\beta (and some more intuition on p_\beta can be helpful too). In which space do they live (the same as x resp. same as y). How can one imagine them geometrically? The polytope defined by the regression constraint is difficult to untangle in the space of x, since one doesn't know which constraints are active. This polytope is an intersection of seminorm balls, which could even be illustrated in 2D. Diagrams, although not easy to make, would help readers enormously.  * between line 189 and 190, does it have to be p_{1, S}?   Minor * l192 "So now, " -> "Now " * l194 hypotheses -> hypothesis * between l200 and l201 put a word between the formulas "or the second line which implies the first by operator inequality" *l202 get -> obtain *l207 get -> obtain *l248 exceeds -> exceed *l255 theoertical -> theoretical


Learning HMMs with Nonparametric Emissions via Spectral Decompositions of Continuous Matrices
Recently, there has been a surge of interest in using spectral methods for estimating latent variable models. However, it is usually assumed that the distribution of the observations conditioned on the latent variables is either discrete or belongs to a parametric family. In this paper, we study the estimation of an m-state hidden Markov model (HMM) with only smoothness assumptions, such as Hölderian conditions, on the emission densities. By leveraging some recent advances in continuous linear algebra and numerical analysis, we develop a computationally efficient spectral algorithm for learning nonparametric HMMs. Our technique is based on computing an SVD on nonparametric estimates of density functions by viewing them as continuous matrices. We derive sample complexity bounds via concentration results for nonparametric density estimation and novel perturbation theory results for continuous matrices. We implement our method using Chebyshev polynomial approximations. Our method is competitive with other baselines on synthetic and real problems and is also very computationally efficient.

The paper proposes a learning approach for HMMs with non-parametric emission distributions that computes the observable representation with the help of kernel density estimates. An approximation of the kernel density estimates with Chebyshev polynomials improves the runtime of the approach and ensures its practical applicability.
Strengths and Weaknesses ======================== Positive aspects: + The paper is well written and has a clear and coherent structure. The discussion of related work is comprehensive. + Non-parametric emission distributions add flexibility to the general HMM framework and reduce bias due to wrong modeling assumptions. Progress in this area should have theoretical and practical impact. + The paper builds upon existing spectral methods for parametric HMMs but introduces novel techniques to extend those approaches to the non-parametric case. + The theoretical bounds (section 5) are interesting, even though most of the results are special cases or straightforward extensions of known results.  Negative aspects: - The restriction to triplets (or a sliding window of length 3) is quite limiting. Is this a fundamental limitation of the approach or is an extension to longer subsequences (without a sliding window) straightforward? - Since the paper mentions the possibility to use Chebyshev polynomials to achieve a speed-up, it would have been interesting to see a runtime comparison at test time. - The presentation is at times too equation-driven and the notation, especially in chapter 3, quite convoluted and hard to follow. An illustrative figure of the key concepts in section 3 would have been helpful. - The experimental evaluation compares the proposed approach to 4 other HMM baselines. Even though NP-SPEC-HMM outperforms those baselines, the experimental evaluation has only toy character (simple length 6 conditional distributions, only one training/test sequence in case of the real datasets).  Minor points ============ * l.22: Only few parametric distributions allow for tractable exact inference in an HMM. * l.183: Much faster approximations than Chebyshev polynomials exist for the evaluation of kernel density estimates, especially in low-dimensional spaces (e.g., based on fast multipole methods). * Figure 1: There is probably a ‘x 10^3’ missing in the plot on the bottom right.  Questions ========= * Eq. (3): Why the restriction to an isotropic bandwidth and a product kernel? Especially a diagonal bandwidth matrix could have been helpful. Would the approximation with Chebyshev polynomials still work? * The paper focuses on learning HMMs with non-parametric emission distributions, but it does not become clear how those emission distributions affect inference. Which of the common inference tasks in a discrete HMM (filtering, smoothing, marginal observation likelihood) can be computed exactly/approximately with an NP-SPEC-HMM? * Is it computationally feasible to use the proposed model in a more realistic application, e.g., action recognition from motion capture sequences?  Conclusion ========== The paper is well written and, from a theoretical perspective, interesting to read. However, the experiments are weak and it remains unclear how practical the proposed model would be in real applications. I’m tending towards accept, but the authors should comment in their rebuttal on the above points.