Computing the Stationary Distribution, Locally
Computing the stationary distribution of a large finite or countably infinite state space Markov Chain (MC) has become central in many problems such as statistical inference and network analysis. Standard methods involve large matrix multiplications as in power iteration, or simulations of long random walks, as in Markov Chain Monte Carlo (MCMC). Power iteration is costly, as it involves computation at every state. For MCMC, it is difficult to determine whether the random walks are long enough to guarantee convergence. In this paper, we provide a novel algorithm that answers whether a chosen state in a MC has stationary probability larger than some ∆ ∈ (0, 1), and outputs an estimate of the stationary probability. Our algorithm is constant time, using information from a local neighborhood of the state on the graph induced by the MC, which has constant size relative to the state space. The multiplicative error of the estimate is upper bounded by a function of the mixing properties of the MC. Simulation results show MCs for which this method gives tight estimates.

The authors present an algorithm to estimate (with
      potentially arbitrary accuracy) whether the stationary distribution of a
      specific state of a Markov chain is greater than or less than a
      pre-specified parameter, \delta. The algorithm uses truncated Monte Carlo
      rollouts to estimate the expected first return time from the specified
      state and then inverts this to get the state's stationary distribution.
      The paper also provides a novel theoretical contribution in the form of a
      proof that return times have exponentially decaying tails for
      countable-space MCs (not just finite-space MCs, which was what was known
      previously). A minimal empirical evaluation is provided as well, to
      demonstrate that this algorithm can be run. Overall, the paper is
      quite well-written and clear with good explanations and intuitions given
      behind specific choices or aspects of the work. The authors address
      potential questions in good fashion and try to give an honest presentation
      of their work. That being said, I have a number of questions regarding the
      work. Regarding the abstract, the authors state that power
      iteration and MCMC run in linear time, but they also give the stationary
      distribution over all states, so if you did that with this algorithm it
      would be at least linear in the state space size as well, no?
      Regarding practicality, could the author's give some intuition
      about what someone would use this procedure for as well as how to set the
      parameters for different types of use cases? The authors indicate that
      there is wide applicability of this algorithm, which is true, but I am
      having a difficult time coming up with useful purposes of this algorithm
      (unless one runs it on every state in the space, which is somewhat
      unreasonable). Could you comment on potential future extensions
      (potentially to improve practicality, e.g., if there was smoothness in the
      space)? Finally, the Borgs et al. [17] algorithm cited in the paper seems
      more practically relevant -- could this algorithm be extended in that
      direction somehow? Why doesn't the algorithm return the estimated
      stationary probability of i, instead of just returning 0/1 based on
      \delta? It seems that that would be more useful. Specifics: -
      it would be helpful if the description of the algorithm was a bit clearer,
      specifically - specify the inputs as the parameters and the state i
      - clearly state that s_k is the length of a sampled trajectory
      - the statement of theorem 3.1 is strangely asymmetrical. Why
      don't the authors just use "With probability greater than (1-\alpha): If
      \pi_i < \delta, the algorithm outputs 0 and if \pi_i > \delta the
      algorithm outputs 1." - on line 261, P^(t)_{ii} is introduced
      without being defined (as far as I can find) -- P in all other instances
      is not time-dependent (and thus Z_ii should tend to +/- \infty, no?) --
      what is this quantity? - furthermore, on line 280, the authors
      state that, in PageRank, Z_ii \approxeq 1, but give no intuition about
      whether this is good/bad or high/low -- what is the range of Z_ii?
      - there is a duplicated sentence on life 653 of the supplementary
      material Results section: - change colors and data markers to
      be distinguishable in black and white - does the x-axis of (a) refer
      to the # of nodes or the node IDs? - is bias correction just working
      better because it is taking more steps (based on (b)) than the biased
      estimate? can you control for this? - please show the actual # steps
      and \delta (not the log of them) but plot these values on a log scale (so
      the reader has a better understanding of the numbers being shown -- figure
      (b) seems to indicate that 8 million steps were taken for some of the data
      points?) - a log scale on the y-axis of (c) would give a better
      indication of the differences between the data points - the figures
      are numbers from a single run on a single graph? why aren't they averaged
      over many random graphs? - a real data example would help highlight
      the efficacy of this algorithm in domains with strange distributions and
      hot spots in the state space - (b) and (d) are said to show the
      results from three different nodes, but what are the results for each and
      how are they combined for the graph? could all three be shown separately?
      - it's not clear that the computation time is upper-bounded by O(1 /
      \delta) because it's a log-log scale -- could the authors draw this line
      on the graph?


Fast Template Evaluation with Vector Quantization
Applying linear templates is an integral part of many object detection systems and accounts for a significant portion of computation time. We describe a method that achieves a substantial end-to-end speedup over the best current methods, without loss of accuracy. Our method is a combination of approximating scores by vector quantizing feature windows and a number of speedup techniques including cascade. Our procedure allows speed and accuracy to be traded off in two ways: by choosing the number of Vector Quantization levels, and by choosing to rescore windows or not. Our method can be directly plugged into any recognition system that relies on linear templates. We demonstrate our method to speed up the original Exemplar SVM detector [1] by an order of magnitude and Deformable Part models [2] by two orders of magnitude with no loss of accuracy.

Authors have proposed a series of 'tricks' to reduce
      the memory storage and computational cost of applying linear templates.
      The presented approximation methods achieve speed-up without significant
      loss of accuracy. Moreover, the authors aim for methods that trade off
      between speed-up and accuracy by: choosing number of quantization levels,
      16-bit or 32-bit arithmetic, and finally choosing to re-score windows or
      not. More precisely, the key idea is to replace convolution by a look-up
      table. To make the latter efficient, the authors use k-means as a
      quantization method effectively reducing dimensionality (authors also
      report excellence of k-means over PCA in Fig. 2). Finally, the authors
      introduce different computational models showing achieved speed-up of the
      proposed methods over the state-of-the-art detection algorithms on Pascal
      VOC 2007. Similar conclusion is drawn based on the experiments on exemplar
      SVM. Reference [2] (see below) seems the most related reference.
      It is not cited in the paper an therefore a comparison is absent. A lot of
      the ground is already covered in that paper - which takes away from the
      novelty. [2] proposes an approximation if the filters in a sparse coding
      way. Also nearest neightbor and pca are considered in [2].
      Although the presented idea is interesting and impactful as it
      contributes to the speeding-up the detection task, the work is too
      incremental. A replacement of dot-product in the convolution operator has
      already been introduced in [1]. Considerable speed-up has also been shown
      in [2] with a sparse intermediate representation and a nearest neighbor
      approach where the closest part filters are retrieved. Also a pca
      representation has been considered there as a baseline already. The
      presented method scales up well as the number of classes grows. Also [3]
      shows significant speed-up, however [3] seems to be complementary and fits
      into the 'cascade' section of the paper. Pros: + Authors
      tackle important problem of speeding up the convolution step that is at
      heart in many detection systems, but not only. + The proposed methods
      trades off between accuracy and speed-up + Simplicity of the method
      + Authors have promised publishing source code. + Paper is
      generally well written. Cons: - Incremental nature of the paper.
      - Experiments shown in Table 1, Table 2, and Table 3 are confusing. In
      particular, why the paper reports different total time for the same method
      in different tables? Authors should also be more explicit how they have
      estimated the running time of unaccessible algorithms. - It is not
      clear which approximation 'trick' contributed the most to the speed-up
      (simple look-up table, packed look-up table, 16-bit arithmetic,
      deformation estimates, ...). Authors should make up experiments showing
      the importance of every 'trick'. - Authors should be more explicit
      about implementation details. For instance about the programming language
      being used in the experiments for every method together with the
      'programming tweaks'. [1] 'Fast, Accurate Detection of 100.000
      Object Classes on a Single Machine' by T. Dean et. al. [2] 'Sparselet
      Models for Efficient Multiclass Object Detection' by H. Song et. al.
      [3] 'Branch and Rank: Efficient, Non-linear Object Detection' by A.
      Lehmann et. al.


Optimistic policy iteration and natural actor-critic: A unifying view and a non-optimality result
Approximate dynamic programming approaches to the reinforcement learning problem are often categorized into greedy value function methods and value-based policy gradient methods. As our first main result, we show that an important subset of the latter methodology is, in fact, a limiting special case of a general formulation of the former methodology; optimistic policy iteration encompasses not only most of the greedy value function methods but also natural actor-critic methods, and permits one to directly interpolate between them. The resulting continuum adjusts the strength of the Markov assumption in policy improvement and, as such, can be seen as dual in spirit to the continuum in TD(λ)-style algorithms in policy evaluation. As our second main result, we show for a substantial subset of softgreedy value function approaches that, while having the potential to avoid policy oscillation and policy chattering, this subset can never converge toward an optimal policy, except in a certain pathological case. Consequently, in the context of approximations (either in state estimation or in value function representation), the majority of greedy value function methods seem to be deemed to suffer either from the risk of oscillation/chattering or from the presence of systematic sub-optimality.

The paper "Optimistic policy iteration and natural
      actor-critic: A unifying view and a non-optimality result" investigates
      policy iteration and natural actor critic (NAC) methods for approximate
      solutions of Markov decision processes. The setting considers Boltzmann
      policies, compatible function approximation, and unbiased estimators. The
      authors introduce a forgetting factor in the NAC update (parameterized by
      a weight kappa), and show that this version is identical to an incremental
      ("optimistic") policy iteration (OPI) with learning rate kappa. By making
      the weight kappa go to zero and decreasing at the same time the policy
      temperature, the two algorithms tend towards a common limit corresponding
      to the original NAC (with policies that tend to fully greedy ones). This
      is exploited to derive insight on the two algorithms and how the
      convergence of OPI might be established using NAC; as well as to interpret
      the policy oscillation phenomenon in policy iteration. Then, the authors
      show that when the temperature is not allowed to decrease to 0, the
      unified algorithm cannot converge to any nontrivial optimal solution, for
      any learning rate. Together, the results suggest a good OPI algorithm
      might be obtained by allowing the parameters to tend to the limits
      discussed above (both weight kappa and the temperature going to zero, with
      the overall algorithm thus tending towards NAC). This is illustrated with
      simulations in a toy problem. I think this is an insightful paper
      and has the potential to make a significant impact in the field of
      approximate reinforcement learning and dynamic programming. The way the
      authors look at the problem is original. The survey of existing
      convergence results is very nice. While the paper is well-written
      in general, the detailed technical discussions are often rather dense and
      not easy to read; therefore, some work to alleviate this would be well
      spent. In particular, getting the intuition behind the discussion on the
      relation between standard NAC and the OPI limiting case (e.g. the
      next-to-last paragraph of Section 3) took a bit of rereading in my case.
      I also have some non-major comments on the formalism. - The
      authors should be careful to introduce all notations properly. E.g. J may
      not be explained in the first paragraph of Section 2; the Qbar notation is
      not explained there either, and also not needed (it can be introduced
      later on near the equation where it is defined). Similarly, the natural
      gradient formula in (6) is left unexplained. - Instead of saying "some
      norm", it could be called as usual a p-norm and denoted ||..||_p. -
      The readability is decreased by the use at some places of negative
      notations rather than positives that are easier to follow, and sometimes
      even easier to write. Example: tau does not tend to 0, instead of tau_k
      > 0 \forall k; or r(s, a) does not tend to +-\infty, instead of |r(s,
      a)| <= R_max which is easier to write. The statement "x < eps for
      all eps > 0" actually defines x = 0. - I suggest introducing some
      different notation for the rescaled versions of the parameters w, such as
      w' or v, since this would make the change to the algorithm easier to
      follow. Some other presentation comments: - Perhaps the
      abstract might make explicit the fact that NAC must be modified to make it
      equivalent to OPI - The keyword "reinforcement learning" could be
      added somewhere early in the introduction - "In addition to that the
      class" doesn't seem to be a well-formed sentence - Regarding
      terminology: in "non-optimistic" policy iteration, it may also be
      necessary to have w^_k be "complete" in the sense explained one para
      below, in addition to kappa = 1. - "no modifications to the existing
      algorithms are needed" -- with the exception of the forgetting factor in
      NAC(kapppa), this should be pointed out - The difference between the
      first and second part of Theorem 3 could be pointed out better/in advance,
      I had to reread the passage a couple of times to understand. Also, the
      progression from Theorem 2 to Theorem 3 might be explained in advance to
      prepare the reader. AFTER AUTHOR FEEDBACK: I am grateful to
      the authors for responding to my comments. My point about "no
      modifications" was not to say that we have new algorithms in the paper,
      but rather suggesting to admit that NAC and OPI are not immediately
      equivalent; before that some changes to NAC to make it more general must
      be made. I still think that saying "x < eps for all eps > 0" is
      mathematically equivalent to saying x=0 (in the context of positive x) so
      as the authors say, it would be good to revise the statement. I
      keep my overall (positive) opinion about the paper.


Distributed Submodular Maximization: Identifying Representative Elements in Massive Data
Many large-scale machine learning problems (such as clustering, non-parametric learning, kernel machines, etc.) require selecting, out of a massive data set, a manageable yet representative subset. Such problems can often be reduced to maximizing a submodular set function subject to cardinality constraints. Classical approaches require centralized access to the full data set; but for truly large-scale problems, rendering the data centrally is often impractical. In this paper, we consider the problem of submodular function maximization in a distributed fashion. We develop a simple, two-stage protocol GREEDI, that is easily implemented using MapReduce style computations. We theoretically analyze our approach, and show, that under certain natural conditions, performance close to the (impractical) centralized approach can be achieved. In our extensive experiments, we demonstrate the effectiveness of our approach on several applications, including sparse Gaussian process inference and exemplar-based clustering, on tens of millions of data points using Hadoop.

This paper provides a novel algorithm for large scale
      submodular function maximization. The main contribution is a distributed
      algorithm, using MapReduce, where the ground set is partitioned and each
      machine operates on a subsets of the ground set. These solutions are
      merged, to find an approximate subset. They provide approximation
      guarantees for their approach and also show that it is tight. They also
      investigate the performance of their algorithm under specific assumptions
      on the datasets or the function. I think overall the authors try
      to solve a very challenging problem, which could have a lot of utility in
      large scale real world applications. I also feel that the experimental
      validation is thorough and extensive. I also appreciate that this problem
      is very challenging and it would really hard to obtain satisfactory
      performance guarantees without additional assumptions. I was
      somewhat disappointed with the theoretical analysis. It seems that the
      guarantees are pretty weak; I acknowledge that the worst case analysis
      shows that the algorithm is tight. However, this is expected particularly
      for a very bad choice of partitions V1, V2, ... I was expecting some kind
      of dependence on the choice of the the distributions, or even a heuristic
      of what choices of distributions might work. The main guarantee (Theorem
      5.1) seems almost a linear factor in m and k, which is somewhat
      discouraging. As the proof technique reveals, and even otherwise, it is
      easy to see that a simple modular upper bound based approximation gives a
      factor k approximation. Given this, it is not immediately clear how GREEDI
      performs theoretically w.r.t a simple modular upper bound particularly for
      large m (which is of practical relevance), though I am sure the modular
      upper bound based algorithm will perform very badly in practice.
      There is one extremely important aspect, however, which is very
      loosely described in the paper. I think this should be clarified much
      better. A number of practical submodular functions are graph based
      submodular functions (this includes, for example, the graph cut like
      function and the exemplar based clustering objective from 3.1 -- A small
      clarification here is this objective is essentially a form of
      facility-location objective with a similarity instead of a distance
      function) In these cases, evaluating the function requires a sum over all
      the elements in V, even though the set under which the function needs to
      be evaluated is considerably smaller. In these cases, it is not clear how
      to evaluate the function. More specifically, a main motivation of this
      approach (lines 122-127) is that the datasets are extremely large and
      would possibly not fit within any single machine. However each of the
      individual machines would still need to compute the outer sum over V in a
      graph based objective (say the facility location objective). Under these
      circumstances, it is not clear how to run the algorithm. Possibly, there
      could be a time reduction due to the distribution, but it is not clear how
      this will help in terms of memory. One solution could be to evaluate the
      function just on the subsets Vi (i.e the outer sum may only be over the
      Vi's), but this would then change the submodular function and the
      guarantees would no longer hold. Overall, I think this is a major issue
      which should be clarified in the rebuttal. Some minor suggestions
      are that the proof of theorem 4.1 (specifically the tight instance) is
      very hard to grasp. Maybe that can be better explained? Also, I think over
      smaller datasets the actual greedy algorithm should be compared to GREEDI
      (just to see how the performance is with respect to the best serial
      algorithm). I would also love to see a timing analysis of the algorithms
      and memory requirements etc. in the experimental results.


Synthesizing Robust Plans under Incomplete Domain Models
Most current planners assume complete domain models and focus on generating correct plans. Unfortunately, domain modeling is a laborious and error-prone task, thus real world agents have to plan with incomplete domain models. While domain experts cannot guarantee completeness, often they are able to circumscribe the incompleteness of the model by providing annotations as to which parts of the domain model may be incomplete. In such cases, the goal should be to synthesize plans that are robust with respect to any known incompleteness of the domain. In this paper, we first introduce annotations expressing the knowledge of the domain incompleteness and formalize the notion of plan robustness with respect to an incomplete domain model. We then show an approach to compiling the problem of finding robust plans to the conformant probabilistic planning problem, and present experimental results with Probabilistic-FF planner.

This paper considers the problem of planning under
      incomplete domain models. The authors formalize domain incompleteness,
      discuss how to generate robust plans using conformant probabilistic
      planning, and compare robustness to solving time and plan quality in two
      test domains. Clarity: The paper is very well written and the
      authors do a great job at explaining all of the details both formally and
      informally. Although I am mostly unfamiliar with planning research, I was
      able to follow along fairly well and understand the contributions.
      Organization is good and the example at the top of page 4 is great, but I
      believe that on line 177, the "fifth candidate model" described is
      actually the second candidate model given by the table in Figure 2.
      Quality: What I like most about this paper is that the
      incompleteness formulation is very intuitive and easy to understand. A
      practioner should be able to read the paper and integrate the ideas into
      their own planning problem without too much worry. Theorems of
      completeness and correctness are also stated. However, perhaps the biggest
      drawback is the lack of proofs, particularly for Theorem 1. While I agree
      with the authors that there is no space in the main body of the paper to
      include proofs, the authors really should have included them as
      supplementary material. Originality: While there are some
      similarities here to other work, the authors do discuss this in the
      related work section and I believe that the proposed formulation for
      incompleteness is adequately original and worthy of publication. The
      authors clearly state how their work differs from previous contributions.
      Significance: I believe the contributions of this paper are
      important to the NIPS community. Robust planning is an under-studied
      problem and the authors address this difficult challenge in a intuitive
      manner. With that said, I am concerned with the applicability of their
      formulation beyond toy problems. Their formulation results in an
      exponential blow up in the size of the problem. In addition, experiments
      are only performed in toy domains that appear to have been created by the
      authors specifically for this paper. I presume that with the lack of
      research into planning in incomplete models, there are no standard domains
      to run experiments in. However, I am concerned that the approach will be
      completely infeasible in large domains that arise in the wild. In very
      large domains, can we ever hope to guarantee a high level of robustness,
      or are we simply going to be satisfied in just finding a valid plan? How
      badly do planners such as PFF suffer from such an exponential blowup
      resulting from the set of all possible complete domain models?
      Typos: - Line 113: "with respects to..." - Line 135:
      "Makov Logic Networks" - Line 430 or 431: "constrasting"
      COMMENTS AFTER AUTHOR RESPONSE: I am glad to hear that
      proofs will be appearing in a later document. If the Ph.D. proposal will
      not be accessible soon, I suggest attaching the proofs as supplementary
      material to the camera-ready paper if this paper is accepted. While I am
      still somewhat concerned about the applicability beyond toy problems, this
      is a good first step and future work with heuristics could potentially
      improve things.


Trading Computation for Communication: Distributed Stochastic Dual Coordinate Ascent
We present and study a distributed optimization algorithm by employing a stochastic dual coordinate ascent method. Stochastic dual coordinate ascent methods enjoy strong theoretical guarantees and often have better performances than stochastic gradient descent methods in optimizing regularized loss minimization problems. It still lacks of efforts in studying them in a distributed framework. We make a progress along the line by presenting a distributed stochastic dual coordinate ascent algorithm in a star network, with an analysis of the tradeoff between computation and communication. We verify our analysis by experiments on real data sets. Moreover, we compare the proposed algorithm with distributed stochastic gradient descent methods and distributed alternating direction methods of multipliers for optimizing SVMs in the same distributed framework, and observe competitive performances.

This paper combines a recent advance in stochastic
      dual coordinate ascent with the standard approach to parallelize training
      which is to assign a mini-batch of example to each process and average the
      resulting gradients. While this combination is novel, its most original
      contribution is the trade-off bounds between communication and
      computation. This is likely to be an influential work, introducing a new
      methodology. The main limitation to its significance are poorly designed
      experiments. In particular, the authors do not discuss the impact of the
      input dimension in the bounds, in particular in the case where the data is
      sparse, which is nearly always the case for big data. 2 major
      limitations appear in the significance of this trade-off: - In the
      discussion on page 5, the authors omit the number of dimensions d. If the
      data is dense, this could be treated as a constant factor as both the
      communication and computation cost are linear in (I assume the main loop
      in computation is W.x which is in O(d)). However the situation becomes
      very different when the data is sparse: the computation cost for a single
      dot product scales as the number of non-zero features we denote d’. Thus
      the total computation cost per iteration is O(m*d’). In the worst case,
      where no features from the m examples overlap, the communication cost will
      also be O(m*d’). We then lose this m ratio between them that was critical
      to the analysis on page 5. Note that in the KDD cup data, one feature out
      of 1 million is non zero. - Experiments only count the number of
      iterations, not the total running time that includes the communication
      latency. On top of that, the time per iteration depends on the parameters.
      So experiments reported on figure 3 that show that increasing m always
      reduce the number of iteration are very misleading, as each single
      iteration scales as O(m*d) in computation cost, and O(d) in communication
      cost. A curve showing the total training time, with values for m and K for
      which this time is minimized would be much more convincing. In
      summary, the experiments are so far of little significance, as they only
      show a reduction of the number of iterations as a function of m and K,
      which is a trivial observation that does not need theorem 1. Note also
      that the authors only say they use openMPI, which does say anything about
      the architecture: - Cluster: how many nodes, how many core per node?
      - Single multicore? Shared memory would mean no communication costs.
      The most interesting observation from theorem 1 is the presence of
      an “effective region” for K and m. But the only thing the experiment show
      is that decreasing lambda gives more room to choose m and K. A effective
      upper threshold of the mK product, supported by actual training times,
      would be a very significant result. Detailed comments: Tens of
      hundreds of CPU cores: do you mean thousands of cores of tens of clusters
      with hundreds of cores? If communication costs are involved, the target
      should be clusters, not multicores with shared memory. Proof of
      theorem 1: while the proof is correct and trivially obtained from the
      bound E(epsilon_D^t), the rest of the proof in not needed. In particular,
      the last sentence is confusing, as there is no T0. “We can complete
      the proof of Theorem 1 for the smooth loss function by plugging the values
      of T and T0.” There seems to be several lines that have nothing to do
      with the proof?? Why do the authors repeat twice DisDCA is
      parameter free? The choice of lambda is critical. P3, l128: I
      thought alpha was the associated dual of w, not x, but this may be
      terminology. Figure 3: varing->varying. Plots in Figure 3 are
      very hard to read.


The Power of Asymmetry in Binary Hashing
When approximating binary similarity using the hamming distance between short binary hashes, we show that even if the similarity is symmetric, we can have shorter and more accurate hashes by using two distinct code maps. I.e. by approximating the similarity between x and x as the hamming distance between f (x) and g(x ), for two distinct binary codes f, g, rather than as the hamming distance between f (x) and f (x ).

This paper suggested constructing different binary
      data representation using different hash functions in the asymmetric theme
      to possibly reduce the encoding length. Encoding data as compactly as
      possible is no doubt important to theory and practice of approximate
      nearest neighbor search. Detailed comments: 1. A major
      concern of the asymmetric design is the consistency of the results. In
      Eqns. (2) (3) and Theorem 1, there is no constraint enforcing Y's
      symmetry, although there was a sentence vaguely mentioning this issue
      below Eqn. (2). Therefore, by this design it is very possible to obtain
      inconsistent results such as < u(x), v(y) > ~= < u(y), v(x) >,
      which means that the learned hash functions CANNOT support a distance
      metric. And in the actual experiments, the authors used whether a certain
      point x is in the database to determine which hash function to use, which
      is ad-hoc and lacks any ground support. Hence, this asymmetric design on
      this hand makes the theoretical model inconsistent and broken, and on the
      other hand weakens the experimental evidence, considering that too many
      parameters were used. 2. Some recent important state-of-the-art
      hashing methods are missing, e.g., Yunchao Gong, S. Lazebnik, A.
      Gordo, and F. Perronnin. Iterative Quantization: A Procrustean Approach to
      Learning Binary Codes for Large-scale Image Retrieval. TPAMI 2012. W.
      Liu, J. Wang, S. Kumar, and S.-F. Chang. Hashing with Graphs. ICML 2011.
      W. Liu, J. Wang, R. Ji, Y.-G. Jiang, and S.-F. Chang. Supervised
      Hashing with Kernels. CVPR 2012. The experiments only compared
      with some baselines known to be problematic in the short-bit setting such
      as LSH, thus are not too convincing. In addition, kernel-based supervised
      hashing (the CVPR12 paper mentioned above) has shown superior performance
      over BRE and MLH. Why not cited and compared? The idea of manipulating
      code inner products was proposed by this work. The authors should give
      clear credit to the CVPR12 paper when using code inner products in their
      method. The authors also need to clarify the relationship and difference
      with the prior asymmetric hashing work [3] (cite the PAMI version).
      3. L084-085: the value for < u, v > seems incorrect.
      4. L142-152: aka, second part of proof to Theorem 1; Line 142, it
      should be ks(S) >= n/2; Line 143, why \theta is in that range? Can the
      authors explain? Line 146-150: how to move to the upper bound, in
      particular for the latter two terms? The current derivation is doubtful.
      5. Part 4-5: There is an important turn from approximation to
      generalization, which makes the main thrust of the paper. My major
      concerns are: 1) Even if we know that compact codes are possible, why
      is a linear threshold function one appropriate choice for forming hash
      functions? 2) Given that in advance we do not know the data space
      much, how to choose k properly in Eqn. (3)? This could be a hard decision
      in practice. I think that a more pertinent version could fix an
      approximation tolerance, and minimize k instead. The presentation in Part
      4 is loose. Overall, I think that the asymmetric
      similarity/distance approximation problem and some results in the paper
      are interesting. When treating practical problems, the proposed methods
      (Part 4-5) stay at the heuristic level and do not correspond well to the
      former claim. The bottom line is that the heuristics are justified by
      sufficient experiments with positive results. In addition, I think that
      the authors should put more efforts on polishing their paper, in view of
      numerous statement problems and grammatical
      errors.


Efficient Exploration and Value Function Generalization in Deterministic Systems
We consider the problem of reinforcement learning over episodes of a finitehorizon deterministic system and as a solution propose optimistic constraint propagation (OCP), an algorithm designed to synthesize efficient exploration and value function generalization. We establish that when the true value function Q ⇤ lies within the hypothesis class Q, OCP selects optimal actions over all but at most dim E [Q] episodes, where dim E denotes the eluder dimension. We establish further efficiency and asymptotic performance guarantees that apply even if Q ⇤ does not lie in Q, for the special case where Q is the span of pre-specified indicator functions over disjoint sets.

*Summary* This paper proposes an algorithm for
      episodic reinforcement learning (RL) in deterministic environments when
      the learner has a set of possible Q-functions at his disposal. Sample
      complexity and regret bounds are given that depend on the so-called
      "eluder dimension" of the function set and are independent of other
      parameters. The agnostic case is considered as well. *Evaluation*
      There is a lot to like about this paper. It is original, well-written,
      and has some nice-looking results. On the other hand, the setting the
      paper considers is very simple (episodic learning, deterministic
      environment), and it is questionable whether the results can be
      generalized to more interesting settings. Further, there are also a few
      things that I found improvable: - With the exception of the
      mentioned linear systems with quadratic cost, I found it not very natural
      to have a set of possible Q-functions at the disposal of the learner.
      Thus, the results for the "tabula rasa" case and agnostic learning seem to
      be the more important ones. However, for the tabula rasa case, it seems
      one does not get anything better than when doing pure exploration until
      all states have been visited. (I think this deserves to be mentioned.)
      Concerning agnostic learning, in Theorem 4 there is an unresolved
      dependence between K and rho, so that it is not clear what can be really
      accomplished in this case. - It would be good to include at least
      some proof sketch for the main theorem to give a more detailed idea why
      the algorithm works. There is enough space for this, even more when the
      special hypothesis classes are dealt with more concisely. In particular,
      the lines 282-295 are simply redundant. - Finally, some related
      work should be mentioned w.r.t. * RL with a set of possible models:
      See recent work of Marcus Hutter and co-authors, and the paper of Maillard
      et al at NIPS 2011 and two subsequent papers by the same author and
      various co-authors at ICML and AISTATS this year. * Deterministic RL:
      Bernstein & Shimkin, COLT 2008. *Minor comments* - In
      l.47, I found it not clear what is precisely meant by "model-based
      algorithms". - In l.82, according to the results near-optimality is
      not so clear, cf. the remark on the dependence of K and rho in Theorem 4
      above. The sentence as stated gives the wrong impression that one gets
      better results with fewer basis functions. - In l.124, I did not see
      why R^(j) is strictly smaller than V*_0(x_j,0). It seems, it should be
      \leq instead. - In lines 125 and 139 L is used with different meaning.
      It would be better to use two different letters instead. - In l.144
      "more recent experience" assumes that there is some temporal order in the
      constraints. This should be made clear before. - In l.168 in Algorithm
      2, it should be made clear that the set C comes from Algorithm 1. - In
      l.177, \cap should be used for set intersection. - In l.209,
      "transition" is used as a verb, which I think is not correct. - In
      l.229, "Q to denote" should be "Q denote". - In l.339, it was not
      clear to me why the condition in Proposition 1 means that the constraints
      are "redundant" (and not just consistent). --- Remarks to
      author's response: - Pure exploration: What I meant is that since
      the system is deterministic, one could first explore till the whole system
      is known and then exploit. - Payoff between K and rho: If one
      chooses a finer discretization (higher K) then the error rho will usually
      be smaller. Hence there seems to be a payoff between K and rho in Theorem
      4, and it is not clear what one can expect for an optimal
      discretization.


Online Robust PCA via Stochastic Optimization
Robust PCA methods are typically based on batch optimization and have to load all the samples into memory during optimization. This prevents them from efficiently processing big data. In this paper, we develop an Online Robust PCA (OR-PCA) that processes one sample per time instance and hence its memory cost is independent of the number of samples, significantly enhancing the computation and storage efficiency. The proposed OR-PCA is based on stochastic optimization of an equivalent reformulation of the batch RPCA. Indeed, we show that OR-PCA provides a sequence of subspace estimations converging to the optimum of its batch counterpart and hence is provably robust to sparse corruption. Moreover, OR-PCA can naturally be applied for tracking dynamic subspace. Comprehensive simulations on subspace recovering and tracking demonstrate the robustness and efficiency advantages of the OR-PCA over online PCA and batch RPCA methods.

The paper presents a iterative algorithm to a robust
      principal component matrix factorization. The data is modeled as a sum of
      a low rank matrix approximation and a sparse noise matrix. Constraining
      the norms of the row and column factors of the former part of the sum
      allows to implement the nuclear norm minimization of the batch robust pca
      algorithm in an online stochastic gradient descent fashion. The
      approach taken by the authors resembles very much the approach taken by
      Marial et al 2009 (ICML) and 2010 (JMLR), only that the objective is
      slightly different (Robust PCA was not dealt with in the JMLR version).
      One difference between the JMLR and the ICML version is that a standard
      stochastic gradient version of the objective function did not perform as
      well as the proposed online dictionary learning approach (in which the
      statistics of the data are accumulated in the matrices A and B) in the
      ICML version, but in the JMLR version, a standard stochastic gradient
      implementation with appropriately chosen learning rate seemed to perform
      ok. As storing and updating the matrices A and B can be costly, it would
      be interesting to see whether a standard first order or quasi-newton
      stochastic gradient approach for optimizing eq. (2) performs as well as
      the solution proposed in algorithms 1 and 2. Quality The
      manuscript is very well written. Clarity All parts of the
      manuscript appear to be very accessible. Problem formulation and
      motivation for the method are clear, objective function and the algorithm
      are comprehensible. Simulations are sensible and well done.
      Originality The approach proposed by the authors seems very
      similar to the one proposed by Marial et al in their 2009 and 2010 paper.
      The proofs appear to be based on the same techniques, in large parts.
      Convergence to a global optimum instead of a local optimum is new.
      Using a low-norm factorization instead of a batch version of nuclear
      norm minimization is fairly well established, see e.g. Rennie and Srebro,
      'Fast Maximum Margin Matrix Factorization for Collaborative Prediction',
      ICML, 2005. Significance Iterative methods are becoming more
      relevant with growing data set sizes. It is important to have scalable
      versions of simple methods such as PCA and robust versions thereof for
      large scale data sets. Although the ingredients are well known, to the
      best of my knowledge, the proposed method is novel and potentially
      relevant to a large part of the research community.


Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions
We investigate three related and important problems connected to machine learning: approximating a submodular function everywhere, learning a submodular function (in a PAC-like setting [28]), and constrained minimization of submodular functions. We show that the complexity of all three problems depends on the "curvature" of the submodular function, and provide lower and upper bounds that refine and improve previous results [2,6,8,27]. Our proof techniques are fairly generic. We either use a black-box transformation of the function (for approximation and learning), or a transformation of algorithms to use an appropriate surrogate function (for minimization). Curiously, curvature has been known to influence approximations for submodular maximization [3,29], but its effect on minimization, approximation and learning has hitherto been open. We complete this picture, and also support our theoretical claims by empirical results.

Curvature and optimal algorithms for learning and
      minimizing submodular functions This paper explores the impact of
      a notion of function curvature on the approximability of problems
      involving submodular functions. While it's not the first work to use this
      notion of curvature to improve bounds for submodular functions (for
      submodular maximization), this paper covers more settings. In particular,
      the problems covered are: - efficiently finding (and representing) a
      function f that approximates the submodular function - PAC-style
      learning - constrained function minimization The paper makes
      the case that many practically interesting functions have small curvature
      and thus admit better approximations that previous bounds would indicate.
      The main approach to solving the approximation problem is to
      separately approximate the cured part of the underlying function. The
      approach to the second problem is to use the Balcan-Harvey PAC-style
      learning algorithm and apply the approximation result. The approach to the
      third problem is less black-box than the previous two, but gives a means
      of choosing an approximation of the original function to analyze; then
      they can apply the previously mentioned results to understand how good an
      approximation this gives. Finally, there is a brief experimental
      section to demonstrate that for functions with small curvature (for the
      "worst case" given that curvature), the empirical constrained minimization
      closely tracks the approximation predicted. These are important
      problems, with lots of different applications, and it's nice to see some
      general tools for understanding what affects (in)approximability.
      It would have been helpful to have more discussion of the
      distinction between the notion of curvature used here and the
      submodularity ratio (is there a way to compare their usefulness in
      improving approximation ratios?) It is a bit difficult to digest
      the new results, since the paper doesn't provide a succinct comparison
      with existing worst-case bounds (for functions with maximum curvature). I
      realize of course that there are a lot of other comparisons with previous
      work one could make (for other specific classes of functions), but this
      particular comparison seems that it would help the reader understand the
      impact of taking curvature into account.


(Nearly) Optimal Algorithms for Private Online Learning in Full-information and Bandit Settings
We give differentially private algorithms for a large class of online learning algorithms, in both the full information and bandit settings. Our algorithms aim to minimize a convex loss function which is a sum of smaller convex loss terms, one for each data point. To design our algorithms, we modify the popular mirror descent approach, or rather a variant called follow the approximate leader. The technique leads to the first nonprivate algorithms for private online learning in the bandit setting. In the full information setting, our algorithms improve over the regret bounds of previous work (due to Dwork, Naor, Pitassi and Rothblum (2010) and Jain, Kothari and Thakurta (2012)). In many cases, our algorithms (in both settings) match the dependence on the input length, T , of the optimal nonprivate regret bounds up to logarithmic factors in T . Our algorithms require logarithmic space and update time.

This work provides new algorithms for differentially
      private online learning in basically all settings where results are known
      in the non-private case. The results very closely parallel (with some
      additional small factors and dependences) the best non-private bounds.
      There has been significant recent interest in differentially private
      learning, and this paper gives a much more comprehensive and compelling
      suite of results than previous work. The approach is actually
      quite simple at its heart---do follow the approximate leader using a
      private (noisy) history that you maintain using standard algorithms for
      maintaining a differentially private counter. The natural suitability of
      follow the leader algorithms for privacy has been observed before, but the
      treatment here and the actual bounds obtained are very nice. I'd
      prefer that you describe this work as giving a class of private learning
      algorithms, or a technique for constructing such algorithms, rather than a
      "general technique for making online algorithms differentially private"
      (since it's not the case that you can take an arbitrary online algorithm
      and make it private using this technique---right?). The practice
      of citing references without listing authors made this paper hard to
      read---I was constantly flipping to the bibliography, since in many cases
      knowing the reference was important to understanding the ideas (e.g. in
      cases where the text did not make sufficiently clear whether the
      referenced work was in the private or non-private case). In Table
      1, it's probably worth reminding the reader that the delta in the first
      column comes from (eps, delta)-DP. Why aren't there citations for
      the non-private results in Table 2? And also, the caption says it's the
      full-information setting, but the results are clearly for the bandit
      setting. And it seems your results for the adaptive case should have a
      T^3/4, not T^2/3, no? Do you think the explicit dependences you
      see on the dimensionality are necessary when preserving privacy? Do you
      think there's a provable (small) gap between what is achievable in the
      dependence on T between private and non-private algorithms (accounting for
      your different poly-log factors)? The second to last paragraph on
      p.6 has strange redundancy.


Bayesian Estimation of Latently-grouped Parameters in Undirected Graphical Models
In large-scale applications of undirected graphical models, such as social networks and biological networks, similar patterns occur frequently and give rise to similar parameters. In this situation, it is beneficial to group the parameters for more efficient learning. We show that even when the grouping is unknown, we can infer these parameter groups during learning via a Bayesian approach. We impose a Dirichlet process prior on the parameters. Posterior inference usually involves calculating intractable terms, and we propose two approximation algorithms, namely a Metropolis-Hastings algorithm with auxiliary variables and a Gibbs sampling algorithm with "stripped" Beta approximation (Gibbs SBA). Simulations show that both algorithms outperform conventional maximum likelihood estimation (MLE). Gibbs SBA's performance is close to Gibbs sampling with exact likelihood calculation. Models learned with Gibbs SBA also generalize better than the models learned by MLE on real-world Senate voting data.

This paper proposes a Bayesian approach to learning
      binary pairwise MRFs with latently tied parameters. A DP prior is used to
      model the grouping of parameters, and two approximate inference methods
      are compared with MLE. Doing Bayesian inference on MRFs is
      difficult, known as doubly intractable. The first algorithm, M-H with
      auxiliary variables, is based on the paper of Moller et.al. 2006. It is
      technically sound (except for the sample generated by Gibbs sampling), but
      could be slow, hard to mix, and therefore not feasible for real
      medium-sized applications. The second algorithm, Gibbs_SBA, is novel to my
      knowledge and seems to be a good candidate for approximate inference. The
      experiments show that the approximate methods are much faster than the
      exact method and obtain similar performance on small problems. It would be
      a meaningful contribution to Bayesian approach to MRFs. My concern
      is on the usage of the Stripped Beta Approximation. The derivation of the
      method is lack of clarity. It is based on two approximations: (1) line
      215: the distribution of all variables other than u and v is independent
      on the value of \theta_i; (2) line 228: approximate the contribution of
      \theta_i in (5) with a Beta distribution. I don’t understand how the
      second approximation is adopted. Apparently Equation 5 is a Beta function
      of \lambda, which is a complicated function of \theta. The authors don’t
      provide sufficient explanation for the beta approximation or the usage of
      the MLE of \theta in the Beta parameters. While the experiments show the
      overall performance of Gibbs_SBA is similar to the exact method in terms
      of the two first metrics, a figure or a paragraph of discussion on the
      accuracy of the approximation on one single step of sampling would be very
      useful. In the experiment of the real problem, the output of the
      parameters does not show any hard grouping because the Bayesian approach
      assumes a posterior distribution. So is the latent grouping prior useful
      only as regularization to prevent over-fitting, or you can make more
      meanings out of it? The improvement on the pseudo-likelihood is not very
      significant. I guess that is because the sampling procedure is initialized
      on the result of PCD. My last concern is about the speed. The
      computation burden is known as a serious problem for Bayesian inference.
      The authors compared the speed of the two approximate methods with the
      exact method. But how are they compared with the MLE algorithm based on
      PCD? It seems that for the fastest method for Bayesian inference,
      Gibbs_SBA, one has to run MLE (PCD) for every parameter on every
      iteration. Would it be applicable for any large problem in the real world?
      Some minor questions and typos: 1. The binary pairwise MRF in
      the paper has pairwise potentials only. Would Gibbs_SBA still applies for
      an MRF with unary bias? 2. For M-H algorithm in section 3.1.1, do you
      also need to integrate over \theta when a new group is created? 3.
      Line 126: Section 3.2.2 -> 3.1.2


PAC-Bayes-Empirical-Bernstein Inequality
We present a PAC-Bayes-Empirical-Bernstein inequality. The inequality is based on a combination of the PAC-Bayesian bounding technique with an Empirical Bernstein bound. We show that when the empirical variance is significantly smaller than the empirical loss the PAC-Bayes-Empirical-Bernstein inequality is significantly tighter than the PAC-Bayes-kl inequality of Seeger (2002) and otherwise it is comparable. Our theoretical analysis is confirmed empirically on a synthetic example and several UCI datasets. The PAC-Bayes-Empirical-Bernstein inequality is an interesting example of an application of the PAC-Bayesian bounding technique to self-bounding functions.

This paper derives a new empircal PAC Bayesian bound
      by combining an existing (non-empircal) PAC Bayesian Berstein bound (i.e.,
      involving the true variance of the loss values) with a PAC Bayesian
      analysis of the concentration of the empirical variance around its true
      value. This new bound has the advantage of being tighter when the
      empirical variance is small compared to the empirical loss. Experiments on
      real and empirical data with simple models compare the new bound with the
      usual empirical PAC Bayesian bound confirming the advantage. While
      this is a well written paper that make a novel contribution to the PAC
      Bayesian literature it is a incremental and only marginally significant
      one. As the authors' claim, the newer bound is only really useful for
      analysing cases when there is a large mismatch between the model class and
      the data generating distribution. Although the idea of applying a
      PAC Bayesian bound to the difference between empirical and true variances
      is original and addresses a shortcoming in the existing work, it appears
      that the steps from there were fairly routine, leaning heavily on
      techniques developed in other recent papers. That said, the definitions
      and proofs all seem to be correct and are logically organised.
      Minor suggestions: 1. Put an article ("a" or "the") before
      all singular uses of "PAC-Bayes-Empirical-Berstein inequality" and the
      like. For example, the opening sentence of the abstract should be "We
      present a PAC-Bayes-Empirical-Bernstein inequality..." 2. You
      should define $KL(\rho\|\pi)$ or otherwise make clear in which direction
      the relative entropy is defined.


Large Scale Distributed Sparse Precision Estimation
We consider the problem of sparse precision matrix estimation in high dimensions using the CLIME estimator, which has several desirable theoretical properties. We present an inexact alternating direction method of multiplier (ADMM) algorithm for CLIME, and establish rates of convergence for both the objective and optimality conditions. Further, we develop a large scale distributed framework for the computations, which scales to millions of dimensions and trillions of parameters, using hundreds of cores. The proposed framework solves CLIME in columnblocks and only involves elementwise operations and parallel matrix multiplications. We evaluate our algorithm on both shared-memory and distributed-memory architectures, which can use block cyclic distribution of data and parameters to achieve load balance and improve the efficiency in the use of memory hierarchies. Experimental results show that our algorithm is substantially more scalable than state-of-the-art methods and scales almost linearly with the number of cores.

Summary of the paper This paper is
      concerned with the resolution of covariance selection problems in a very
      large scale setup (up to millions of features and trillions of
      parameters). The estimator considered is the CLIME, which is known to
      provide an estimator of the precision matrix of a Gaussian vector that can
      be solved column by column, thus more easily amenable to distributed
      computations. Theoretical analysis is also claimed to be easier than for
      the Graphical-Lasso. The authors developed a new algorithm to fit the
      CLIME by solving the problem block wise rather than column wise, couple to
      an inexact direction method of multipliers. Theoretical results on the
      convergence rate of this algorithm are stated, in term of both objective
      function and distance to optimality. Special attention is paid for special
      matrix structures (sparse and low rank) along the computation to reduce
      the computational burden. The algorithm is implemented within a scalable
      parallel computation framework for both shared and distributed memory
      systems. In the numerical studies, comparison are made in term of runtime
      with its direct competitors on both synthetic and real data. A detailed
      study concerning this new algorithm is also provided to evaluate the
      speedup regarding the block sizes and the number of cores on both shared
      memory and distributed memory systems. Comments This paper
      is well written and very clear. Novelty and contributions brought by the
      method are clearly stated in the introduction and connexion to existing
      works clearly established, with sounded bibliographical references. The
      algorithm is nicely introduced with a good balance between technical
      points of numerical analysis/algebra and theoretical guarantees for
      convergence rates. The numerical analysis is appropriate and the
      implementation of the algorithm shows very impressive performances.
      My only concern is the availability of such a method: if the
      source code (mostly based upon open source libraries) is not made
      available to the community, their is no point in such a work.


Adaptivity to Local Smoothness and Dimension in Kernel Regression
We present the first result for kernel regression where the procedure adapts locally at a point x to both the unknown local dimension of the metric space X and the unknown Hölder-continuity of the regression function at x. The result holds with high probability simultaneously at all points x in a general metric space X of unknown structure.

This paper considers the noisy regression setting with
      n data (X_i,Y_i) such that \E(Y | X=x) =f(x). The underlying function f is
      locally Holder smooth with associated coefficient \alpha_x. The
      distribution \mu underlying X is locally d_x dimensional. The paper
      proposes a method to adapt locally the bandwidth in kernel regression to
      the unknown \alpha_x and d_x. The authors propose an adaptive estimate
      \hat f of f that is such that, if n large enough, with probability 1 -
      \delta, for any x |\hat f(x) - f| < C
      (\log(n/\delta)/n)^{\alpha_x/(2\alpha_x + d)}. They assume however
      that: (i) An upper bound on \sup_x d_x is available (ii) For each
      x, the dimension d_x is "attained" provided that we consider a small
      enough neighbourhood This paper is a combination of results in [4]
      and [5] (see bibliography in the paper), i.e. local Lepski's method, and
      [9], i.e. adaptation to the intrinsic dimension: the results are
      interesting but not surprising in view of these works. I did not read the
      proofs in full details, but what I checked is correct and the results are
      plausible. I have some questions: - Results concerning adaptive
      (to the smoothness) methods such as Lepski's method are in general proved
      uniformly on a class of function (The alpha Holder functions in your
      case). Could you do the same? - What about confidence intervals for
      your adaptive estimate? - You assume (ii) for the intrinsic dimension.
      Do you really need this? It is not necessary in Lepski's method to assume
      that the smoothness \alpha is available: adaptive methods will tend to
      over-smooth, but this is not a problem. Could you have something similar
      with the intrinsic dimension? To me, the main problem with this
      paper is the writing, which is not precise and not intuitive. Here is an
      un-exhaustive list of the issues I spotted: -p1l43: the minimax
      optimal oracle rate is indeed n^{-2\alpha/(2\alpha+d)}, but you should
      definitely mention the minimax optimal adaptive rate, which is more
      relevant to your problem, i.e. \min_{\hat f} \max_{\alpha} \max_{f
      \alpha Holder} \E \|\hat f - f\|_{\infty}
      (n/\log(n))^{2\alpha/(2\alpha+d)} > C where C is a constant. In
      other words, the minimax optimal oracle rate is n^{-2\alpha/(2\alpha+d)},
      but the minimax optimal rate is (n/\log(n))^{2\alpha/(2\alpha+d)}
      (inevitable additional (\log(n))^{2\alpha/(2\alpha+d)} to pay for the
      non-knowledge of \alpha). Also, it is the rate you obtain in your result.
      -p2l64 and 76: Are \rho and \|.\| the same connected/ So \rho does
      actually define a norm? -p2l90: What is B(x,r)? I assumed it is the
      ball of centre x and radius r but you should state it. -p2l98: What is
      w.l.o.g.? Without loss of generality? -p3l150: the constant N is very
      important. Could you give it explicitly? -p3l161 (main equation): You
      have C_{n,\delta} = C\log(n/\delta): there is a dependency in
      \log(n)^{2\alpha/(2\alpha+d)} in your bound. Please make it explicit. This
      is anyway matching the lower (non-oracle) bound. -p5l228: Please
      provide a proof for Lemma 2: shouldn't an additional
      log(1/(\epsilon\delta)) appear in the bound due to union bound on
      S_{\epsilon}? -p6l286: citation for McDiarmid inequ please
      -p7l153: The proof of Thm 1 is to me correct but redacted in a non
      precise way. Instead of saying "Assume that the statement of Lemma ..." is
      verified, it would be more clear to precise exactly on which event the
      Theorem holds. -p8l395: Also the proof seems un-necessary complicated:
      you could maybe skip the definition of \bar h and prove that: (1) if
      c\tilde h < \hat h, then |\hat f_{n, \hat h}(x) - f(x)|< \hat
      \sigma_{\tilde h} < C \tilde R(\tilde h) and (2) the probability that
      c\tilde h > \hat h is very small since it is not likely that |\hat
      f_{n, c\tilde h} - \hat f_{n, \hat h}| > \hat \sigma_{\hat h} for such
      a \hat h. Unify the notation log and ln. Is your log a log in base
      2? And also it would be great if you could extend the discussion
      on your results, and the intuitive explanations. The paper is to my mind
      quite hard to read, because very technical (sometimes without necessity).
      Also you should add some references on Lepski's method: the upper and
      lower bounds for this method were not provided in [4], [5] but
      before.


Speeding up Permutation Testing in Neuroimaging *
Multiple hypothesis testing is a significant problem in nearly all neuroimaging studies. In order to correct for this phenomena, we require a reliable estimate of the Family-Wise Error Rate (FWER). The well known Bonferroni correction method, while simple to implement, is quite conservative, and can substantially under-power a study because it ignores dependencies between test statistics. Permutation testing, on the other hand, is an exact, non-parametric method of estimating the FWER for a given α-threshold, but for acceptably low thresholds the computational burden can be prohibitive. In this paper, we show that permutation testing in fact amounts to populating the columns of a very large matrix P. By analyzing the spectrum of this matrix, under certain conditions, we see that P has a low-rank plus a low-variance residual decomposition which makes it suitable for highly sub-sampled -on the order of 0.5% -matrix completion methods. Based on this observation, we propose a novel permutation testing methodology which offers a large speedup, without sacrificing the fidelity of the estimated FWER. Our evaluations on four different neuroimaging datasets show that a computational speedup factor of roughly 50× can be achieved while recovering the FWER distribution up to very high accuracy. Further, we show that the estimated α-threshold is also recovered faithfully, and is stable.

general comments: i love this paper, and i
      think it makes a great contribution. in particular, the matrix completion
      and optimization communities have mostly focused on estimation problems,
      rather than testing problems. this is a beautiful example of how the novel
      developments in one field can make substantial contributions to another
      field. note that i haven't read the supplement and i am not an expert in
      matrix completion, though i do know some stats and neuroscience.
      major comments: -- the authors comment, "We believe this
      is due to the approximate estimate of bias from training module." there
      are several possible sources of error, including model misspecification,
      approximation error (even given the true model), and sampling error. it
      will be interesting to investigate when these errors arise, their relative
      impact, how to mitigate and parse them, etc. some synthetic data analysis
      would be a most welcome addition; i imagine the full journal length
      version of this work would include that. -- t-tests are ubiquitous
      and great, and obviously problematic. here, perhaps less problematic, as
      rather than modeling the distribution of the data, the authors are
      modeling the distribution of the residuals. thus, already, perhaps this
      test is "robust" in certain ways (to the assumption of gaussian
      distributed data). yet, robust isn't mentioned in the manuscript. i think
      this method is robust to certain very simple assumptions, and that point
      should be highlighted and explored if i am correct. -- the data
      sets that were used are perfectly fine. however, no ground truth is
      available for them. so, to complement a synthetic data analysis, a real
      data analysis with ground truth would be excellent. my favorite example of
      such an endeavor is from: @article{Eklund2012a, author = {Eklund,
      Anders and Andersson, Mats and Josephson, Camilla and Johannesson, Magnus
      and Knutsson, Hans}, journal = {NeuroImage}, month = apr,
      number = {3}, pages = {565--578}, pmid = {22507229},
      publisher = {Elsevier Inc.}, title = {{Does parametric fMRI
      analysis with SPM yield valid results?-An empirical study of 1484 rest
      datasets.}}, url = {http://www.ncbi.nlm.nih.gov/pubmed/22507229},
      volume = {61}, year = {2012} } using this data, one
      can check how accurate the permutation tests are, and then how much
      accuracy degrades via this method. -- the figures could be
      improved. the checklist available at:
      http://buddhabaki.tumblr.com/post/44143989765/top-20-figure-checklist
      could help. specifically, fonts are too small, graphics are not vector
      graphics, symbols are not so obviously different, color choices might be
      hard for color blind people, axis labels and titles are missing or could
      be improved. streaming comments: -- Bonferoni, FWER,
      and other statistical techniques should be defined prior to their
      introduction, as many in this community are relatively unfamiliar with
      testing. -- permutation testing is not necessarily sampling from
      the global null, only in the multiple hypothesis testing scenario.
      probably worth clarifying. -- "permutation testing ensures"
      does it really "ensure" this? i haven't thought carefully about
      whether it is possible in the multiple testing scenario for permutation
      tests to not capture some dependencies, but i imagine it does not. for
      example, 3rd order dependencies, given finite samples, wouldn't be
      captured typically. this is not to say that it is not unbiased, but
      rather, unbiased doesn't ensure relevant dependencies are carried through;
      those that are hard to capture might still be missed. please cite
      something or soften or explain. -- " are required" don't love
      that either. required for what? high power? yes. unbiased results, not
      necessarily. please clarify or soften. -- "Observe that if
      voxel-wise statistics were completely i.i.d., then regardless of their
      distribution, we could simply compute the distribution of the sample
      maximum analytically by exponentiating." what do you mean
      exponentiating the univariate CDF? multiplying? please clarify. also,
      i think CDF is not defined, only the acronym is given. -- why is
      "Null" capital? and "Global" seems non-standard to me, but maybe is
      standard in some literatures. certainly not in the stats textbooks i'm
      familiar with, eg 'testing statistical hypotheses' is the standard used at
      many top programs, and is my default for notations & definitions.
      -- "this subspace. [15, 5]." extra period. -- "In
      other words, our algorithm does the same job as permutation testing,
      099 but takes anywhere from a few minutes up to a few hours, rather
      than days or weeks." but is approximate, and its accuracy will depend
      on the matrix completion assumptions and algorithms. --
      Theorem 3.1 of which paper? i figured it out eventually, but was confused
      at first, fyi. also, you fluctuate between Thm and Theorem; i prefer
      consistency. -- "In order to preserve anonymity," while i
      appreciate preserving anonymity, referring to the studies by a letter does
      not anonymize, afaict. anonymizing involves stripping headers, and
      possibly faces, generating unique random IDs, etc. also, punctuation is a
      bit weird in the following sentence, with a space before the colon only
      sometimes. -- "How sensitive is the estimated α-level threshold
      with respect to the recovered Null distribution" how sensitive to
      what? -- fig 1 shows the KL and BD divergence/distance between the
      subsampled and the truth? i guess that is implied, but i don't see that
      explicitly stated. for example, y-axis just gives scale, not quantity. no
      titles. and the caption doesn't make it clear to me either. --
      does 'e^{-5}' mean '10^{-5}'? or really 'e'? i guess that means the log
      scale is natural log scale. maybe those things are safe to assume in this
      community, though i wasn't sure. -- "We believe this is due to the
      approximate estimate of bias from training module." it would be great
      to investigate the source of this error. simulations would obviously be
      appropriate for such an investigation. a synthetic data analysis, where
      the simulation parameters were chosen to match the realized data to the
      extent possible would be especially interesting. -- i think in
      your tex file, when writing things like Fig. blah, if you instead write
      'Fig.\ ' the spacing will look cleaner (i'm just guessing that you are
      not) -- i think fig 2 might be more impressive on a log scale? the
      black box thing is weird to me. i've often seen a line break with some
      twiddles separating, know what i mean? also, the y-axis is 'time', the
      units are '(minutes)'. on a log scale, the speed-up factor may be explicit
      in the image? -- i don't think 'upto' is a word. -- "Our
      experiments suggest that the threshold is stable." i think this
      sentence belongs *after* showing it, not before? QUALITY:
      great CLARITY: great ORIGINALITY & SIGNIFICANCE:
      great


Action from Still Image Dataset and Inverse Optimal Control to Learn Task Specific Visual Scanpaths
Human eye movements provide a rich source of information into the human visual information processing. The complex interplay between the task and the visual stimulus is believed to determine human eye movements, yet it is not fully understood, making it difficult to develop reliable eye movement prediction systems. Our work makes three contributions towards addressing this problem. First, we complement one of the largest and most challenging static computer vision datasets, VOC 2012 Actions, with human eye movement recordings collected under the primary task constraint of action recognition, as well as, separately, for context recognition, in order to analyze the impact of different tasks. Our dataset is unique among the eyetracking datasets of still images in terms of large scale (over 1 million fixations recorded in 9157 images) and different task controls. Second, we propose Markov models to automatically discover areas of interest (AOI) and introduce novel sequential consistency metrics based on them. Our methods can automatically determine the number, the spatial support and the transitions between AOIs, in addition to their locations. Based on such encodings, we quantitatively show that given unconstrained read-world stimuli, task instructions have significant influence on the human visual search patterns and are stable across subjects. Finally, we leverage powerful machine learning techniques and computer vision features in order to learn task-sensitive reward functions from eye movement data within models that allow to effectively predict the human visual search patterns based on inverse optimal control. The methodology achieves state of the art scanpath modeling results.

This paper presents work on human eye movements when
      viewing still images. A new dataset is collected, measuring eye fixations
      under task-specific instructions -- the same images are viewed by subjects
      instructed to determine the human action or the scene context. The paper
      presents analyses of these data, a model for scanpaths, a model for
      fixations, and a temporal model of saccades. The analyses measure
      consistency of different subjects' viewing of an image-task pair. The
      model for scanpaths builds a task-image pair-specific HMM. The model for
      fixations uses HOG descriptors and an SVM to predict salient regions of an
      image. The temporal model uses reinforcement learning to generate a model
      for viewing of an image. Overall, the paper makes contributions to
      eye movement analysis -- expanding on previous work [16] to clearly show
      task matters for gaze. In my view this is the strongest part of the paper.
      On the other hand, various computational models are described, the HMM for
      scanpaths, the HOG-SVM saliency predictor, and the reinforcement learning.
      These are reasonable, but there are concerns about their
      applicability/importance (details below). On the whole, the paper is
      solid, and could be accepted as a poster. 1) The HMM for scanpaths
      seems quite limited in its use. The model is a task-image pair-specific
      HMM -- given an image for which many subjects' eye movements have been
      observed given a task, an HMM that predicts how other people will view
      this image when asked to do this task is learned. While HMMs are a
      reasonable model, this task doesn't seem very common. Further, it seems
      any temporal model would be appropriate for this task, and no particular
      baselines are presented. 2) A HOG-SVM saliency predictor is
      defined. This uses dataset and task-specific training data to build a
      model of where a subject will look. The performance is good, and better
      than baselines. However, it seems only marginally better than [12], and
      this could either be due to the use of different features or a non-linear
      classifier. It is necessary to clarify from which of these the performance
      gain comes. 3) A similar comment applies to the reinforcement
      learning and comparison to baselines. Different features (HOG) and
      learning strategy are used in this method. While overall performance is
      important, understanding where these gains come from is important too.
      Minor comments: - The paper needs proofreading, there
      are many typos and spelling errors. - Why does the dataset have 3s
      exposures for action recognition and 2.5 for context? Won't this
      difference bias comparative studies? - What are the "three key
      presses" (line 135)? From the description, it seems only one key press is
      made by a subject. - Line 200 "locating people"? - +/-
      std. deviation should be added to numbers in Table 1. - I don't
      think the cross-stimulus baseline for gaze prediction is defined in the
      text. A sentence could be added, especially pointing out that the
      performance is significantly lower, which bolsters a main message of the
      paper.


Spectral methods for neural characterization using generalized quadratic models
We describe a set of fast, tractable methods for characterizing neural responses to high-dimensional sensory stimuli using a model we refer to as the generalized quadratic model (GQM). The GQM consists of a low-rank quadratic function followed by a point nonlinearity and exponential-family noise. The quadratic function characterizes the neuron's stimulus selectivity in terms of a set linear receptive fields followed by a quadratic combination rule, and the invertible nonlinearity maps this output to the desired response range. Special cases of the GQM include the 2nd-order Volterra model [1,2] and the elliptical Linear-Nonlinear-Poisson model [3]. Here we show that for "canonical form" GQMs, spectral decomposition of the first two response-weighted moments yields approximate maximumlikelihood estimators via a quantity called the expected log-likelihood. The resulting theory generalizes moment-based estimators such as the spike-triggered covariance, and, in the Gaussian noise case, provides closed-form estimators under a large class of non-Gaussian stimulus distributions. We show that these estimators are fast and provide highly accurate estimates with far lower computational cost than full maximum likelihood. Moreover, the GQM provides a natural framework for combining multi-dimensional stimulus sensitivity and spike-history dependencies within a single model. We show applications to both analog and spiking data using intracellular recordings of V1 membrane potential and extracellular recordings of retinal spike trains.

This paper makes two contributions. First, this paper
      continues a trend of recent work by Park and Pillow (2011) and Ramirez
      & Paninski (2012) to provide a firm grounding of the spike-triggered
      methods STA and STC within forward, likelihood-based modelling. STA and
      STC are calculated as moments from the stimulus/response ensemble, and are
      popular for constructing stimulus/response models due to their speed and
      simplicity. However, as explored extensively in previous work (e.g.
      Paninski 2003, Sharpee et al 2004, Samengo & Gollisch 2013), the
      validity and utility of these moments depends on particular restrictive
      conditions on the stimulus set. The prior work of Park & Pillow (2011)
      was able to shed light on this, by relating these moments to the
      parameters of a Generalised Linear Model (GLM), a popular forward model
      for neural responses. While GLM parameter estimation typically requires
      log-likelihood maximisation over a full dataset (a potentially slow
      operation), if one instead maximises the expected log-likelihood (EL)
      under the stimulus distribution, the STA/STC moments provide a fast and
      asymptotically consistent means to estimate the GLM. Here, the authors
      extend this work, by asking how moments such as STA and STC can assist in
      the estimation of a more complex forward model -- the Generalised
      Quadratic Model (GQM) -- via the EL framework. In doing so, the paper
      makes its second contribution, in that it introduces the GQM as a new,
      general class of models for characterising neural responses. The GQM
      extends the popular Generalised Linear Model (GLM) by allowing for
      quadratic (rather than just linear) relationships between stimulus and
      response (together with a point nonlinearity and exponential-family
      noise). For Gaussian likelihoods, the authors derive moment-based
      estimators for the GQM under the conditions of Gaussian-distributed
      stimuli, and axis-symmetric stimuli. A similar derivation is provided for
      Poisson likelihoods under a set of assumptions for the stimulus
      distribution. Low-rank estimates of the parameters are also shown to be
      asymptotically consistent. Finally, the authors demonstrate an application
      of these results to real neural data (intracellular voltage and spiking
      cells), demonstrating a performance gain of the GQM over the GLM. They
      also show a synthetic example where the vast speed improvements from using
      moments, as opposed to maximising the likelihood over a full dataset, are
      evident. The paper is technically sound, giving thorough
      derivations of the relationships between the GQM and stimulus/response
      moments under the set of conditions studied. While this aspect of the work
      is technical in nature, its significance -- as for that of recent work on
      this topic raised above -- is in establishing the theoretical groundwork
      that underlies the workings of a common set of analytical techniques.
      These results also build incrementally upon these previous papers,
      although the relationship between the exponentiated-quadratic model
      presented in Park & Pillow (2011) section 2.2, and that given here in
      section 3.2, should be made clearer. The GQM may generally be a promising
      tool for neural characterisation, although this paper does not really
      provide a thorough enough evaluation of the realistic data requirements to
      properly assess this. This paper, in its current form, does fall a
      little short in its clarity. Part of my confusion in reading it stems from
      the fact that half of the time the paper is selling the GQM, and half of
      the time the paper is demonstrating the deeper relationships between
      moment-based estimators and the GQM parameters. In addition, the title
      seems to emphasise spectral methods (I assume this refers to the
      dimensionality reduction in Section 4), but this only figures as a small
      part of the paper. Because of the jumps in focus between the EL approach
      in section 3, dimensionality reduction in section 4, and the GQM demo in
      section 5, it took me a number of reads just to get a handle on the
      narrative of the paper. Some bracketing commentary at the start and the
      end of each section to ground the reader would really be helpful. Also,
      section 3.3 should really be a part of section 4. Some minor
      comments: - line 207: E[x_i . x_j^3] is zero. This should be E[x_i
      . x_j^2] or E[x_j^3] or something like that in order to be a third moment.
      - line 209: what happens when the stimulus is not white? Also, an
      explanation of how the general fourth-order moments at the end of the
      expression for Lambda (line 204), reduce to the matrix M = E[x_i^2 .
      x_j^2] would be helpful (I admit I didn't follow this logic). -
      lines 283-292: Typesetting -- mu_x has the subscript in math italics on
      line 283, but bold in (17) and on line 291. - Section 4. I'm not
      sure why the notation here was changed from vector (bold) x to matrix
      (bold) X, and scalar (italic non-bold) y to matrix (bold) Y. It makes this
      section needlessly confusing (X and Y are matrices now?) and harder to
      connect with the work in previous sections. Surely the authors could stick
      with vector x and scalar y. - Fig 4: r^2 = 0.55 in the figure
      legend, but r^2 = 0.50 in the panel. - line 122: unfinished
      sentence


Provable Subspace Clustering: When LRR meets SSC
Sparse Subspace Clustering (SSC) and Low-Rank Representation (LRR) are both considered as the state-of-the-art methods for subspace clustering. The two methods are fundamentally similar in that both are convex optimizations exploiting the intuition of "Self-Expressiveness". The main difference is that SSC minimizes the vector 1 norm of the representation matrix to induce sparsity while LRR minimizes nuclear norm (aka trace norm) to promote a low-rank structure. Because the representation matrix is often simultaneously sparse and low-rank, we propose a new algorithm, termed Low-Rank Sparse Subspace Clustering (LRSSC), by combining SSC and LRR, and develops theoretical guarantees of when the algorithm succeeds. The results reveal interesting insights into the strength and weakness of SSC and LRR and demonstrate how LRSSC can take the advantages of both methods in preserving the "Self-Expressiveness Property" and "Graph Connectivity" at the same time.

SUMMARY: This paper proposes a new subspace clustering
      algorithm called Low Rank Sparse Subspace Clustering (LRSSC) and aims to
      study the conditions under which it is guaranteed to produce a correct
      clustering. The correctness is defined in terms of two properties. The
      self-expressiveness property (SEP) captures whether a data point is
      expressed as a linear combination of other points in the same subspace.
      The graph connectivity property (GCP) captures whether the points in one
      subspace form a connected component of the graph formed by all the data
      points. The LRSSC algorithm builds on two existing subspace clustering
      algorithms, SSC and LRR, which have complementary properties. The solution
      of LRR is guaranteed to satisfy the SEP under the strong assumption of
      independent subspaces and the GCP under weak assumptions (shown in this
      paper). On the other hand, the solution of SSC is guaranteed to satisfy
      the SEP under milder conditions, even with noisy data or data corrupted
      with outliers, but the solution of SSC need not satisfy the GCP. This
      paper combines the objective functions of both methods with the hope of
      obtaining a method that satisfies both SEP and GEP for some range of
      values of the relative weight between the two objective functions. Theorem
      1 derives conditions under which LRSSC satisfies SEP in the deterministic
      case. These conditions are natural generalizations of existing conditions
      for SSC. But they are actually weaker than existing conditions. Theorem 2
      derives conditions under which LRSSC satisfies SEP in the random case
      (data drawn at random from randomly drawn subspaces). Overall, it is shown
      that when the weight of the SSC term is large enough and the ratio of the
      data dimension to the subspace dimension grows with the log of the number
      of points, then LRSSC is guaranteed to satisfy SEP with high probability.
      I say high, because it does not tend to 1. Finally, Proposition 1 and
      Lemma 4 show that LRR satisfies GCP (presumably almost surely).
      Experiments support that for a range of the SSC weight, LRSCC works.
      Additional experiments on model selection show the usefulness of the
      analysis. QUALITY: This is an excellent paper. It addresses a
      difficult and important problem. It derives new conditions that advance
      the state of the art for the subspace clustering problem. For the first
      time, easy-to-verify conditions for the case of non-independent subspaces
      are presented. The conditions are clean, intuitive and useful. Moreover,
      for the first time an attempt to study both the SEP and the GCP properties
      together is done. While the story is not complete yet, this is a clear
      step in the right direction. That said, I do have some comments to improve
      the paper. Most notably, the discussion in Section 4 needs improvement.
      Also, there are a number of imprecisions in the literature review that
      need to be corrected. Finally, the experiments on Hopkins 155 are not
      entirely clear. My detailed comments are as follows: - For data
      drawn from a single subspace, I was expecting that graph connectivity
      would be defined as second eigenvalue of the Laplacian being larger than
      zero, where the Laplacian = Degree - Affinity, and Affinity = |C|+|C|^T.
      Somehow, the authors limit themselves to showing that all the entries of C
      are nonzero. It is probably trivial, but I would have preferred making the
      connection with connectedness (based on the eigenvalue of the Laplacian)
      more explicit. - The whole paper is predicated on the idea that as
      one varies lambda from 0 to infinity one goes from LRR to SSC. But this is
      not the case, since for lambda = 0, the constraint diag(C) = 0 is not
      present in standard LRR. Moreover, I don't think there is a value of
      lambda for which one gets LRR. I was disappointed that this is not
      mentioned explicitly till the very end of section 4. Can one write the
      constraint as lambda diag(C) = 0 and the analysis follows? -
      Following on the point above, it was disappointing that no result on the
      graph connectivity is shown for lambda > 0. In other words, the paper
      does not shed light on graph connectivity for SSC or LRSSC. Only for LRR.
      - The Low Rank Subspace Clustering (LRSC) work of Favaro deserves
      a discussion. Equation in (6) is in fact an extension of LRSC, not of LRR,
      because LRSC uses the Frobeniuos norm of the error, while LRR uses the L21
      norm. Of course they coincide in the noiseless case. Arguably, the name
      LRSC is more appropriate than LRR throughout, since we are not talking
      about a low rank representation in general, but one specifically designed
      for the subspace clustering problem. - The proof of the success of
      LRR for independent subspaces was not shown in Vidal et al. PAMI 2005
      [23]. It was shown in Section 2.3 of Vidal et al. IJCV 2008, " Multiframe
      motion segmentation with missing data using PowerFactorization and GPCA"
      - To the best of my knowledge, the first application of subspace
      clustering to hybrid system identification is Vidal et al. CDC 2003, which
      is much earlier than Bako, Automatica 2011 [2]. - The experiments
      on Figure 9 seem to be showing a performance that is worse than that of
      SSC and LRR. As lambda goes to infinity, this should approximately
      converge to the SSC performance, which is 1.52% for 2 motions, 4.40% for 3
      motions, and 2.18% for two motions. The reported number of nearly 9% for 3
      motions seems way off. I understand that the parameter setting may be
      slightly different. But I wonder what happens if one uses the parameter
      settings of Elhamifar. Would the behavior of getting a smaller error for
      some lambda still hold in practice? Overall, I think the experiments needs
      to be redone to be made comparable with the state of the art. -
      The quality of English decreases as a function of the page number. --
      The articles "the" and "a" are missing in several places
      (100,123,153,154,205,271,281) -- It should be "a d-dimensional
      subspace" (085), -- There should be a space before references
      (040,041,048,221,353), -- It should be "these subspaces, we say a
      matrix C obeys the SEP" in line 100, -- What is the \perp symbol in
      115. -- it should be "the only difference being the definition of mu"
      in 179 -- It should be "Assume random sampling. If ..." in 207 --
      It should be "Combining Lemmas 1 and 2, we get" in line 229 -- It
      should be "and random sampling. It holds" in 240 -- What is
      probabilistic theorem? in line 245 -- It should be "ignoring constant
      terms" or "ignoring the constant C" or something like that in 256 --
      The grammar of "The graph connectivity problem concerns when SEP is
      satisfied, whether each block of the solution C to LRSSC represents a
      connected graph" needs to be revised. -- It should be "If the
      subspaces are independent, X is not full rank, and the data points" in 274
      -- It should be "et al." not "et.al" -- There is a "the the" and
      "if if" somewhere in the paper CLARITY: The paper very clearly
      conveys the essence, originality and novelty of the contribution. But I
      had read the previous papers on the subject, and in particular the papers
      from Candes with the conditions for the deterministic case (inradius),
      etc. I think that a reader unfamiliar with that paper may have
      difficulties with definition 3 and the examples that follow. Now, the
      technical details inside the proofs are of course harder to follow. One
      minor thing is that the range of lambda values or lambda above a threshold
      is mentioned earlier in the paper (e.g., 182), but such a range does not
      appear till Remark 4, so the reader is left wondering what threshold is
      the paper referring to. Another issue is that it is not clear what "vice
      versa" refers to in Remark 4. Are you trying to say when lambda is smaller
      than a threshold versus larger than a threshold? ORIGINALITY: The
      paper is highly original. The core contributions are to find weaker
      conditions for the correctness of SSC, new conditions for the correctness
      of LRR, and conditions for the correctness of a new method LRSSC. To the
      best of my knowledge, all the material presented in the paper is new.
      IMPACT: Subspace clustering is receiving increased attention in
      the community, and sparse and low rank methods are showing excellent
      performance in many datasets. Understanding why these methods work is in
      my view fundamental. This paper presents a fundamental step towards
      achieving this understanding.