Paper ID: | 3303 |
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Title: | Towards a Zero-One Law for Column Subset Selection |

Originality: To the best of my knowledge the results are new. The methods used on the other hand are simple and to my understanding relatively standard in the field. Yet, the fact that they work for a broad class of loss functions is interesting. Quality: The statements of the results and proofs appear to be correct. The motivation to study more general functions is clear and the experiment helps significantly with it (I would even discuss about the experiment earlier in the paper). Yet, I dislike the use of ``necessary" conditions for the approx. monotonicity and approx. triangle inequality. Indeed, at the complete absence of any of the two conditions the authors provide counterexamples. To my understanding that is not enough for a necessary characterization of the error losses: the condition is necessary only if any function (and not at least one function) which does not satisfy this property cannot be approximated in the appropriate sense. Also, I am not sure what is the zero-one law the authors are referring at the title. It will be great if the authors offer an explanation for it. Clarity: For the most part, the paper is well-written and the arguments relatively clean. One comment is that I consider important is to explain for which choice of r Algorithm 1 works for the general theorem (the initialization step says that r is of order log n which is rather confusing: can I choose r to be 0 for example?) Also I consider important the authors to explain the use of Cramer's rule in the beginning of Subsection 2.1. Significance: The paper points to an interesting direction of classifying the loss functions for which an approximation algorithm with appropriate guarantees can be constructed. They provide certain natural sufficient assumptions under which this is happening, which generalize much beyond the standard \ell_p norms. Their direction can definitely impact future theoretical and potentially empirical research.

As a caveat, I am not an expert in the literature surrounding low-rank reconstruction, and may not be entirely correct in my evaluation of the originality and significance of the contributions. Originality:This paper builds upon previous work, in particular [62], which developed column-subset selection for low-rank approximation under the l_p norm. This paper expands upon [62], obtaining results for a broader class of functions and furthermore tightening and fixing some results from [62]. These expansions seem very valuable to the machine learning community. However, the authors may want to further motivate their work by providing specific examples of loss functions to which they extend previous theory, and which have found successful applications in machine learning. Quality: This works appears to be of high quality. The authors provide the intuition behind their theoretical contribution as well as the motivation in a way that is very accessible; the results seem theoretically important, and I can imagine several implications for the machine learning field, for example in using low-rank approximations for problems that are prone to outlier data. Clarity: This paper is somewhat clear, but could in my opinion be improved upon in a few ways that I describe in the relevant section. Overall, the authors guide the reader in understanding the motivation and reason for the hypotheses and class of functions that they investigate. Although I have not carefully read over the proofs, I understand the underlying reasoning of this paper. Significance: This paper appears to be significant due to the importance of low-rank approximation and reconstruction in machine learning, from both a computational efficiency and memory usage perspective (among others). *********** Post-rebuttal comments *********** Based on the author rebuttal and reviewer discussions, I recommend the following for the camera-ready version if the paper is accepted: (i) The issue regarding the choice of r in Algorithm 1 should be clarified. It seems that r simply indicates that the loop in Alg. 1 should be run until |T_r| < 1000k, and that the number of loop iterations required is O(log n); I recommend the authors make this fact obvious within the algorithm itself, for example by changing the first for loop to ''while |T_i| < 1000k''. (ii) Following Reviewer 2's review, I recommend the authors change the title. (iii) Similarly, I agree that the approx. monotone/necessary conditions have been proven to be *sufficient* but not *necessary*; the paper should be updated to clarify this.

This paper studies a generalized low-rank approximation problem. In this problem, we are given a matrix $A$ and a cost function $g(.)$. The goal is to find a rank $k$ matrix $B$ such that $\sum_{i, j}g(A_{i,j} - B_{i, j})$ is minimized. In the approximation version of the problem, we aim to find a rank $k polylog (n, k)$ matrix $B'$ so that $\sum_{i, j}g(A_{i,j} - B'_{i, j})$ is a constant times worse than $\sum_{i, j}g(A_{i,j} - B_{i, j})$. The paper's main goal is to understand what kind of $g$ admits approximation solutions. It focuses on column selection algorithms, i.e., the output $B'$ is a linear combination of a subset of columns from $A$. The paper gives a tight characterization of $g(.)$, i.e., it needs to satisfy approximate triangle inequality, and monotone property. Their upper bound result is a generalization of [62]. [62] states that if the cost is $\ell_p$ norm, then there is a good subset of columns to approximate $A$. Their major observations are that (i) [62]'s technique can be used to build a recursive method to find columns that can approximately span $A$, and (ii) [62] is a general technique so many assumptions made there can be relaxed. The authors showed that using only triangle inequality and monotone property suffices to generalize [62]. For the lower bound analysis, the authors claim that both approximate triangle inequality and monotone properties are necessary but the paper has only one lower bound result for the function $H(x)$. They argue that when the cost function is $H(x)$, we can construct a matrix $A$ so that it is impossible to choose a small number of columns to approximate $A$. I think the lower bound result is weaker than what the authors advertised. The generalized low rank approximation problem is an important problem in statistics. The authors gave a non-trivial generalization of [62]. The analysis looks believable. I have two major concerns: 1. Lower bound: I am not able to see why Theorem 1.3 implies that both triangle inequality and monotone property is necessary. 2. Presentation: the authors made a serious effort to explain the intuition for the analysis (Sec 1.1.2) but some of the intuitions still appear to be incomprehensible (e.g., line 158 to line 160; line 168 to line 179). In addition, I feel some of the key definitions are not explained properly. For example, for the equation between line 232 and 233, I cannot understand how Q is defined (e.g., is Q a variable to be optimized or we need a worst-case Q). My concerns could be fixable; I am willing to change my rating if the above two questions are properly addressed/answered.