Optimal Shrinkage of Singular Values Under Random Data Contamination
A low rank matrix X has been contaminated by uniformly distributed noise, missing values, outliers and corrupt entries. Reconstruction of X from the singular values and singular vectors of the contaminated matrix Y is a key problem in machine learning, computer vision and data science. In this paper, we show that common contamination models (including arbitrary combinations of uniform noise, missing values, outliers and corrupt entries) can be described efficiently using a single framework. We develop an asymptotically optimal algorithm that estimates X by manipulation of the singular values of Y , which applies to any of the contamination models considered. Finally, we find an explicit signal-to-noise cutoff, below which estimation of X from the singular value decomposition of Y must fail, in a welldefined sense.
This paper consider the problem of matrix de-noising under the mean square-loss via estimators that shrink the singular values of a contaminated version of the matrix to be estimated. Specifically, the authors seek asymptotically optimal shrinkers under various random contamination models, extending beyond additive white noise.
Unfortunately, I could not determine the novelty of the techniques used as there is no clear indication in the paper of what results are new or rather simple extensions of previous results, such as in Nadakuditi (2014) and Gavish & Donoho (2017). As a non-expert, as far as I can see, the results in this paper are rather simple extensions of those given in Gavish and Donoho. A clear indication of what is new and a detailed comparison to what is already known is needed. In addition, I found many sentences to be unclear and inadequate, making it a difficult read.
Some more specific questions/comments:
In line 213 it is not clear whether you prove that the a.s. limit exists or assume it exists.
It is unclear under what class of estimators is yours optimal. Only a clue is given in the very last sentence of the paper.
In line 64, the variance of A's entries is taken to be the same as that of B, namely \sigma_B, while later it is changed to sigma_A. Thm. 2 however does not involve sigma_A. Is there a specific reason for choosing sigma_B = sigma_A?
In addition, it is unclear how can one estimate mu_A, sigma_A and sigma_B, which are needed for the method.
Lines 89 and 90 should come before Thm. 1.
The hard threshold shrinker is missing a multiplicative y_i all over the paper.
There are many typos.