Informationtheoretic bounds and phase transitions in clustering, sparse PCA, and submatrix localization
Abstract
We study the problem of detecting a structured, lowrank signal matrix corrupted with additive Gaussian noise. This includes clustering in a Gaussian mixture model, sparse PCA, and submatrix localization. Each of these problems is conjectured to exhibit a sharp informationtheoretic threshold, below which the signal is too weak for any algorithm to detect. We derive upper and lower bounds on these thresholds by applying the first and second moment methods to the likelihood ratio between these "planted models" and null models where the signal matrix is zero. Our bounds differ by at most a factor of root two when the rank is large (in the clustering and submatrix localization problems, when the number of clusters or blocks is large) or the signal matrix is very sparse. Moreover, our upper bounds show that for each of these problems there is a significant regime where reliable detection is information theoretically possible but where known algorithms such as PCA fail completely, since the spectrum of the observed matrix is uninformative. This regime is analogous to the conjectured 'hard but detectable' regime for community detection in sparse graphs.
 Publication:

arXiv eprints
 Pub Date:
 July 2016
 arXiv:
 arXiv:1607.05222
 Bibcode:
 2016arXiv160705222B
 Keywords:

 Mathematics  Statistics Theory;
 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Statistical Mechanics;
 Computer Science  Information Theory;
 Mathematics  Probability
 EPrint:
 For sparse PCA and submatrix localization, we determine the informationtheoretic threshold exactly in the limit where the number of blocks is large or the signal matrix is very sparse based on a conditional second moment method, closing the factor of root two gap in the first version