Abstract
We consider the problem of accurately recovering a matrix B of size M by M, which represents a probability distribution over M^2 outcomes, given access to an observed matrix of "counts" generated by taking independent samples from the distribution B. How can structural properties of the underlying matrix B be leveraged to yield computationally efficient and information theoretically optimal reconstruction algorithms? When can accurate reconstruction be accomplished in the sparse data regime? This basic problem lies at the core of a number of questions that are currently being considered by different communities, including building recommendation systems and collaborative filtering in the sparse data regime, community detection in sparse random graphs, learning structured models such as topic models or hidden Markov models, and the efforts from the natural language processing community to compute "word embeddings". Many aspects of this problemboth in terms of learning and property testing/estimation and on both the algorithmic and information theoretic sidesremain open.
Our results apply to the setting where B has a low rank structure. For this setting, we propose an efficient (and practically viable) algorithm that accurately recovers the underlying M by M matrix using O(M) samples} (where we assume the rank is a constant). This linear sample complexity is optimal, up to constant factors, in an extremely strong sense: even testing basic properties of the underlying matrix (such as whether it has rank 1 or 2) requires Omega(M) samples. Additionally, we provide an even stronger lower bound showing that distinguishing whether a sequence of observations were drawn from the uniform distribution over M observations versus being generated by a wellconditioned Hidden Markov Model with two hidden states requires Omega(M) observations, while our positive results for recovering B immediately imply that Omega(M) observations suffice to learn such an HMM. This lower bound precludes sublinearsample hypothesis tests for basic properties, such as identity or uniformity, as well as sublinear sample estimators for quantities such as the entropy rate of HMMs.
BibTeX  Entry
@InProceedings{huang_et_al:LIPIcs:2018:8362,
author = {Qingqing Huang and Sham M. Kakade and Weihao Kong and Gregory Valiant},
title = {{Recovering Structured Probability Matrices}},
booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
pages = {46:146:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770606},
ISSN = {18688969},
year = {2018},
volume = {94},
editor = {Anna R. Karlin},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8362},
URN = {urn:nbn:de:0030drops83625},
doi = {10.4230/LIPIcs.ITCS.2018.46},
annote = {Keywords: Random matrices, matrix recovery, stochastic block model, Hidden Markov Models}
}
Keywords: 

Random matrices, matrix recovery, stochastic block model, Hidden Markov Models 
Collection: 

9th Innovations in Theoretical Computer Science Conference (ITCS 2018) 
Issue Date: 

2018 
Date of publication: 

12.01.2018 