Definition: In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.
Input: We consider the problem of accurately recovering a matrix B of size M ×M , which represents a probability distribution overM 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 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 problem—both in terms of learning and property testing/estimation and on both the algorithmic and information theoretic sides—remain open. Our results apply to the setting where B has a particular rank 2 structure. For this setting, we propose an efficient (and practically viable) algorithm that accurately recovers the underlying M × M matrix using Θ(M) samples. This result easily translates to Θ(M) sample algorithms for learning topic models with two topics over dictionaries of size M , and learning hidden Markov Models with two hidden states and observation distributions supported on M elements. These linear sample complexities are 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 Ω(M) samples. Furthermore, we provide an even stronger lower bound where distinguishing whether a sequence of observations were drawn from the uniform distribution over M observations versus being generated by an HMM with two hidden states requires Ω(M) observations. This precludes sublinear-sample 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. This impossibility of sublinear-sample property testing in these settings is intriguing and underscores the significant differences between these structured settings and the standard setting of drawing i.i.d samples from an unstructured distribution of support size M . ∗MIT. Email: qqh@mit.edu. †University of Washington. Email: sham@cs.washington.edu ‡Stanford University. Email: kweihao@gmail.com §Stanford University. Email: valiant@stanford.edu. Gregory and Weihao’s contributions were supported by NSF CAREER Award CCF-1351108, and a research grant from the Okawa Foundation. ar X iv :1 60 2. 06 58 6v 1 [ cs .L G ] 2 1 Fe b 20 16

Recovering Structured Probability Matrices