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 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 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 well-conditioned 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 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.
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Learning to Sample from Censored Markov Random Fields
We study the problem of learning Censored Markov Random Fields (abbreviated CMRFs), which are Markov Random Fields where some of the nodes are censored (i.e. not observed). We assume the CMRF is high temperature but, crucially, make no assumption about its structure. This makes structure learning impossible. Nevertheless we introduce a new definition, which we call learning to sample, that circumvents this obstacle. We give an algorithm that can learn to sample from a distribution within đťś–đť‘› earthmover distance of the target distribution for any đťś–>0. We obtain stronger results when we additionally assume high girth, as well as computational lower bounds showing that these are essentially optimal.
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- Award ID(s):
- 2031883
- NSF-PAR ID:
- 10344258
- Date Published:
- Journal Name:
- Proceedings of Thirty Fourth Conference on Learning Theory
- Volume:
- 134
- Page Range / eLocation ID:
- 3419-3451
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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