Information-theoretic bounds and phase transitions in clustering, sparse PCA, and submatrix localization

We study the problem of detecting a structured, low-rank 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 information-theoretic 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.Comment: For sparse PCA and submatrix localization, we determine the information-theoretic 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 versio

MMSE of probabilistic low-rank matrix estimation: Universality with respect to the output channel

This paper considers probabilistic estimation of a low-rank matrix from non-linear element-wise measurements of its elements. We derive the corresponding approximate message passing (AMP) algorithm and its state evolution. Relying on non-rigorous but standard assumptions motivated by statistical physics, we characterize the minimum mean squared error (MMSE) achievable information theoretically and with the AMP algorithm. Unlike in related problems of linear estimation, in the present setting the MMSE depends on the output channel only trough a single parameter - its Fisher information. We illustrate this striking finding by analysis of submatrix localization, and of detection of communities hidden in a dense stochastic block model. For this example we locate the computational and statistical boundaries that are not equal for rank larger than four.Comment: 10 pages, Allerton Conference on Communication, Control, and Computing 201

Sigma Point Belief Propagation

The sigma point (SP) filter, also known as unscented Kalman filter, is an attractive alternative to the extended Kalman filter and the particle filter. Here, we extend the SP filter to nonsequential Bayesian inference corresponding to loopy factor graphs. We propose sigma point belief propagation (SPBP) as a low-complexity approximation of the belief propagation (BP) message passing scheme. SPBP achieves approximate marginalizations of posterior distributions corresponding to (generally) loopy factor graphs. It is well suited for decentralized inference because of its low communication requirements. For a decentralized, dynamic sensor localization problem, we demonstrate that SPBP can outperform nonparametric (particle-based) BP while requiring significantly less computations and communications.Comment: 5 pages, 1 figur

Inference And Learning: Computational Difficulty And Efficiency

In this thesis, we mainly investigate two collections of problems: statistical network inference and model selection in regression. The common feature shared by these two types of problems is that they typically exhibit an interesting phenomenon in terms of computational difficulty and efficiency. For statistical network inference, our goal is to infer the network structure based on a noisy observation of the network. Statistically, we model the network as generated from the structural information with the presence of noise, for example, planted submatrix model (for bipartite weighted graph), stochastic block model, and Watts-Strogatz model. As the relative amount of ``signal-to-noise\u27\u27 varies, the problems exhibit different stages of computational difficulty. On the theoretical side, we investigate these stages through characterizing the transition thresholds on the ``signal-to-noise\u27\u27 ratio, for the aforementioned models. On the methodological side, we provide new computationally efficient procedures to reconstruct the network structure for each model. For model selection in regression, our goal is to learn a ``good\u27\u27 model based on a certain model class from the observed data sequences (feature and response pairs), when the model can be misspecified. More concretely, we study two model selection problems: to learn from general classes of functions based on i.i.d. data with minimal assumptions, and to select from the sparse linear model class based on possibly adversarially chosen data in a sequential fashion. We develop new theoretical and algorithmic tools beyond empirical risk minimization to study these problems from a learning theory point of view

Mutual Information in Rank-One Matrix Estimation

We consider the estimation of a n-dimensional vector x from the knowledge of noisy and possibility non-linear element-wise measurements of xxT , a very generic problem that contains, e.g. stochastic 2-block model, submatrix localization or the spike perturbation of random matrices. We use an interpolation method proposed by Guerra and later refined by Korada and Macris. We prove that the Bethe mutual information (related to the Bethe free energy and conjectured to be exact by Lesieur et al. on the basis of the non-rigorous cavity method) always yields an upper bound to the exact mutual information. We also provide a lower bound using a similar technique. For concreteness, we illustrate our findings on the sparse PCA problem, and observe that (a) our bounds match for a large region of parameters and (b) that it exists a phase transition in a region where the spectum remains uninformative. While we present only the case of rank-one symmetric matrix estimation, our proof technique is readily extendable to low-rank symmetric matrix or low-rank symmetric tensor estimationComment: 8 pages, 1 figure