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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 estimation
Factorizing low-rank matrices has many applications in machine learning and statistics. For probabilistic models in the Bayes optimal setting, a general expression for the mutual information has been proposed using heuristic statistical physics compu
A reconfigurable intelligent surface (RIS) consists of massive meta elements, which can improve the performance of future wireless communication systems. Existing RIS-aided channel estimation methods try to estimate the cascaded channel directly, inc
We point out a limitation of the mutual information neural estimation (MINE) where the network fails to learn at the initial training phase, leading to slow convergence in the number of training iterations. To solve this problem, we propose a faster
The Mutual Information (MI) is an often used measure of dependency between two random variables utilized in information theory, statistics and machine learning. Recently several MI estimators have been proposed that can achieve parametric MSE converg
We prove that mutual information is actually negative copula entropy, based on which a method for mutual information estimation is proposed.