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Sparse Principal Component Analysis (PCA) is a dimensionality reduction technique wherein one seeks a low-rank representation of a data matrix with additional sparsity constraints on the obtained representation. We consider two probabilistic formulations of sparse PCA: a spiked Wigner and spiked Wishart (or spiked covariance) model. We analyze an Approximate Message Passing (AMP) algorithm to estimate the underlying signal and show, in the high dimensional limit, that the AMP estimates are information-theoretically optimal. As an immediate corollary, our results demonstrate that the posterior expectation of the underlying signal, which is often intractable to compute, can be obtained using a polynomial-time scheme. Our results also effectively provide a single-letter characterization of the sparse PCA problem.
We study optimal estimation for sparse principal component analysis when the number of non-zero elements is small but on the same order as the dimension of the data. We employ approximate message passing (AMP) algorithm and its state evolution to ana
We study the problem of recovering a block-sparse signal from under-sampled observations. The non-zero values of such signals appear in few blocks, and their recovery is often accomplished using a $ell_{1,2}$ optimization problem. In applications suc
In this paper study the problem of signal detection in Gaussian noise in a distributed setting. We derive a lower bound on the size that the signal needs to have in order to be detectable. Moreover, we exhibit optimal distributed testing strategies that attain the lower bound.
We revisit the universal outlier hypothesis testing (Li emph{et al.}, TIT 2014) and derive fundamental limits for the optimal test. In outlying hypothesis testing, one is given multiple observed sequences, where most sequences are generated i.i.d. fr
This work studies the recursive robust principal components analysis (PCA) problem. Here, robust refers to robustness to both independent and correlated sparse outliers, although we focus on the latter. A key application where this problem occurs is