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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 analyze what is the information theoretically minimal mean-squared error and the one achieved by AMP in the limit of large sizes. For a special case of rank one and large enough density of non-zeros Deshpande and Montanari [1] proved that AMP is asymptotically optimal. We show that both for low density and for large rank the problem undergoes a series of phase transitions suggesting existence of a region of parameters where estimation is information theoretically possible, but AMP (and presumably every other polynomial algorithm) fails. The analysis of the large rank limit is particularly instructive.
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 exhi
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 formulat
We study the statistical problem of estimating a rank-one sparse tensor corrupted by additive Gaussian noise, a model also known as sparse tensor PCA. We show that for Bernoulli and Bernoulli-Rademacher distributed signals and emph{for all} sparsity
This paper is concerned with the problem of recovering a structured signal from a relatively small number of corrupted random measurements. Sharp phase transitions have been numerically observed in practice when different convex programming procedure
We consider the problem of decoding a discrete signal of categorical variables from the observation of several histograms of pooled subsets of it. We present an Approximate Message Passing (AMP) algorithm for recovering the signal in the random dense