34,423 research outputs found
Information-theoretically Optimal Sparse PCA
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.Comment: 5 pages, 1 figure, conferenc
: Robust Principal Component Analysis for Exponential Family Distributions
Robust Principal Component Analysis (RPCA) is a widely used method for
recovering low-rank structure from data matrices corrupted by significant and
sparse outliers. These corruptions may arise from occlusions, malicious
tampering, or other causes for anomalies, and the joint identification of such
corruptions with low-rank background is critical for process monitoring and
diagnosis. However, existing RPCA methods and their extensions largely do not
account for the underlying probabilistic distribution for the data matrices,
which in many applications are known and can be highly non-Gaussian. We thus
propose a new method called Robust Principal Component Analysis for Exponential
Family distributions (), which can perform the desired
decomposition into low-rank and sparse matrices when such a distribution falls
within the exponential family. We present a novel alternating direction method
of multiplier optimization algorithm for efficient
decomposition. The effectiveness of is then demonstrated in
two applications: the first for steel sheet defect detection, and the second
for crime activity monitoring in the Atlanta metropolitan area
Scalable Group Level Probabilistic Sparse Factor Analysis
Many data-driven approaches exist to extract neural representations of
functional magnetic resonance imaging (fMRI) data, but most of them lack a
proper probabilistic formulation. We propose a group level scalable
probabilistic sparse factor analysis (psFA) allowing spatially sparse maps,
component pruning using automatic relevance determination (ARD) and subject
specific heteroscedastic spatial noise modeling. For task-based and resting
state fMRI, we show that the sparsity constraint gives rise to components
similar to those obtained by group independent component analysis. The noise
modeling shows that noise is reduced in areas typically associated with
activation by the experimental design. The psFA model identifies sparse
components and the probabilistic setting provides a natural way to handle
parameter uncertainties. The variational Bayesian framework easily extends to
more complex noise models than the presently considered.Comment: 10 pages plus 5 pages appendix, Submitted to ICASSP 1
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