On the Estimation Performance of Generalized Power Method for Heteroscedastic Probabilistic PCA

• URL: http://arxiv.org/abs/2312.03438v1
• Date: Wed, 6 Dec 2023 11:41:17 GMT
• Title: On the Estimation Performance of Generalized Power Method for Heteroscedastic Probabilistic PCA
• Authors: Jinxin Wang, Chonghe Jiang, Huikang Liu, Anthony Man-Cho So
• Abstract summary: We show that, given a suitable iterate between the GPMs generated by at least geometrically bound to some threshold, the GPMs decrease to some threshold with the residual part of certain "-resi decomposition" In this paper, we demonstrate the superior performance with sub-Gaussian noise settings using the PCA technique.
• Score: 21.9585534723895
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The heteroscedastic probabilistic principal component analysis (PCA) technique, a variant of the classic PCA that considers data heterogeneity, is receiving more and more attention in the data science and signal processing communities. In this paper, to estimate the underlying low-dimensional linear subspace (simply called \emph{ground truth}) from available heterogeneous data samples, we consider the associated non-convex maximum-likelihood estimation problem, which involves maximizing a sum of heterogeneous quadratic forms over an orthogonality constraint (HQPOC). We propose a first-order method -- generalized power method (GPM) -- to tackle the problem and establish its \emph{estimation performance} guarantee. Specifically, we show that, given a suitable initialization, the distances between the iterates generated by GPM and the ground truth decrease at least geometrically to some threshold associated with the residual part of certain "population-residual decomposition". In establishing the estimation performance result, we prove a novel local error bound property of another closely related optimization problem, namely quadratic optimization with orthogonality constraint (QPOC), which is new and can be of independent interest. Numerical experiments are conducted to demonstrate the superior performance of GPM in both Gaussian noise and sub-Gaussian noise settings.

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