SRPCA: Sparse Reverse of Principal Component Analysis for Fast Low-Rank Matrix Completion

Wael Hassanieh, Abdallah Chehade

Keywords: matrix completion, low rank, PCA, collaborative filtering, image inpainting, time-series imputation

Abstract:

Supervised and unsupervised learning methods experience a decline in performance when applied to incomplete, corrupted, or noisy datasets. Matrix completion is a common task to impute the missing values in sparsely observed matrices. Given a matrix $\mathbf{X} \in \mathbb{R}^{m \times n}$, low-rank matrix completion computes a rank-$r$ approximation of $\mathbf{X}$, where $r\ll\min\\{m,n\\}$, by only observing a few random entries of $\mathbf{X}$. It is commonly applied for recommender systems, image processing, and multi-output collaborative modeling. Existing matrix completion methods suffer either from slow convergence or failure under significant missing data levels.
This paper proposes a novel approach, the Sparse Reverse of Principal Component Analysis (SRPCA), that reformulates matrix factorization based low-rank completion $(\min_{\mathbf{U},\mathbf{V}}\Vert\mathcal{P}_{\mathbf{\Omega}}(\mathbf{X}-\mathbf{U}\mathbf{V}^T)\Vert_F^2)$
to iteratively learn a single low-rank subspace representation by solving the convex optimization problem
$\min\_{\mathbf{V}}\Vert\mathcal{P}\_{\mathbf{\Omega}}(\mathbf{X}-\mathbf{P}\mathbf{V}^T)\Vert_F^2$ under the principal component analysis framework, resulting in a significant convergence acceleration. SRPCA converges iteratively and is computationally tractable with a proven controllable upper bound on the number of iterations until convergence. Unlike existing matrix completion algorithms, the proposed SRPCA applies iterative pre-processing resets that maintain smoothness across the reconstructed matrix, which results in a performance boost for smooth matrices. The performance of the proposed technique is validated on case studies for image processing, multivariate time-series imputation, and collaborative filtering. SRPCA is also compared with state-of-the-art benchmarks for matrix completion.

Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

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Submission Number: 13518