Structured Variational $D$-Decomposition for Accurate and Stable Low-Rank Approximation

• URL: http://arxiv.org/abs/2506.08535v1
• Date: Tue, 10 Jun 2025 07:59:34 GMT
• Title: Structured Variational $D$-Decomposition for Accurate and Stable Low-Rank Approximation
• Authors: Ronald Katende,
• Abstract summary: We introduce the $D$-decomposition, a non-orthogonal matrix factorization of the form $A approx P D Q$.<n>The decomposition is defined variationally by minimizing a regularized Frobenius loss.<n> Benchmarks against truncated SVD, CUR, and nonnegative matrix factorization show improved reconstruction accuracy on MovieLens, MNIST, Olivetti Faces, and gene expression matrices.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce the $D$-decomposition, a non-orthogonal matrix factorization of the form $A \approx P D Q$, where $P \in \mathbb{R}^{n \times k}$, $D \in \mathbb{R}^{k \times k}$, and $Q \in \mathbb{R}^{k \times n}$. The decomposition is defined variationally by minimizing a regularized Frobenius loss, allowing control over rank, sparsity, and conditioning. Unlike algebraic factorizations such as LU or SVD, it is computed by alternating minimization. We establish existence and perturbation stability of the solution and show that each update has complexity $\mathcal{O}(n^2k)$. Benchmarks against truncated SVD, CUR, and nonnegative matrix factorization show improved reconstruction accuracy on MovieLens, MNIST, Olivetti Faces, and gene expression matrices, particularly under sparsity and noise.

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