提出一种新型矩阵分解方法，能更准确稳定地低秩逼近。
Structured Variational $D$-Decomposition for Accurate and Stable Low-Rank Approximation
- 用变分法优化非正交分解P D Q，可控制秩、稀疏性和条件数。
- 在电影评分、手写数字等数据上，重建精度优于传统方法。
- 适合处理带噪声或稀疏的数据，计算效率高，适合大规模应用。
我们提出D-分解，一种非正交矩阵分解形式A ≈ P D Q，其中P ∈ ℝ^(n×k)，D ∈ ℝ^(k×k)，Q ∈ ℝ^(k×n)。该分解通过最小化正则化的Frobenius损失进行变分定义，可调控秩、稀疏性与条件数。不同于LU或SVD等代数分解，其通过交替最小化求解。我们证明了解的存在性与扰动稳定性，且每次更新复杂度为𝒪(n²k)。在MovieLens、MNIST、Olivetti Faces和基因表达矩阵上的基准测试表明，相比截断SVD、CUR和非负矩阵分解，其在稀疏和噪声环境下重建精度显著提升。
原文摘要 · Abstract (English)
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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