基于核心条件的正则化矩阵三因子分解用于高维结构系统
Core-Conditioned Regularized Matrix Tri-Factorization for High-Dimensional Structured Systems
AI总结：
本文研究了正则化的矩阵三因子分解，探讨了在正则化和核心条件设置下的重建精度、因子尺度、数值条件等，通过分析建立了PDQ表示的代数范围，证明了在强制正则化下的最小化器存在性，并提供了二次全数据目标的块更新方法。
摘要：
本文研究了一种正则化的矩阵三因子分解$A\approx PDQ$，其中$P$和$Q$为边因子，$D$为核心，其条件可以显式正则化或约束。该方法是一种结构化低秩近似框架，而非LU、QR、Cholesky或奇异值分解的替代方案。在无正则化的全数据Frobenius秩$r$问题中，截断SVD仍是最优基准。本文的贡献在于正则化和核心条件设置下，研究了重建精度、因子尺度、数值条件、扰动行为和加权近似。分析建立了PDQ表示的代数范围，证明了在强制正则化下的最小化器存在性，识别了由潜在空间变换引起的非唯一性，推导了二次全数据目标的块更新方法，并给出了产品级扰动界。对于全数据二次情况下的精确交替最小化，证明了下降性、迭代的有界性和在标准Kurdyka--Łojasiewicz假设下收敛到临界点。全多种子验证表明，在噪声和病态低秩近似中表现竞争，同时报告了纯谱基线未提供的诊断信息，包括学习到的核心条件数和块系统条件数。验证还澄清了该方法的局限性：随机SVD在纯谱压缩中仍更快，而当前加权缺失项变体在矩阵补全基线中不具普遍竞争力。因此，该框架应视为一种正则化且诊断透明的三因子分解方法，适用于需联合控制近似质量和数值条件的场景。
英文摘要：
This paper studies a regularized matrix tri-factorization \(A\approx PDQ\), where \(P\) and \(Q\) are side factors and \(D\) is a central core whose conditioning can be explicitly regularized or constrained. The formulation is a structured low-rank approximation framework, not a replacement for LU, QR, Cholesky, or the singular value decomposition. In the unregularized full-data Frobenius rank-\(r\) problem, truncated SVD remains the optimal benchmark. The contribution here concerns the regularized and core-conditioned setting, where reconstruction accuracy is treated together with factor scale, numerical conditioning, perturbation behavior, and weighted approximation. The analysis establishes the algebraic scope of the \(PDQ\) representation, proves existence of minimizers under coercive regularization, identifies the non-uniqueness induced by latent-space transformations, derives well-posed block updates for the quadratic full-data objective, and gives product-level perturbation bounds. For exact alternating minimization in the full-data quadratic case, it proves descent, boundedness of iterates, and convergence to a critical point under standard Kurdyka--Łojasiewicz assumptions. A full multi-seed validation indicates competitive behavior in noisy and ill-conditioned low-rank approximation while reporting diagnostics not provided by purely spectral baselines, including the learned core condition number and block-system conditioning. The validation also clarifies the method's limits: randomized SVD remains faster for pure spectral compression, and the current weighted missing-entry variant is not uniformly competitive with matrix-completion baselines. The framework is therefore best viewed as a regularized and diagnostically transparent tri-factorization for settings where approximation quality and numerical conditioning must be controlled jointly.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
