提出一种约束子空间的二次矩阵分解法，有效捕捉数据低维结构。
Subspace-Constrained Quadratic Matrix Factorization: Algorithm and Applications
- 联合学习切空间、法向子空间与二次映射关系
- 在合成与真实数据集上优于现有方法，具强鲁棒性
- 适合低秩数据建模与流形学习任务
矩阵分解已成为建模低秩结构数据的通用框架。针对流形学习中的挑战，本文提出一种子空间约束的二次矩阵分解模型，旨在联合学习关键低维结构，包括切空间、法向子空间以及连接切空间与低维表示的二次形式。通过交替最小化求解该分解模型，深入研究非线性回归与投影子问题。同时分析了二次投影问题的理论性质及交替策略的收敛特性。为验证方法有效性，在合成数据和真实数据集上进行数值实验，结果表明所提模型优于现有方法，展现出在捕捉核心低维结构方面的鲁棒性与高效性。
原文摘要 · Abstract (English)
Matrix Factorization has emerged as a widely adopted framework for modeling data exhibiting low-rank structures. To address challenges in manifold learning, this paper presents a subspace-constrained quadratic matrix factorization model. The model is designed to jointly learn key low-dimensional structures, including the tangent space, the normal subspace, and the quadratic form that links the tangent space to a low-dimensional representation. We solve the proposed factorization model using an alternating minimization method, involving an in-depth investigation of nonlinear regression and projection subproblems. Theoretical properties of the quadratic projection problem and convergence characteristics of the alternating strategy are also investigated. To validate our approach, we conduct numerical experiments on synthetic and real-world datasets. Results demonstrate that our model outperforms existing methods, highlighting its robustness and efficacy in capturing core low-dimensional structures.
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