矩阵三分解中的稀疏诱导可识别性
Sparsity Induced Identifiability in Matrix Tri-Factorisation
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中文总结 AI 辅助
本文针对一般实值矩阵三分解，开展稀疏诱导可识别性的严格理论研究，提出新颖分解策略推导恢复保证等结果，经蒙特卡洛实验验证，填补了相关理论空白。
中文摘要 AI 辅助
矩阵分解是挖掘高维数据低维结构的基础工具，应用于数据压缩、去噪、结构发现、可解释表示学习和降维等场景。与传统双因子模型相比，矩阵三分解提供了更强的建模灵活性，而稀疏约束通常可同时提升可解释性与恢复性能。尽管稀疏性在双因子矩阵分解中的作用已被广泛研究，但针对一般实值矩阵三分解的严格理论保证仍大多未被探索。为填补这一空白，我们开展了据我们所知首个针对一般实值矩阵三分解中稀疏诱导可识别性的严格理论研究。我们的分析得益于一种新颖的分解策略，该策略将原问题转化为两个耦合的辅助分解问题，同时保留了从观测值恢复原始因子矩阵所需的结构信息。基于此分解，我们推导了恢复保证与结构一致性结果，这些结果刻画了系数稀疏性如何影响充分恢复条件、收敛行为、谱近似误差、高概率界及结构保留情况。全面的蒙特卡洛实验验证了所提理论，证明理论结果与经验观测值高度一致。
英文摘要
Matrix factorisation is a fundamental tool for exploiting low-dimensional structure in high-dimensional data, with applications such as data compression, denoising, structure discovery, interpretable representation learning, and dimensionality reduction. Compared to conventional two-factor models, matrix tri-factorisation provides greater modelling flexibility, while sparsity constraints often improve both interpretability and recovery performance. Although the role of sparsity has been extensively studied for two-factor matrix factorisation, rigorous theoretical guarantees for general real-valued matrix tri-factorisation remain largely unexplored. To address this gap, we establish, to the best of our knowledge, the first rigorous theoretical study for sparsity-induced identifiability in general real-valued matrix tri-factorisation. Our analysis is enabled by a novel decomposition strategy that transforms the original problem into two coupled auxiliary factorisation problems, while preserving the structural information necessary to the recovery of the original factor matrices from the observations. Building upon this decomposition, we derive recovery guarantees and structural consistency results that characterise how coefficient sparsity influences the sufficient recovery conditions, convergence behaviour, spectral approximation error, high-probability bounds, and structure preservation. Comprehensive Monte Carlo experiments validate the proposed theory and demonstrate close agreement between the theoretical results and empirical observations.
