用阶梯损失提升矩阵分类鲁棒性，同时保留全局结构。
Heaviside Low-Rank Support Matrix Machine
- 采用阶梯损失替代传统损失函数，增强抗噪能力。
- 引入低秩约束，准确捕捉数据全局结构特征。
- 算法各子问题有闭式解，适合处理高维矩阵数据。
支持矩阵机（SMM）是一种新兴的分类框架，可直接处理矩阵结构观测数据，避免向量化带来的空间相关性破坏。然而，现有大多数SMM变体依赖凸或非凸代理损失函数，对噪声敏感。为此，本文提出一种新型的阶梯低秩支持矩阵机（HL-SMM），采用阶梯损失而非常见的铰链或坡道损失以提升鲁棒性。同时，引入低秩约束以精确刻画数据内在全局结构。理论上，分析了KKT点并严格证明了充分必要条件。算法上，设计了一种高效的近端交替最小化（PAM）方案，所有子问题均有闭式解。在多个基准数据集上的大量实验表明，所提HL-SMM在分类准确率和鲁棒性方面均优于现有先进方法。
原文摘要 · Abstract (English)
Support matrix machine (SMM) is an emerging classification framework that directly handles matrix-structured observations, thereby avoiding the spatial correlations destroyed by vectorization. However, most existing SMM variants rely on convex or nonconvex surrogate loss functions, which may lead to high sensitivity to noise. To address this issue, we propose a novel Heaviside low-rank SMM model called HL-SMM, which leverages the Heaviside loss instead of the common hinge or ramp losses for robustness. Moreover, the low-rank constraint is adopted to accurately characterize the inherent global structure. In theory, we analyze the Karush-Kuhn-Tucker (KKT) points and rigorously prove the sufficient and necessary conditions. In algorithms, we develop an effective proximal alternating minimization (PAM) scheme, where all subproblems have closed-form solutions. Extensive experiments on benchmark datasets validate that the proposed HL-SMM achieves superior classification accuracy and robustness compared to state-of-the-art methods.
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