Heaviside Low-Rank Support Matrix Machine

The support vector machine (SVM) is a widely used classifier, but choosing an appropriate loss function remains difficult. Convex losses such as the hinge loss and least-squares loss are sensitive to outliers, while bounded non-convex losses often lead to high computational cost. To address this, we propose a hybrid truncated loss function ($L_{\mathrm{ht}}$) that is both sparse and bounded, and build the $L_{\mathrm{ht}}$-SVM model for single-view classification. We introduce the P-stationary point and use it to establish the first-order necessary and sufficient optimality conditions. Based on these conditions, we design an alternating direction method of multipliers with a working-set strategy that reduces computational cost and achieves global convergence. We further extend $L_{\mathrm{ht}}$-SVM to multi-view learning by adding structural information and view weights, resulting in Mv$L_{\mathrm{ht}}$-SVM, which follows both the consensus and complementarity principles. Experiments on synthetic, real-world, and image datasets show that $L_{\mathrm{ht}}$-SVM achieves higher accuracy with fewer support vectors and better noise robustness than five single-view methods, while Mv$L_{\mathrm{ht}}$-SVM outperforms six multi-view methods in accuracy, precision, recall, and F1-score.

In this paper, we consider asymptotic properties of the support vector machine (SVM) in high-dimension, low-sample-size (HDLSS) settings under a spiked model. The existing theory of the SVM in the HDLSS context relies on the geometric representation of HDLSS data, which requires that the eigenvalues of the covariance matrices are not dominant. We first show that the geometric representation does not hold under the spiked model. We show that the Gram matrix of HDLSS data converges in distribution to a random matrix, namely, the HDLSS data converge to a random configuration in a finite-dimensional space whose dimension is given by the number of the spikes. We show that the misclassification rates of the SVM do not tend to zero, that is, the SVM does not hold the consistency property. We also show that the bias-corrected SVM (BC-SVM) does not give preferable performance in this setting because the bias term itself should be modified. In order to overcome such difficulties, we propose a spike-corrected SVM (SC-SVM). We show that the SC-SVM holds the consistency property when the sample size goes to infinity, and that the growth of the sample size is essential in the sense that any projection-based procedure fails when the sample size is fixed. Finally, we check the performance of the classifiers by numerical simulations.

Nonconvex methods have emerged as a dominant approach for low-rank matrix estimation, a problem that arises widely in machine learning and AI for learning and representing high-dimensional data. Existing analyses for these methods often require additional regularization to mitigate nonconvexity, even though such regularization is often unnecessary in practice. Moreover, most analyses rely on problem-specific arguments that are difficult to generalize to more complex settings. In this paper, we develop a theoretical framework for studying nonconvex procedures across a broad class of low-rank matrix estimation problems. Rather than focusing on a specific model, we reveal a fundamental mechanism that explains why nonconvex procedures can behave well in low-rank estimation. Our key device is a {\it benign regularizer} that does not alter the original update rule, but yields an equivalent locally strongly convex formulation of the algorithm. This perspective uncovers a disguised convexity inherent in the nonconvex procedure and provides a new route to theoretical guarantees for nonconvex low-rank matrix estimation.