WW hinge loss and so on.

The support vector machine (SVM) and minimum Euclidean norm least squares regression are two fundamentally different approaches to fitting linear models, but they have recently been connected in models for very high-dimensional data through a phenomenon of support vector proliferation, where every training example used to fit an SVM becomes a support vector. In this paper, we explore the generality of this phenomenon and make the following contributions. First, we prove a super-linear lower bound on the dimension (in terms of sample size) required for support vector proliferation in independent feature models, matching the upper bounds from previous works. We further identify a sharp phase transition in Gaussian feature models, bound the width of this transition, and give experimental support for its universality. Finally, we hypothesize that this phase transition occurs only in much higher-dimensional settings in the $\ell_1$ variant of the SVM, and we present a new geometric characterization of the problem that may elucidate this phenomenon for the general $\ell_p$ case.

Support vector machines are widely used in machine learning classification tasks, but traditional SVM models suffer from sensitivity to outliers and instability in resampling, which limits their performance in practical applications. To address these issues, this paper proposes a novel rescaled Huberized pinball loss function with asymmetric, non-convex, and smooth properties. Based on this loss function, we develop a corresponding SVM model called RHPSVM (Rescaled Huberized Pinball Loss Support Vector Machine). Theoretical analyses demonstrate that RHPSVM conforms to Bayesian rules, has a strict generalization error bound, a bounded influence function, and controllable optimality conditions, ensuring excellent classification accuracy, outlier insensitivity, and resampling stability. Additionally, RHPSVM can be extended to various advanced SVM variants by adjusting parameters, enhancing its flexibility. We transform the non-convex optimization problem of RHPSVM into a series of convex subproblems using the concave-convex procedure (CCCP) and solve it with the ClipDCD algorithm, which is proven to be convergent. Experimental results on simulated data, UCI datasets, and small-sample crop leaf image classification tasks show that RHPSVM outperforms existing SVM models in both noisy and noise-free scenarios, especially in handling high-dimensional small-sample data.

Despite this difference in formulation, we recast adversarial classification under zero-one loss as an ERM method with a novel prescribed loss function.

Those points in the first two sets are classified to a single class as by traditional classifiers.

The hard-margin SVM is a linear classification model that finds the separating hyperplane that maximizes the minimum margin of error for every training sample.

Empirical risk minimization (ERM) may be used to generalize the result from train to test data.

Equation discovery provides a grey-box approach to system identification by uncovering governing dynamics directly from observed data. However, a persistent challenge lies in ensuring that identified models generalise across operating conditions rather than over-fitting to specific datasets. This work investigates this issue by applying a Bayesian relevance vector machine (RVM) within a multi-task learning (MTL) framework for simultaneous parameter identification across multiple datasets. In this formulation, responses from the same structure under different excitation levels are treated as related tasks that share model parameters but retain task-specific noise characteristics. A simulated single degree-of-freedom oscillator with linear and cubic stiffness provided the case study, with datasets generated under three excitation regimes. Standard single-task RVM models were able to reproduce system responses but often failed to recover the true governing terms when excitations insufficiently stimulated non-linear dynamics. By contrast, the MTL-RVM combined information across tasks, improving parameter recovery for weakly and moderately excited datasets, while maintaining strong performance under high excitation. These findings demonstrate that multi-task Bayesian inference can mitigate over-fitting and promote generalisation in equation discovery. The approach is particularly relevant to structural health monitoring, where varying load conditions reveal complementary aspects of system physics.