Our problem can be understood as a two-sided NP classification problem.

To demonstrate the utility ofPREF-SHAP, we apply our method to a variety of synthetic and real-worlddatasets andshowthatricher andmoreinsightful explanations canbe obtainedoverthebaseline.

Kernel methods represent a fundamental class of machine learning techniques and have gained widespread adoption across diverse domains [32], including computer vision [22, 13], natural language processing (NLP) [36, 4], and bioinformatics [29], among others. The core idea underlying kernel methods is to employ a kernel function that implicitly maps the input data into a high-dimensional feature space, thereby enabling the use of linear models to solve nonlinear learning tasks in the original space. Consequently, the selection of an appropriate kernel function is critical to the performance of the method. Traditional kernel methods predominantly rely on positive definite (PD) kernels, such as the polynomial kernel and the Gaussian kernel. According to Mercer's Theorem, a PD kernel ensures that the resulting kernel matrix is positive semidefinite (PSD), thereby facilitating the analysis of the learning problem within the framework of reproducing kernel Hilbert spaces (RKHS) [9]. The PSD property guarantees that the corresponding optimization problem is convex and thus tractable. These authors contributed equally to this work.

In comparison to Figure 6, we see that ZN is no longer the top predictor.

First-order methods in convex optimization offer low per-iteration cost but often suffer from slow convergence, while second-order methods achieve fast local convergence at the expense of costly Hessian inversions. In this paper, we highlight a middle ground: minimizing a quadratic majorant with fixed curvature at each iteration. This strategy strikes a balance between per-iteration cost and convergence speed, and crucially allows the reuse of matrix decompositions, such as Cholesky or spectral decompositions, across iterations and varying regularization parameters. We introduce the Quadratic Majorization Minimization with Extrapolation (QMME) framework and establish its sequential convergence properties under standard assumptions. The new perspective of our analysis is to center the arguments around the induced norm of the curvature matrix $H$. To demonstrate practical advantages, we apply QMME to large-scale kernel regularized learning problems. In particular, we propose a novel Sylvester equation modelling technique for kernel multinomial regression. In Julia-based experiments, QMME compares favorably against various established first- and second-order methods. Furthermore, we demonstrate that our algorithms complement existing kernel approximation techniques through more efficiently handling sketching matrices with large projection dimensions. Our numerical experiments and real data analysis are available and fully reproducible at https://github.com/qhengncsu/QMME.jl.

We propose Kernel Logistic Regression (KLR) learning. Unlike linear methods, KLR uses kernels to implicitly map patterns to high-dimensional feature space, enhancing separability. By learning dual variables, KLR dramatically improves storage capacity, achieving perfect recall even when pattern numbers exceed neuron numbers (up to ratio 1.5 shown), and enhances noise robustness. KLR demonstrably outperforms Hebbian and linear logistic regression approaches.

The support vector machine (SVM) is known for its good performance in binary classification, but its extension to multi-class classification is still an on-going research issue. In this paper, we propose a new approach for classification, called the import vector machine (IVM), which is built on kernel logistic regression (KLR). We show that the IVM not only per- forms as well as the SVM in binary classification, but also can naturally be generalized to the multi-class case. Furthermore, the IVM provides an estimate of the underlying probability. Similar to the "support points" of the SVM, the IVM model uses only a fraction of the training data to index kernel basis functions, typically a much smaller fraction than the SVM.

Traditionally, kernel learning methods requires positive definitiveness on the kernel, which is too strict and excludes many sophisticated similarities, that are indefinite, in multimedia area. To utilize those indefinite kernels, indefinite learning methods are of great interests. This paper aims at the extension of the logistic regression from positive semi-definite kernels to indefinite kernels. The model, called indefinite kernel logistic regression (IKLR), keeps consistency to the regular KLR in formulation but it essentially becomes non-convex. Thanks to the positive decomposition of an indefinite matrix, IKLR can be transformed into a difference of two convex models, which follows the use of concave-convex procedure. Moreover, we employ an inexact solving scheme to speed up the sub-problem and develop a concave-inexact-convex procedure (CCICP) algorithm with theoretical convergence analysis. Systematical experiments on multi-modal datasets demonstrate the superiority of the proposed IKLR method over kernel logistic regression with positive definite kernels and other state-of-the-art indefinite learning based algorithms.

In this paper, we study the problem of large-scale Kernel Logistic Regression (KLR). A straightforward approach is to apply stochastic approximation to KLR. We refer to this approach as non-conservative online learning algorithm because it updates the kernel classifier after every received training example, leading to a dense classifier. To improve the sparsity of the KLR classifier, we propose two conservative online learning algorithms that update the classifier in a stochastic manner and generate sparse solutions. With appropriately designed updating strategies, our analysis shows that the two conservative algorithms enjoy similar theoretical guarantee as that of the non-conservative algorithm. Empirical studies on several benchmark data sets demonstrate that compared to batch-mode algorithms for KLR, the proposed conservative online learning algorithms are able to produce sparse KLR classifiers, and achieve similar classification accuracy but with significantly shorter training time. Furthermore, both the sparsity and classification accuracy of our methods are comparable to those of the online kernel SVM.