# Sparse Learning and Class Probability Estimation with Weighted Support Vector Machines

arXiv: https://arxiv.org/abs/2312.10618 · HTML: https://mlsanity.com/paper/2312.10618/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 49 references that are in the map.
- First posted on arXiv 17 Dec 2023.

## Builds on (most cited first, 4 of 4)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [OSQP: An Operator Splitting Solver for Quadratic Programs](https://mlsanity.com/paper/1711.08013/) (2017)
- [Linear Algorithms for Robust and Scalable Nonparametric Multiclass Probability Estimation](https://mlsanity.com/paper/2205.12460/) (2022)
- [Robust Brain MRI Image Classification with SIBOW-SVM](https://mlsanity.com/paper/2311.08908/) (2023)

## Built on by (most cited first, 1 of 1)

- [Binary Kernel Logistic Regression: a sparsity-inducing formulation and a convergent decomposition training algorithm](https://mlsanity.com/paper/2512.19440/) (2025)

## Abstract (arXiv)

Classification and probability estimation are fundamental tasks with broad applications across modern machine learning and data science, spanning fields such as biology, medicine, engineering, and computer science. Recent development of weighted Support Vector Machines (wSVMs) has demonstrated considerable promise in robustly and accurately predicting class probabilities and performing classification across a variety of problems (Wang et al., 2008). However, the existing framework relies on an ℓ2-norm regularized binary wSVMs optimization formulation, which is designed for dense features and exhibits limited performance in the presence of sparse features with redundant noise. Effective sparse learning thus requires prescreening of important variables for each binary wSVM to ensure accurate estimation of pairwise conditional probabilities. In this paper, we propose a novel class of wSVMs frameworks that incorporate automatic variable selection with accurate probability estimation for sparse learning problems. We developed efficient algorithms for variable selection by solving either the ℓ1-norm or elastic net regularized wSVMs optimization problems. Class probability is then estimated either via the ℓ2-norm regularized wSVMs framework applied to the selected variables, or directly through elastic net regularized wSVMs. The two-step approach offers a strong advantage in simultaneous automatic variable selection and reliable probability estimators with competitive computational efficiency. The elastic net regularized wSVMs achieve superior performance in both variable selection and probability estimation, with the added benefit of variable grouping, at the cost of increases compensation time for high dimensional settings. The proposed wSVMs-based sparse learning methods are broadly applicable and can be naturally extended to K-class problems through ensemble learning.