GEMSS: A Variational Method for Discovering Multiple Sparse Solutions in Classification and Regression Problems

Abstract

In underdetermined regression and classification problems, multiple feature subsets often yield equivalent predictive performance. In applied settings, especially with $n \ll p$, high dimension or collinearities, it is valuable to provide a domain expert with a menu of statistically plausible explanations, rather than one arbitrary solution. This creates the need for appropriate methods. We present Gaussian Ensemble for Multiple Sparse Solutions (GEMSS), a method that uses a single variational mixture to approximate the corresponding multimodal posterior. Its evidence lower bound contains a built-in repulsion between the mixture's components, enabling the model to simultaneously produce several distinct sparse solutions. We evaluate GEMSS on a novel, reusable benchmark. The ground-truth solution set and its structure are known by construction and set-level recovery metrics are evaluated. GEMSS consistently outperforms dedicated multiplicity methods (Enumeration LASSO, ALFESE), two strong sampling baselines that approximate the same posterior (Randomized-LASSO ensemble, BB-SSL), and naive iterative masking. As solutions' overlap increases, the gap widens and additional ensemble restarts cannot close it. Only ALFESE proves competitive. Further, GEMSS is validated on real-world datasets, producing multiple distinct and highly predictive solutions: the practical goal that existing methods struggle to meet. The open-source Python package 'gemss' is available (github.com/kat-er-ina/gemss) and democratized through a free online application at huggingface.co/spaces/kat-er-ina/gemss.