{"ID":2878454,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.17783","arxiv_id":"2508.17783","title":"Algebraic Approach to Ridge-Regularized Mean Squared Error Minimization in Minimal ReLU Neural Network","abstract":"This paper investigates a perceptron, a simple neural network model, with ReLU activation and a ridge-regularized mean squared error (RR-MSE). Our approach leverages the fact that the RR-MSE for ReLU perceptron is piecewise polynomial, enabling a systematic analysis using tools from computational algebra. In particular, we develop a Divide-Enumerate-Merge strategy that exhaustively enumerates all local minima of the RR-MSE. By virtue of the algebraic formulation, our approach can identify not only the typical zero-dimensional minima (i.e., isolated points) obtained by numerical optimization, but also higher-dimensional minima (i.e., connected sets such as curves, surfaces, or hypersurfaces). Although computational algebraic methods are computationally very intensive for perceptrons of practical size, as a proof of concept, we apply the proposed approach in practice to minimal perceptrons with a few hidden units.","short_abstract":"This paper investigates a perceptron, a simple neural network model, with ReLU activation and a ridge-regularized mean squared error (RR-MSE). Our approach leverages the fact that the RR-MSE for ReLU perceptron is piecewise polynomial, enabling a systematic analysis using tools from computational algebra. In particular...","url_abs":"https://arxiv.org/abs/2508.17783","url_pdf":"https://arxiv.org/pdf/2508.17783v1","authors":"[\"Ryoya Fukasaku\",\"Yutaro Kabata\",\"Akifumi Okuno\"]","published":"2025-08-25T08:24:20Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.AI\",\"cs.LG\",\"stat.CO\"]","methods":"[]","has_code":false}