{"ID":2827051,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.17473","arxiv_id":"2512.17473","title":"Alternating Direction Method of Multipliers for Nonlinear Matrix Decompositions","abstract":"We present an algorithm based on the alternating direction method of multipliers (ADMM) for solving nonlinear matrix decompositions (NMD). Given an input matrix $X \\in \\mathbb{R}^{m \\times n}$ and a factorization rank $r \\ll \\min(m, n)$, NMD seeks matrices $W \\in \\mathbb{R}^{m \\times r}$ and $H \\in \\mathbb{R}^{r \\times n}$ such that $X \\approx f(WH)$, where $f$ is an element-wise nonlinear function. We evaluate our method on several representative nonlinear models: the rectified linear unit activation $f(x) = \\max(0, x)$, suitable for nonnegative sparse data approximation, the component-wise square $f(x) = x^2$, applicable to probabilistic circuit representation, and the MinMax transform $f(x) = \\min(b, \\max(a, x))$, relevant for recommender systems. The proposed framework flexibly supports diverse loss functions, including least squares, $\\ell_1$ norm, and the Kullback-Leibler divergence, and can be readily extended to other nonlinearities and metrics. We illustrate the applicability, efficiency, and adaptability of the approach on real-world datasets, highlighting its potential for a broad range of applications.","short_abstract":"We present an algorithm based on the alternating direction method of multipliers (ADMM) for solving nonlinear matrix decompositions (NMD). Given an input matrix $X \\in \\mathbb{R}^{m \\times n}$ and a factorization rank $r \\ll \\min(m, n)$, NMD seeks matrices $W \\in \\mathbb{R}^{m \\times r}$ and $H \\in \\mathbb{R}^{r \\times...","url_abs":"https://arxiv.org/abs/2512.17473","url_pdf":"https://arxiv.org/pdf/2512.17473v2","authors":"[\"Atharva Awari\",\"Nicolas Gillis\",\"Arnaud Vandaele\"]","published":"2025-12-19T11:40:06Z","proceeding":"eess.SP","tasks":"[\"eess.SP\",\"cs.LG\",\"math.OC\",\"stat.ML\"]","methods":"[]","has_code":false}