{"ID":2866446,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.19886","arxiv_id":"2509.19886","title":"Sparse Regularization by Smooth Non-separable Non-convex Penalty Function Based on Ultra-discretization Formula","abstract":"In sparse optimization, the $\\ell_{1}$ norm is widely adopted for its convexity, yet it often yields solutions with smaller magnitudes than expected. To mitigate this drawback, various non-convex sparse penalties have been proposed. Some employ non-separability, with ordered weighting as an effective example, to retain large components while suppressing small ones. Motivated by these approaches, we propose ULPENS, a non-convex, non-separable sparsity-inducing penalty function that enables control over the suppression of elements. Derived from the ultra-discretization formula, ULPENS can continuously interpolate between the $\\ell_{1}$ norm and a non-convex selective suppressing function by adjusting parameters inherent to the formula. With the formula, ULPENS is smooth, allowing the use of efficient gradient-based optimization algorithms. We establish key theoretical properties of ULPENS and demonstrate its practical effectiveness through numerical experiments.","short_abstract":"In sparse optimization, the $\\ell_{1}$ norm is widely adopted for its convexity, yet it often yields solutions with smaller magnitudes than expected. To mitigate this drawback, various non-convex sparse penalties have been proposed. Some employ non-separability, with ordered weighting as an effective example, to retain...","url_abs":"https://arxiv.org/abs/2509.19886","url_pdf":"https://arxiv.org/pdf/2509.19886v1","authors":"[\"Natsuki Akaishi\",\"Koki Yamada\",\"Kohei Yatabe\"]","published":"2025-09-24T08:34:49Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"eess.SP\"]","methods":"[]","has_code":false}