External Division of Two Bregman Proximity Operators for Poisson Inverse Problems

This paper presents a novel method for recovering sparse vectors from linear models corrupted by Poisson noise. The contribution is twofold. First, an operator defined via the external division of two Bregman proximity operators is introduced to promote sparse solutions while mitigating the estimation bias induced by classical $\ell_1$-norm regularization. This operator is then embedded into the already established NoLips algorithm, replacing the standard Bregman proximity operator in a plug-and-play manner. Second, the geometric structure of the proposed external-division operator is elucidated through two complementary reformulations, which provide clear interpretations in terms of the primal and dual spaces of the Poisson inverse problem. Numerical tests show that the proposed method exhibits more stable convergence behavior than conventional Kullback-Leibler (KL)-based approaches and achieves significantly superior performance on synthetic data and an image restoration problem.

Nonconvex Regularization for Feature Selection in Reinforcement Learning

This work proposes an efficient batch algorithm for feature selection in reinforcement learning (RL) with theoretical convergence guarantees. To mitigate the estimation bias inherent in conventional regularization schemes, the first contribution extends policy evaluation within the classical least-squares temporal-difference (LSTD) framework by formulating a Bellman-residual objective regularized with the sparsity-inducing, nonconvex projected minimax concave (PMC) penalty. Owing to the weak convexity of the PMC penalty, this formulation can be interpreted as a special instance of a general nonmonotone-inclusion problem. The second contribution establishes novel convergence conditions for the forward-reflected-backward splitting (FRBS) algorithm to solve this class of problems. Numerical experiments on benchmark datasets demonstrate that the proposed approach substantially outperforms state-of-the-art feature-selection methods, particularly in scenarios with many noisy features.

Linearly-involved Moreau-Enhanced-over-Subspace Model: Debiased Sparse Modeling and Stable Outlier-Robust Regression

We present an efficient mathematical framework based on the linearly-involved Moreau-enhanced-over-subspace (LiMES) model. Two concrete applications are considered: sparse modeling and robust regression. The popular minimax concave (MC) penalty for sparse modeling subtracts, from the $\ell_1$ norm, its Moreau envelope, inducing nearly unbiased estimates and thus yielding remarkable performance enhancements. To extend it to underdetermined linear systems, we propose the projective minimax concave penalty using the projection onto the input subspace, where the Moreau-enhancement effect is restricted to the subspace for preserving the overall convexity. We also present a novel concept of stable outlier-robust regression which distinguishes noise and outlier explicitly. The LiMES model encompasses those two specific examples as well as two other applications: stable principal component pursuit and robust classification. The LiMES function involved in the model is an ``additively nonseparable'' weakly convex function but is defined with the Moreau envelope returning the minimum of a ``separable'' convex function. This mixed nature of separability and nonseparability allows an application of the LiMES model to the underdetermined case with an efficient algorithmic implementation. Two linear/affine operators play key roles in the model: one corresponds to the projection mentioned above and the other takes care of robust regression/classification. A necessary and sufficient condition for convexity of the smooth part of the objective function is studied. Numerical examples show the efficacy of LiMES in applications to sparse modeling and robust regression.