Mathematical foundations for imaging, learning, and optimization.

Plug-and-Play (PnP) algorithms are a class of iterative methods for inverse imaging. Within an optimization algorithm, they combine a flexible fidelity term, encoding the forward operator, and a pretrained image denoiser, in order to deal with more severe corruptions such as blurring or downsampling when reconstructing an image. This work studies provably convergent PnP methods for mixed-noise forward processes by using the infimal convolution as a fidelity term, providing a statistical interpretation as a joint maximum a-posteriori estimator over both noises, and preserving the Bayesian MAP interpretation of PnP methods. Independently, we extend the PnP formulation to multiple prior terms using the Davis–Yin three-operator splitting. This extension can be combined with either standard fidelities or the proposed mixed-noise infimal-convolution fidelities. We verify that these generalized PnP methods are convergent under standard Kurdyka–Lojasiewicz conditions. Numerical experiments on Laplace-Gaussian and Poisson–Gaussian noise demonstrate stable convergence of single-prior and multiple-prior PnP methods, and divergence under fidelity mismatch. Furthermore, PnP with infimal convolution fidelities are able to scale to noise up to 38% standard deviation, with multiple priors reaching different stationary points that qualitatively preserve more textural properties.

Optimal transport (OT) compares distributions and aligns datasets in machine learning, yet unregularized discrete OT requires a linear program with quadratically many transport variables. We propose HELLO, a hierarchical solver that casts large-scale discrete OT as edge localization and uses dual potentials to guide both coarse-to-fine initialization and within-level refinement. Initialization propagates coarse dual potentials across a recursive subsampling hierarchy to assign candidate edges. Refinement then iteratively inserts the largest dual violators in each row and column until the relative KKT residual meets a prescribed tolerance, while budgeted pruning ensures linear memory complexity. For exact-arithmetic refinement, we prove finite termination at a global optimum under a symbolic lexicographic rule. At the million-point scale, HELLO attains lower transport objectives with order-of-magnitude runtime improvements over strong baselines across feature dimensions from single digits to thousands. It further scales to 1.28 million samples per marginal in 8192 dimensions on a single H100, using 41.6 GiB peak GPU memory while satisfying a full relative KKT residual below 10^-6. Beyond standard discrete OT, the framework supports general pairwise costs and serves as a scalable balanced-OT oracle for semi-discrete OT, Gromov–Wasserstein, unbalanced OT, and OT-based Flow Matching.

Reinforcement learning (RL), particularly RL with Verifiable Rewards (RLVR), has recently emerged as a central paradigm for enhancing large language models’ (LLMs) reasoning abilities, demonstrating remarkable effectiveness across reasoning tasks. Recent studies suggest that high-entropy tokens play an exceptionally important role in model training, since training with only the highest 20% entropy tokens yields significant performance gains. However, why such high-entropy tokens are beneficial remains insufficiently understood. In this work, we find that although high-entropy tokens within one answer tend to correlate with large gradient magnitude, entropy alone fails to consistently reflect token importance across different answers, considering the variations in the answer-level reward signals. Based on this observation, we introduce the Gradient Magnitude-based Token Selection (GMTS) method to quantify token importance, which leverages the entropy-gradient connection to approximate gradient-magnitude rankings for token selection. We find that training on the top 20% tokens ranked by GMTS consistently outperforms entropy-based token selection across three reasoning domains and various model sizes, suggesting that GMTS provides a more fine-grained estimate of token contribution for RLVR training.

Typical dimension reduction techniques for nonoverlapping sparse optimization involve screening or sieving strategies based on a dual certificate derived from the first-order optimality condition, approximating the gradients or exploiting certain inherent low-dimensional structure of the sparse solution. In comparison, dimension reduction rules for overlapping group sparsity are generally less developed because the subgradient structure is more complex, making the link between sparsity pattern and the dual variable indirect due to the non-separability. In this work, we propose new dual certificates for overlapping group sparsity and a novel adaptive scheme for identifying the support of the overlapping group LASSO. We demonstrate how this scheme can be integrated into and significantly accelerate existing algorithms, including Primal-Dual splitting method, alternating direction method of multipliers and a recently developed variable projection scheme based on over-parameterization. We provide convergence analysis of the method and verify its practical effectiveness through experiments on standard datasets.