Hierarchical clustering is a powerful tool for exploratory data analysis, organizing data into a tree of clusterings from which a partition can be chosen. This paper generalizes these ideas by proving that, for any reasonable hierarchy, one can optimally solve any center-based clustering objective over it (such as k-means). Moreover, these solutions can be found exceedingly quickly and are themselves necessarily hierarchical. Thus, given a cluster tree, we show that one can quickly access a plethora of new, equally meaningful hierarchies. Just as in standard hierarchical clustering, one can then choose any desired partition from these new hierarchies. We conclude by verifying the utility of our proposed techniques across datasets, hierarchies, and partitioning schemes.

Recent advances in false discovery rate (FDR)-controlled feature selection methods have improved reliability by effectively limiting false positives, making them wellsuited for complex applications. A popular FDR-controlled framework called data splitting uses the "mirror statistics" to select features. However, we find that the unit variance assumption on mirror statistics could potentially limit the feature selection power. To address this, we generalize the mirror statistics in the Gaussian mirror framework and introduce a new approach called "generalized Gaussian mirror" (G2M), which adaptively learns the variance and forms new test statistics. We demonstrate both theoretically and empirically that the proposed test statistics achieve higher power than those of Gaussian mirror and data splitting. Comparisons with other FDR-controlled frameworks on synthetic, semi-synthetic, and real datasets highlight the superior performance of the G2M method in achieving higher power while maintaining FDR control. These findings suggest the potential for the G2M method for practical applications in real-world problems. Code is available at: https://github.com/skyve2012/G2M.

Reinforcement learning (RL) in continuous state-action spaces remains challenging in scientific computing due to poor sample efficiency and lack of pathwise physical consistency. We introduce Differential Reinforcement Learning (Differential RL), a novel framework that reformulates RL from a continuous-time control perspective via a differential dual formulation. This induces a Hamiltonian structure that embeds physics priors and ensures consistent trajectories without requiring explicit constraints. To implement Differential RL, we develop Differential Policy Optimization (dfPO), a pointwise, stage-wise algorithm that refines local movement operators along the trajectory for improved sample efficiency and dynamic alignment. We establish pointwise convergence guarantees, a property not available in standard RL, and derive a competitive theoretical regret bound of O(K5/6). Empirically, dfPO outperforms standard RL baselines on representative scientific computing tasks, including surface modeling, grid control, and molecular dynamics, under low-data and physics-constrained conditions.

This paper studies sampling error bounds for denoising diffusion probabilistic models (DDPMs) in the 2-Wasserstein distance. Our contributions are threefold. (i) Under general Lipschitz-type conditions on the score function and for a broad class of variance schedules, including the cosine schedule, we establish sharp upper bounds that are optimal in both the dimension and the number of steps, and recover several sharp error bounds previously obtained in the literature. (ii) We prove that the same Lipschitz-type conditions, which encompass those commonly imposed on the (learned) score, imply a logarithmic Sobolev inequality and hence a quadratic transportation cost inequality for the DDPM. As a consequence, in settings covered by existing work, an optimal Wasserstein bound, up to a logarithmic factor, follows from the recently obtained sharp error bound in the Kullback-Leibler divergence under geometric-type variance schedules. (iii) We show that for general log-concave target distributions, the optimal Wasserstein error bound remains attainable even without a quadratic transportation cost inequality for the target. Our analysis is based on viewing the DDPM sampler as a discretization of the Föllmer process rather than the conventional reverse Ornstein-Uhlenbeck process.

Bayesian inference provides principled uncertainty quantification, but accurate posterior sampling with MCMC can be computationally prohibitive for modern applications. Variational inference (VI) offers a scalable alternative and often yields accurate predictive distributions, but cheap variational families such as mean-field (MF) can produce over-concentrated approximations that miss posterior dependence. We propose variational predictive resampling (VPR), a scalable posterior sampling method that exploits VI's predictive strength within a predictive-resampling framework to better approximate the Bayesian posterior. Given a prior-likelihood pair, VPR repeatedly imputes future observations from the current variational predictive, updates the variational approximation after each imputation, and records the parameter value implied by the completed sample. We establish conditions under which the law of the parameter returned by VPR is well defined and show that its finite-horizon approximation converges to this limit. In a tractable Gaussian location model, we show that VPR with MF variational predictives converges to the exact Bayesian posterior, whereas the optimal MF-VI approximation retains a non-vanishing asymptotic gap. Experiments on linear regression, logistic regression, and hierarchical linear mixed-effects models demonstrate that VPR substantially improves posterior uncertainty quantification and recovers posterior dependence missed by MF-VI, while remaining computationally competitive with, and often more efficient than, MCMC.

We proceed to show the sparsistency510 of the estimated parameters. First, suppose that Θ t;ij 6= 0 for some time tand index (i,j). Due to 0 < γ < 1, the above inequality implies that bΘt;ij = 0521 for every t and (i,j) 6 St, and bΘt;ij bΘt 1;ij = 0 for every t > 0 and (i,j) 6 Dt. The proof is inspired527 by Corollary 1 in [47]. First, we present the following key lemmas.528

In this paper, we investigate the question: Given a small number of datapoints, for example N = 30, how tight can PAC-Bayes and test set bounds be made? For such small datasets, test set bounds adversely affect generalisation performance by withholding data from the training procedure. In this setting, PAC-Bayes bounds are especially attractive, due to their ability to use all the data to simultaneouslylearn a posterior and bound its generalisation risk. We focus on the case of i.i.d.