We prove that conditional diffusion models whose reverse kernels are finite Gaussian mixtures with ReLU-network logits can approximate suitably regular target distributions arbitrarily well in context-averaged conditional KL divergence, up to an irreducible terminal mismatch that typically vanishes with increasing diffusion horizon. A path-space decomposition reduces the output error to this mismatch plus per-step reverse-kernel errors; assuming each reverse kernel factors through a finite-dimensional feature map, each step becomes a static conditional density approximation problem, solved by composing Norets' Gaussian-mixture theory with quantitative ReLU bounds. Under exact terminal matching the resulting neural reverse-kernel class is dense in conditional KL.

Estimating optimal individualized treatment rules (ITRs) via outcome weighted learning (OWL) often relies on observed rewards that are noisy or optimistic proxies for the true latent utility. Ignoring this reward uncertainty leads to the selection of policies with inflated apparent performance, yet existing OWL frameworks lack the finite-sample guarantees required to systematically embed such uncertainty into the learning objective. To address this issue, we propose PAC-Bayesian Reward-Certified Outcome Weighted Learning (PROWL). Given a one-sided uncertainty certificate, PROWL constructs a conservative reward and a strictly policy-dependent lower bound on the true expected value. Theoretically, we prove an exact certified reduction that transforms robust policy learning into a unified, split-free cost-sensitive classification task. This formulation enables the derivation of a nonasymptotic PAC-Bayes lower bound for randomized ITRs, where we establish that the optimal posterior maximizing this bound is exactly characterized by a general Bayes update. To overcome the learning-rate selection problem inherent in generalized Bayesian inference, we introduce a fully automated, bounds-based calibration procedure, coupled with a Fisher-consistent certified hinge surrogate for efficient optimization. Our experiments demonstrate that PROWL achieves improvements in estimating robust, high-value treatment regimes under severe reward uncertainty compared to standard methods for ITR estimation.

Nonrigid registration is conventionally divided into point set registration, which aligns sparse geometries, and image registration, which aligns continuous intensity fields on regular grids. However, this dichotomy creates a critical bottleneck for emerging scientific data, such as spatial transcriptomics, where high-dimensional vector-valued functions, e.g., gene expression, are defined on irregular, sparse manifolds. Consequently, researchers currently face a forced choice: either sacrifice single-cell resolution via voxelization to utilize image-based tools, or ignore the critical functional signal to utilize geometric tools. To resolve this dilemma, we propose Domain Elastic Transform (DET), a grid-free probabilistic framework that unifies geometric and functional alignment. By treating data as functions on irregular domains, DET registers high-dimensional signals directly without binning. We formulate the problem within a rigorous Bayesian framework, modeling domain deformation as an elastic motion guided by a joint spatial-functional likelihood. The method is fully unsupervised and scalable, utilizing feature-sensitive downsampling to handle massive atlases. We demonstrate that DET achieves 92\% topological preservation on MERFISH data where state-of-the-art optimal transport methods struggle ($<$5\%), and successfully registers whole-embryo Stereo-seq atlases across developmental stages -- a task involving massive scale and complex nonrigid growth. The implementation of DET is available on {https://github.com/ohirose/bcpd} (since Mar, 2025).

In practical machine learning, the environments encountered during the model development and deployment phases often differ, especially when a model is used by many users in diverse settings. Learning models that maintain reliable performance across plausible deployment environments is known as distributionally robust (DR) learning. In this work, we study the problem of distributionally robust feature selection (DRFS), with a particular focus on sparse sensing applications motivated by industrial needs. In practical multi-sensor systems, a shared subset of sensors is typically selected prior to deployment based on performance evaluations using many available sensors. At deployment, individual users may further adapt or fine-tune models to their specific environments. When deployment environments differ from those anticipated during development, this strategy can result in systems lacking sensors required for optimal performance. To address this issue, we propose safe-DRFS, a novel approach that extends safe screening from conventional sparse modeling settings to a DR setting under covariate shift. Our method identifies a feature subset that encompasses all subsets that may become optimal across a specified range of input distribution shifts, with finite-sample theoretical guarantees of no false feature elimination.

For classification tasks, the performance of a deep neural network is determined by the structure of its decision boundary, whose geometry directly affects essential properties of the model, including accuracy and robustness. Motivated by a classical tube formula due to Weyl, we introduce a method to measure the decision boundary of a neural network through local surface volumes, providing a theoretically justifiable and efficient measure enabling a geometric interpretation of the effectiveness of the model applicable to the high dimensional feature spaces considered in deep learning. A smaller surface volume is expected to correspond to lower model complexity and better generalisation. We verify, on a number of image processing tasks with convolutional architectures that decision boundary volume is inversely proportional to classification accuracy. Meanwhile, the relationship between local surface volume and generalisation for fully connected architecture is observed to be less stable between tasks. Therefore, for network architectures suited to a particular data structure, we demonstrate that smoother decision boundaries lead to better performance, as our intuition would suggest.

Your Out-of-Distribution Detection Method is Not Robust!

Reinforcement learning (RL) has become a standard technique for post-training diffusion-based image synthesis models, as it enables learning from reward signals to explicitly improve desirable aspects such as image quality and prompt alignment. In this paper, we propose an online RL variant that reduces the variance in the model updates by sampling paired trajectories and pulling the flow velocity in the direction of the more favorable image. Unlike existing methods that treat each sampling step as a separate policy action, we consider the entire sampling process as a single action. We experiment with both high-quality vision language models and off-the-shelf quality metrics for rewards, and evaluate the outputs using a broad set of metrics. Our method converges faster and yields higher output quality and prompt alignment than previous approaches.