Multiview datasets are common in scientific and engineering applications, yet existing fusion methods offer limited theoretical guarantees, particularly in the presence of heterogeneous and high-dimensional noise. We propose Generalized Robust Adaptive-Bandwidth Multiview Diffusion Maps (GRAB-MDM), a new kernel-based diffusion geometry framework for integrating multiple noisy data sources. The key innovation of GRAB-MDM is a {view}-dependent bandwidth selection strategy that adapts to the geometry and noise level of each view, enabling a stable and principled construction of multiview diffusion operators. Under a common-manifold model, we establish asymptotic convergence results and show that the adaptive bandwidths lead to provably robust recovery of the shared intrinsic structure, even when noise levels and sensor dimensions differ across views. Numerical experiments demonstrate that GRAB-MDM significantly improves robustness and embedding quality compared with fixed-bandwidth and equal-bandwidth baselines, and usually outperform existing algorithms. The proposed framework offers a practical and theoretically grounded solution for multiview sensor fusion in high-dimensional noisy environments.

Causal inference is paramount for understanding the effects of interventions, yet extracting personalized insights from increasingly complex data remains a significant challenge for modern machine learning. This is the case, in particular, when considering functional outcomes observed over a continuous domain (e.g., time, or space). Estimation of heterogeneous treatment effects, known as CATE, has emerged as a crucial tool for personalized decision-making, but existing meta-learning frameworks are largely limited to scalar outcomes, failing to provide satisfying results in scientific applications that leverage the rich, continuous information encoded in functional data. Here, we introduce FOCaL (Functional Outcome Causal Learning), a novel, doubly robust meta-learner specifically engineered to estimate a functional heterogeneous treatment effect (F-CATE). FOCaL integrates advanced functional regression techniques for both outcome modeling and functional pseudo-outcome reconstruction, thereby enabling the direct and robust estimation of F-CATE. We provide a rigorous theoretical derivation of FOCaL, demonstrate its performance and robustness compared to existing non-robust functional methods through comprehensive simulation studies, and illustrate its practical utility on diverse real-world functional datasets. FOCaL advances the capabilities of machine intelligence to infer nuanced, individualized causal effects from complex data, paving the way for more precise and trustworthy AI systems in personalized medicine, adaptive policy design, and fundamental scientific discovery.

Sampling equilibrium molecular configurations from the Boltzmann distribution is a longstanding challenge. Boltzmann Generators (BGs) address this by combining exact-likelihood generative models with importance sampling, but their practical scalability is limited. Meanwhile, coarse-grained surrogates enable the modeling of larger systems by reducing effective dimensionality, yet often lack the reweighting process required to ensure asymptotically correct statistics. In this work, we propose Coarse-Grained Boltzmann Generators (CG-BGs), a principled framework that unifies scalable reduced-order modeling with the exactness of importance sampling. CG-BGs act in a coarse-grained coordinate space, using a learned potential of mean force (PMF) to reweight samples generated by a flow-based model. Crucially, we show that this PMF can be efficiently learned from rapidly converged data via force matching. Our results demonstrate that CG-BGs faithfully capture complex interactions mediated by explicit solvent within highly reduced representations, establishing a scalable pathway for the unbiased sampling of larger molecular systems.

This work addresses the problem of offline safe imitation learning (IL), where the goal is to learn safe and reward-maximizing policies from demonstrations that do not have per-timestep safety cost or reward information. In many real-world domains, online learning in the environment can be risky, and specifying accurate safety costs can be difficult. However, it is often feasible to collect trajectories that reflect undesirable or unsafe behavior, implicitly conveying what the agent should avoid. We refer to these as non-preferred trajectories. We propose a novel offline safe IL algorithm, OSIL, that infers safety from non-preferred demonstrations. We formulate safe policy learning as a Constrained Markov Decision Process (CMDP). Instead of relying on explicit safety cost and reward annotations, OSIL reformulates the CMDP problem by deriving a lower bound on reward maximizing objective and learning a cost model that estimates the likelihood of non-preferred behavior. Our approach allows agents to learn safe and reward-maximizing behavior entirely from offline demonstrations. We empirically demonstrate that our approach can learn safer policies that satisfy cost constraints without degrading the reward performance, thus outperforming several baselines.

We develop a theoretical analysis of LLM-guided formal theorem proving in interactive proof assistants (e.g., Lean) by modeling tactic proposal as a stochastic policy in a finite-horizon deterministic MDP. To capture modern representation learning, we treat the state and action spaces as general compact metric spaces and assume Lipschitz policies. To explain the gap between worst-case hardness and empirical success, we introduce problem distributions generated by a reference policy $q$, including a latent-variable model in which proofs exhibit reusable cut/lemma/sketch structure represented by a proof DAG. Under a top-$k$ search protocol and Tsybakov-type margin conditions, we derive lower bounds on finite-horizon success probability that decompose into search and learning terms, with learning controlled by sequential Rademacher/covering complexity. Our main separation result shows that when cut elimination expands a DAG of depth $D$ into a cut-free tree of size $Ω(Λ^D)$ while the cut-aware hierarchical process has size $O(λ^D)$ with $λ\llΛ$, a flat (cut-free) learner provably requires exponentially more data than a cut-aware hierarchical learner. This provides a principled justification for subgoal decomposition in recent agentic theorem provers.

For the moment, it is worth noting that such sets of functions (e.g., Haar wavelets, Daubechies wavelets) are readily We are now in a position to present the main theorem of this subsection. To avoid repetition, we defer further discussions on the rates observed in Theorem A.1 to Remark 2.7 where a holistic In fact, by Proposition 1.1, there exists an optimal transport map Based on (B.2), the natural plug-in estimator of ρ Suppose that the same assumptions from Theorem 2.2 hold. B.2 Nonparametric independence testing: Optimal transport based Hilbert-Schmidt independence criterion Proposition B.2 shows that the test based on Further, when the sampling distribution is fixed, Proposition B.2 shows that In the following result (see Appendix C.2 for a proof), we show that if This section is devoted to proving our main results and is organized as follows: In Appendix C.1, we Further by Lemma D.2, we also have: ϕ Note that (C.10) immediately yields the following conclusions: S By (1.5) and some simple algebra, the following holds: null null null S Combining the above display with (C.9), we further have: null null null null 1 2 W Combining the above observation with Theorem 2.1, we have: lim sup For the next part, to simplify notation, let us begin with some notation. By using the exponential Markov's inequality coupled with the standard union Now by using [7, Theorem 2.10], we have P (B We are now in a position to complete the proof of Theorem 2.2 using steps I-III. Therefore, it is now enough to bound the right hand side of (C.17).

As several industries are moving towards modeling massive 3D virtual worlds, theneed forcontent creation tools that can scale interms ofthequantity,quality, and diversity of 3D content is becoming evident.