We introduce the Riemannian Proximal Sampler, a method for sampling from densities defined on Riemannian manifolds. The performance of this sampler critically depends on two key oracles: the Manifold Brownian Increments (MBI) oracle and the Riemannian Heat-kernel (RHK) oracle. We establish high-accuracy sampling guarantees for the Riemannian Proximal Sampler, showing that generating samples with ε-accuracy requires O(log(1/ε)) iterations in Kullback-Leibler divergence assuming access to exact oracles and O(log2(1/ε))iterations in the total variation metric assuming access to sufficiently accurate inexact oracles.

Can algebraic geometry enhance the sharpness, robustness, and interpretability of modern neural reasoning models by equipping them with a mathematically grounded inductive bias? To answer this, we introduce Tropical Attention, an attention mechanism grounded in tropical geometry that lifts the attention kernel into tropical projective space, where reasoning is piecewise-linear and 1-Lipschitz, thus preserving the polyhedral decision structure inherent to combinatorial reasoning. We prove that Multi-Head Tropical Attention (MHTA) stacks universally approximate tropical circuits and realize tropical transitive closure through composition, achieving polynomial resource bounds without invoking recurrent mechanisms. These guarantees explain why the induced polyhedral decision boundaries remain sharp and scale-invariant, rather than smoothed by Softmax. Empirically, we show that Tropical Attention delivers stronger out-of-distribution generalization in both length and value, with high robustness against perturbative noise, and substantially faster inference with fewer parameters compared to Softmax-based and recurrent attention baselines, respectively. For the first time, we push the domain of neural algorithmic reasoning beyond PTIME problems to NP-hard/complete problems, paving the way toward sharper and more expressive Large Reasoning Models (LRMs) capable of tackling complex combinatorial challenges in Phylogenetics, Cryptography, Particle Physics, and Mathematical Discovery.

While being disturbed by environmental noises, the acoustic masking technique is a conventional way to reduce the annoyance in audio engineering that seeks to cover up the noises with other dominant yet less intrusive sounds. However, misalignment between the dominant sound and the noise--such as mismatched downbeats--often requires an excessive volume increase to achieve effective masking. Motivated by recent advances in cross-modal generation, in this work, we introduce an alternative method to acoustic masking, aiming to reduce the noticeability of environmental noises by blending them into personalized music generated based on user-provided text prompts. Following the paradigm of music generation using mel-spectrogram representations, we propose a Blending Noises into Personalized Music (BNMusic) framework with two key stages.

This paper investigates the limitations of the normalization in attention mechanisms. We begin with a theoretical framework that enables the identification of the model's selective ability and the geometric separation involved in token selection. Our analysis includes explicit bounds on distances and separation criteria for token vectors under softmax scaling. Through experiments with pre-trained GPT-2 model, we empirically validate our theoretical results and analyze key behaviors of the attention mechanism. Notably, we demonstrate that as the number of selected tokens increases, the model's ability to distinguish informative tokens declines, often converging toward a uniform selection pattern. We also show that gradient sensitivity under softmax normalization presents challenges during training, especially at low temperature settings. These findings advance current understanding of softmax-based attention mechanism and motivate the need for more robust normalization and selection strategies in future attention architectures.

Recently, Graph Neural Controlled Differential Equations (Graph Neural CDEs) successfully adapted Neural CDEs from paths on Euclidean domains to paths on graph domains. Building on this foundation, we introduce Permutation Equivariant Neural Graph CDEs, which project Graph Neural CDEs onto permutation equivariant function spaces. This significantly reduces the model's parameter count without compromising representational power, resulting in more efficient training and improved generalisation. We empirically demonstrate the advantages of our approach through experiments on simulated dynamical systems and real-world tasks, showing improved performance in both interpolation and extrapolation scenarios.

Data errors, corruptions, and poisoning attacks during training pose a major threat to the reliability of modern AI systems. While extensive effort has gone into empirical mitigations, the evolving nature of attacks and the complexity of data require a more principled, provable approach to robustly learn on such data--and to understand how perturbations influence the final model. Hence, we introduce MIBPCert, a novel certification method based on mixed-integer bilinear programming (MIBP) that computes sound, deterministic bounds to provide provable robustness even under complex threat models. By computing the set of parameters reachable through perturbed or manipulated data, we can predict all possible outcomes and guarantee robustness. To make solving this optimization problem tractable, we propose a novel relaxation scheme that bounds each training step without sacrificing soundness. We demonstrate the applicability of our approach to continuous and discrete data, as well as different threat models--including complex ones that were previously out of reach.

Most multimodal models treat every negative pair alike, ignoring the ambiguous negatives that differ from the positive by only a small detail. We propose BoundaryA ware Curriculum with Local Attention(BACL), a lightweight add-on that turns these borderline cases into a curriculum signal. ABoundary-aware Negative Sampler gradually raises difficulty, while a Contrastive Local Attention loss highlights where the mismatch occurs. The two modules are fully differentiable and work with any off-the-shelf dual encoder. Theory predicts a fast O(1/n) error rate; practice shows up to +32 % R@1 over CLIP and new SOTA on four large-scale benchmarks, all without extra labels.

Is the war entering a new phase? Russia has renewed its strikes on Ukraine, killing three people including an eight-year-old girl, Ukrainian officials said. The Russian strikes on Friday come a day after Ukraine launched its biggest-ever drone attack on Moscow, killing a different eight-year-old girl and sparking an inferno at a major oil refinery, according to Russian officials. Between late Thursday and early Friday, Russia launched 90 drones at Ukraine, according to the Ukrainian air force. "An eight-year-old girl was killed. These are the consequences of this morning's enemy attack on Pavlohrad," Oleksandr Ganzha, the governor of Ukraine's central Dnipropetrovsk region, said.

Generalized policy and execution efficiency constitute the two critical challenges in robotic manipulation. While recent foundation policies benefit from the commonsense reasoning capabilities of internet-scale pretrained vision-language models (VLMs), they often suffer from low execution frequency. To mitigate this dilemma, dual-system approaches have been proposed to leverage a VLM-based System 2 module for handling high-level decision-making, and a separate System 1 action module for ensuring real-time control. However, existing designs maintain both systems as separate models, limiting System 1 from fully leveraging the rich pretrained knowledge from the VLM-based System 2. In this work, we propose Fast-in-Slow (FiS), a unified dual-system vision-language-action (VLA) model that embeds the System 1 execution module within the VLM-based System 2 by partially sharing parameters. This innovative paradigm not only enables high-frequency execution in System 1, but also facilitates coordination between multimodal reasoning and execution components within a single foundation model of System 2. Given their fundamentally distinct roles within FiS-VLA, we design the two systems to incorporate heterogeneous modality inputs alongside asynchronous operating frequencies, enabling both fast and precise manipulation. To enable coordination between the two systems, a dual-aware co-training strategy is proposed that equips System 1 with action generation capabilities while preserving System 2's contextual understanding to provide stable latent conditions for System 1. For evaluation, FiS-VLA outperforms previous state-of-the-art methods by 8% in simulation and 11% in realworld tasks in terms of average success rate, while achieving a 117.7 Hz control frequency with action chunk set to eight.

In recent years, artificial intelligence has significantly advanced medical image segmentation. Nonetheless, challenges remain, including efficient 3D medical image processing across diverse modalities and handling data variability. In this work, we introduce Hierarchical Soft Mixture-of-Experts (HoME), a two-level token-routing layer for efficient long-context modeling, specifically designed for 3D medical image segmentation. Built on the Mamba Selective State-Space Model (SSM) backbone, HoME enhances sequential modeling through adaptive expert routing.