Topological descriptors have been increasingly utilized for capturing multiscale structural information in relational data. In this work, we consider various filtrations on the (box) product of graphs and the effect on their outputs on the topological descriptors - the Euler characteristic (EC) and persistent homology (PH). In particular, we establish a complete characterization of the expressive power of EC on general color-based filtrations. We also show that the PH descriptors of (virtual) graph products contain strictly more information than the computation on individual graphs, whereas EC does not. Additionally, we provide algorithms to compute the PH diagrams of the product of vertex-and edge-level filtrations on the graph product. We also substantiate our theoretical analysis with empirical investigations on runtime analysis, expressivity, and graph classification performance. Overall, this work paves way for powerful graph persistent descriptors via product filtrations.

Recent advances in large language models (LLMs) and audio language models have significantly improved music generation, particularly in lyrics-to-song generation.

Based on our analysis of over 1,400 language model checkpoints on over 110,000 tokens of English, we find that up to 98% of the variance in language model behavior at the word level can be explained by three simple heuristics: the unigram probability (frequency) of a given word, the n-gram probability of the word, and the semantic similarity between the word and its context. Furthermore, we see consistent behavioral phases in all language models, with their predicted probabilities for words overfitting to those words' n-gram probabilities for increasing n over the course of training. Taken together, these results suggest that learning in neural language models may follow a similar trajectory irrespective of model details.

Federated Multi-Task Learning (FL-MTL) enables clients with heterogeneous data to collaboratively train models capable of handling multiple downstream tasks. However, FL-MTL faces key challenges, including statistical heterogeneity, task interference, and the need to balance local learning with global knowledge sharing. Traditional methods like FedAvg struggle in such settings due to the lack of explicit mechanisms to address these issues. In this paper, we propose FedRAM, a threestep framework that progressively updates two scalar hyperparameters: the task importance weight and the client aggregation coefficient. FedRAM introduces a reference-proxy-agent strategy, where the proxy model serves as an intermediate between the local reference model and the global agent model. This design reduces the need for repeated local training while preserving local performance. Extensive experiments on six real-world FL-MTL benchmarks show that FedRAM improves performance by at least 3% over the most baseline on both in-domain and outof-domain tasks, while reducing computational cost by 15 . These results make FedRAM a robust and practical solution for large-scale FL-MTL applications. The code is available at https://github.com/wwffvv/FedRAM.

Stochastic optimal control methods often struggle in complex non-convex landscapes, frequently becoming trapped in local optima due to their inability to learn from historical trajectory data. This paper introduces Memory-Augmented Potential Field Theory, a unified mathematical framework that integrates historical experience into stochastic optimal control. Our approach dynamically constructs memory-based potential fields that identify and encode key topological features of the state space, enabling controllers to automatically learn from past experiences and adapt their optimization strategy. We provide a theoretical analysis showing that memory-augmented potential fields possess non-convex escape properties, asymptotic convergence characteristics, and computational efficiency. We implement this theoretical framework in a Memory-Augmented Model Predictive Path Integral (MPPI) controller that demonstrates significantly improved performance in challenging non-convex environments. The framework represents a generalizable approach to experience-based learning within control systems (especially robotic dynamics), enhancing their ability to navigate complex state spaces without requiring specialized domain knowledge or extensive offline training.

Existing Embodied Question Answering (EQA) benchmarks primarily focus on household environments, often overlooking safety-critical aspects and reasoning processes pertinent to industrial settings. This drawback limits the evaluation of agent readiness for real-world industrial applications. To bridge this, we introduce IndustryEQA, the first benchmark dedicated to evaluating embodied agent capabilities within safety-critical warehouse scenarios. Built upon the NVIDIA Isaac Sim platform, IndustryEQA provides high-fidelity episodic memory videos featuring diverse industrial assets, dynamic human agents, and carefully designed hazardous situations inspired by real-world safety guidelines. The benchmark includes rich annotations covering six categories: equipment safety, human safety, object recognition, attribute recognition, temporal understanding, and spatial understanding. Besides, it also provides extra reasoning evaluation based on these categories. Specifically, it comprises 971 question-answer pairs generated from small warehouse and 373 pairs from large ones, incorporating scenarios with and without human. We further propose a comprehensive evaluation framework, including various baseline models, to assess their general perception and reasoning abilities in industrial environments. IndustryEQA aims to steer EQA research towards developing more robust, safety-aware, and practically applicable embodied agents for complex industrial environments.

Test-Time Scaling (TTS) improves the performance of Large Language Models (LLMs) by using additional inference-time computation to explore multiple reasoning paths through search. Yet how to allocate a fixed rollout budget most effectively during search remains underexplored, often resulting in inefficient use of compute at test time. To bridge this gap, we formulate test-time search as a resource allocation problem and derive the optimal allocation strategy that maximizes the probability of obtaining a correct solution under a fixed rollout budget. Within this formulation, we reveal a core limitation of existing search methods: solution-level allocation tends to favor reasoning directions with more candidates, leading to theoretically suboptimal and inefficient use of compute. To address this, we propose Direction-Oriented Resource Allocation (DORA), a provably optimal method that mitigates this bias by decoupling direction quality from candidate count and allocating resources at the direction level. To demonstrate DORA's effectiveness, we conduct extensive experiments on challenging mathematical reasoning benchmarks including MATH500, AIME2024, and AIME2025. The empirical results show that DORA consistently outperforms strong baselines with comparable computational cost, achieving state-of-the-art accuracy. We hope our findings contribute to a broader understanding of optimal TTS for LLMs.1

Training data attribution (TDA) methods aim to measure how training data impacts a model's predictions. While gradient-based attribution methods, such as influence functions, offer theoretical grounding, their computational costs make them impractical for large-scale applications. Representation-based approaches are far more scalable, but typically rely on heuristic embeddings that are not optimized for attribution, limiting their fidelity. To address these challenges, we propose AirRep, a scalable, representation-based approach that closes this gap by learning task-specific and model-aligned representations optimized explicitly for TDA. AirRep introduces two key innovations: a trainable encoder tuned for attribution quality, and an attention-based pooling mechanism that enables accurate estimation of group-wise influence. We train AirRep using a ranking objective over automatically constructed training subsets labeled by their empirical effect on target predictions. Experiments on instruction-tuned LLMs demonstrate that AirRep achieves performance on par with state-of-the-art gradient-based approaches while being nearly two orders of magnitude more efficient at inference time. Further analysis highlights its robustness and generalization across tasks and models.

Traditional reinforcement learning (RL) assumes the agents make decisions based on Markov decision processes (MDPs) with one-step transition models. In many real-world applications, such as energy management and stock investment, agents can access multi-step predictions of future states, which provide additional advantages for decision making. However, multi-step predictions are inherently high-dimensional: naively embedding these predictions into an MDP leads to an exponential blow-up in state space and the curse of dimensionality. Moreover, existing RL theory provides few tools to analyze prediction-augmented MDPs, as it typically works on one-step transition kernels and cannot accommodate multi-step predictions with errors or partial action-coverage. We address these challenges with three key innovations: First, we propose the Bayesian value function to characterize the optimal prediction-aware policy tractably. Second, we develop a novel BellmanJensen Gap analysis on the Bayesian value function, which enables characterizing the value of imperfect predictions. Third, we introduce BOLA (Bayesian Offline Learning with Online Adaptation), a two-stage model-based RL algorithm that separates offline Bayesian value learning from lightweight online adaptation to real-time predictions. We prove that BOLA remains sample-efficient even under imperfect predictions.

Large language models have been widely applied, but can inadvertently encode sensitive or harmful information, raising significant safety concerns. Machine unlearning has emerged to alleviate this concern; however, existing training-time unlearning approaches, relying on coarse-grained loss combinations, have limitations in precisely separating knowledge and balancing removal effectiveness with model utility. In contrast, we propose Fine-grained Activation manipuLation by Contrastive Orthogonal uNalignment (FALCON), a novel representation-guided unlearning approach that leverages information-theoretic guidance for efficient parameter selection, employs contrastive mechanisms to enhance representation separation, and projects conflict gradients onto orthogonal subspaces to resolve conflicts between forgetting and retention objectives. Extensive experiments demonstrate that FALCON achieves superior unlearning effectiveness while maintaining model utility, exhibiting robust resistance against knowledge recovery attempts.