We develop a general framework for analyzing representation costs of parametric data-fitting methods through their parameter-space regularizers. From this abstract perspective, we define representation costs for arbitrary parametric models and reveal their induced (native) function spaces. This unifies recent function-space views of data-fitting methods. We also prove that many natural results hold in this abstract setting, including representer theorems for parametric methods on their native spaces. The framework also rigorously connects parametric methods with their equivalent nonparametric descriptions under sufficient overparameterization. Classical methods and their native spaces, such as kernel methods / reproducing kernel Hilbert spaces, wavelets / Besov spaces, and shallow neural networks / variation spaces emerge as special cases of our abstract framework. A byproduct of "axiomatizing" the study of representation costs is that we also immediately obtain new results for deep neural networks: For depth-$L$ feedforward ReLU networks, their induced native spaces are $p$-normable quasi-Banach spaces with $p = 2/L$. This reveals that the inductive bias of deep neural networks (as given by the representation cost) cannot be captured by norms for depths $L > 2$.

Multi-agent debate improves the reliability of large language models (LLMs) through iterative peer critiques. However, fixed topologies often introduce persistent positional biases, amplify unreliable agents, and cause high sensitivity to role assignments. We introduce \textit{Permutation-Equivariant Adaptive Routing Multi-Agent Debate (PEAR)}, an inference-time protocol that dynamically reconfigures communication roles and sparse topologies across consecutive debate rounds. By strategically switching agent-to-role assignments based on evolving agent states, PEAR prevents any agent from permanently occupying a privileged network position or distributes influence more evenly across the debate. We theoretically characterize PEAR as an equivariant sparse router: it preserves accuracy under agent relabeling while reducing routing complexity and improving generalization. Comprehensive empirical evaluations across four reasoning benchmarks and six diverse LLM backbones demonstrate PEAR significantly improves average accuracy over the strongest debate baselines. The code is at https://github.com/EVIEHub/PEAR.

The scaling law for large language models (LLMs) depicts that the path towards machine intelligence necessitates training at large scale. Thus, companies continuously build large-scale GPU clusters, and launch training jobs that span over thousands of computing nodes. However, LLM pre-training presents unique challenges due to its complex communication patterns, where GPUs exchange data in sparse yet high-volume bursts within specific groups. Inefficient resource scheduling exacerbates bandwidth contention, leading to suboptimal training performance. This paper presents Arnold, a scheduling system summarizing our experience to effectively align LLM communication patterns with data center topology at scale. An in-depth characteristic study is performed to identify the impact of physical network topology to LLM pre-training jobs. Based on the insights, we develop a scheduling algorithm to effectively align communication patterns with the physical network topology in modern data centers. Through simulation experiments, we show the effectiveness of our algorithm in reducing the maximum spread of communication groups by up to 1.67x. In production training, our scheduling system improves the end-to-end performance by 10.6% when training with more than 9600 GPUs, a significant improvement for our training pipeline.

Inspired by the workings of biological brains, humans have designed artificial neural networks (ANNs), sparking profound advancements across various fields. However, the biological brain possesses high plasticity, enabling it to develop simple, efficient, and powerful structures to cope with complex external environments. In contrast, the superior performance of ANNs often relies on meticulously crafted architectures, which can make them vulnerable when handling complex inputs. Moreover, overparameterization often characterizes the most advanced ANNs. This paper explores the path toward building streamlined and plastic ANNs.

Rare events in time series are critical to model but hard to learn due to data scarcity. Current generative models struggle with extreme values. We observe that rare events leave distinct topological fingerprints - transitions in Betti numbers from point-cloud embeddings - that are more stable and discriminative than statistical moments. We introduce PHINN, a flow-matching framework using dynamic Betti curves as conditioning signals and a persistence landscape loss for homology consistency. It scales to multivariate data, includes a natural-language interface to set Betti targets, supports cross-domain meta-learning and few-shot generation, and provides certified adversarial robustness. On financial, epidemiological, and multi-modal benchmarks, PHINN outperforms statistical and diffusion baselines in topological fidelity (beta-RMSE down 41-63%, transition accuracy up 84%) and matches jump-diffusion models in tail coverage while exceeding them in shape fidelity. All results have 95% confidence intervals.

Predicting the behavior of complex systems is critical in many scientific and engineering domains, and hinges on the model's ability to capture their underlying dynamics. Existing methods encode the intrinsic dynamics of high-dimensional observations through latent representations and predict autoregressively. However, these latent representations lose the inherent spatial structure of spatiotemporal dynamics, leading to the predictor's inability to effectively model spatial interactions and neglect emerging dynamics during long-term prediction. In this work, we propose SparseDiff, introducing a test-time adaptation strategy to dynamically update the encoding scheme to accommodate emergent spatiotemporal structures during the long-term evolution of the system.

Analysing how neural networks represent data features in their activations can help interpret how they perform tasks. Hence, a long line of work has focused on mathematically characterising the geometry of such "neural representations." In parallel, machine learning has seen a surge of interest in understanding how dynamical systems perform computations on time-varying input data. Yet, the link between computation-through-dynamics and representational geometry remains poorly understood. Here, we hypothesise that recurrent neural networks (RNNs) perform computations by dynamically warping their representations of task variables. To test this hypothesis, we develop a Riemannian geometric framework that enables the derivation of the manifold topology and geometry of a dynamical system from the manifold of its inputs. By characterising the time-varying geometry of RNNs, we show that dynamic warping is a fundamental feature of their computations.

Federated Learning (FL) enables collaborative model training across distributed clients without data sharing, but its high computational and communication demands strain resource-constrained devices.

This paper explores the use of Hermite transform based reproducing kernel Banach space methods to construct exact or un-approximated models of feedforward neural networks of arbitrary width, depth and topology, including ResNet and Transformers networks, assuming only a feedforward topology, finite energy activations and finite (spectral-) norm weights and biases. Using this model, two straightforward but surprisingly tight bounds on Rademacher complexity are derived, precisely (1) a general bound that is width-independent and scales exponentially with depth; and (2) a width-and depth-independent bound for networks with appropriately constrained (below threshold) weights and biases.

Multi-agent language systems are often built as hand-designed workflows, where agents are assigned semantic roles and communication protocols are specified in advance. We propose NeuroMAS, a method that first treats a multi-agent language system as a trainable and scalable neural-network-like architecture with LLM agents as nodes and intermediate textual signals as edges. In NeuroMAS, agent nodes are role-free but structure-aware: the topology only determines how information can flow in general, while reinforcement learning training determines how nodes communicate, specialize, and coordinate. This formulation shifts multi-agent design from workflow engineering toward architecture design, where depth, width, connectivity, and growth protocol become scalable sources of capability. Further, we provide a theoretical perspective showing why such modular textual computation is more parameter-efficient when tasks admit hierarchical decompositions. Experiments show that NeuroMAS improves significantly over both inference-time and trained multi-agent baselines. We further find that organizational scaling is path-dependent: larger systems can be challenging to train from scratch, but become feasible when grown progressively from smaller trained systems. These results suggest that learned neural multi-agent systems are a promising scaling axis for LLMs.