Computing the polar decomposition and the related matrix sign function has been a well-studied problem in numerical analysis for decades. Recently, it has emerged as an important subroutine within the Muon algorithm for training deep neural networks. However, the requirements of this application differ sharply from classical settings: deep learning demands GPU-friendly algorithms that prioritize high throughput over high precision. We introduce Polar Express, a new method for computing the polar decomposition. Like Newton–Schulz and other classical polynomial methods, our approach uses only matrix-matrix multiplications, making it very efficient on GPUs. Inspired by earlier work of Chen & Chow and Nakatsukasa & Freund, Polar Express adapts the update rule at each iteration by solving a minimax optimization problem. We prove that this strategy minimizes error in a worst-case sense, allowing Polar Express to converge as rapidly as possible both in the early iterations and asymptotically. We also address finite-precision issues, making it practical to use in bfloat16. When integrated into Muon, our method yields consistent improvements in validation loss for a GPT-2 model on one to ten billion tokens from the FineWeb dataset, outperforming recent alternatives across a range of learning rates.

Understanding how populations of neurons represent information is a central challenge across machine learning and neuroscience. Recent work in both fields has begun to characterize the representational geometry and functionality underlying complex distributed activity. For example, artificial neural networks trained on data with more features than neurons compress data by representing features non-orthogonally in so-called superposition. However, the effect of time (or memory), an additional capacity-constraining pressure, on underlying representational geometry in recurrent models is not well understood. Here, we study how memory demands affect representational geometry in recurrent neural networks (RNNs), introducing the concept of temporal superposition. We develop a theoretical framework in RNNs with linear recurrence trained on a delayed serial recall task to better understand how properties of the data, task demands, and network dimensionality lead to different representational strategies, and show that these insights generalize to nonlinear RNNs. Through this, we identify an effectively linear, dense regime and a sparse regime where RNNs utilize an interference-free space, characterized by a phase transition in the angular distribution of features and decrease in spectral radius. Finally, we analyze the interaction of spatial and temporal superposition to observe how RNNs mediate different representational tradeoffs. Overall, our work offers a mechanistic, geometric explanation of representational strategies RNNs learn, how they depend on capacity and task demands, and why.

Neural scaling laws underlie many of the recent advances in deep learning, yet their theoretical understanding remains largely confined to linear models. In this work, we present a systematic analysis of scaling laws for quadratic and diagonal neural networks in the feature learning regime. Leveraging connections with matrix compressed sensing and LASSO, we derive a detailed phase diagram for the scaling exponents of the excess risk as a function of sample complexity and weight decay. This analysis uncovers crossovers between distinct scaling regimes and plateau behaviors, mirroring phenomena widely reported in the empirical neural scaling literature. Furthermore, we establish a precise link between these regimes and the spectral properties of the trained network weights, which we characterize in detail. As a consequence, we provide a theoretical validation of recent empirical observations connecting the emergence of power-law tails in the weight spectrum with network generalization performance, yielding an interpretation from first principles.

As AI increasingly shapes daily life, energy consumption and data privacy have become pressing concerns. On-device learning trains models directly on edge devices, cutting energy consumption and safeguarding data privacy. However, the expanding scale of modern neural networks creates a major obstacle for on-device training. Although prior work has concentrated on compact convolutional architectures, we instead apply subspace-based training to transformer models. Motivated by the idea that a model's essential information lies in a fixed subspace, we introduce Weight-Activation Subspace Iteration (WASI), a method that mitigates the memory bottleneck of backpropagation and boosts inference efficiency in transformer models by restricting training to this subspace. Our results demonstrate that WASI maintains accuracy comparable to vanilla training while reducing memory usage by up to $62\times$ and computational cost (FLOPs) by up to $2\times$. On a Raspberry Pi 5, WASI achieves roughly $1.4\times$ faster training and inference than vanilla training. The code is available at https://github.com/Le-TrungNguyen/ICLR2026-WASI.git.

The pursuit of computational efficiency has driven the adoption of low-precision formats for training transformer models. However, this progress is often hindered by notorious training instabilities. This paper provides the first mechanistic explanation for a long-standing and unresolved failure case where training with flash attention in low-precision settings leads to catastrophic loss explosion. Our in-depth analysis reveals that the failure is not a random artifact but caused by two intertwined phenomena: the emergence of similar low-rank representations within the attention mechanism and the compounding effect of biased rounding errors inherent in low-precision arithmetic. We demonstrate how these factors create a vicious cycle of error accumulation that corrupts weight updates, ultimately derailing the training dynamics. To validate our findings, we introduce a minimal modification to the flash attention that mitigates the bias in rounding errors. This simple change stabilizes the training process, confirming our analysis and offering a practical solution to this persistent problem. Code is available at https://github.com/ucker/why-low-precision-training-fails.

At NeurIPS 2024, Kera (2311.12904) introduced the use of transformers for computing Groebner bases, a central object in computer algebra with numerous practical applications. In this paper, we improve this approach by applying Hierarchical Attention Transformers (HATs) to solve systems of multivariate polynomial equations via Groebner bases computation. The HAT architecture incorporates a tree-structured inductive bias that enables the modeling of hierarchical relationships present in the data and thus achieves significant computational savings compared to conventional flat attention models. We generalize to arbitrary depths and include a detailed computational cost analysis. Combined with curriculum learning, our method solves instances that are much larger than those in Kera (2311.12904).