arXiv subjects

Noah Amsel

Publications and source records attributed to Noah Amsel.

16 recordsLinked to original sources

arXiv2026

Dion3: Full-Stack Orthogonal Updates

The Muon optimizer incurs a significant overhead cost due to its cubic-time Newton-Schulz orthogonalization step. When weights are sharded, communication overhead compounds this computational cost, eroding the benefits of Muon in many settings. We present Dion3, a revision of Muon that targets this overhead at every level of the stack. Our Gram Newton-Schulz algorithm reduces the FLOP cost of orthogonalization, our CuteDSL kernels accelerate it by exploiting symmetry, and our megabatching strategy reduces communication overhead. Moreover, we propose a simple change to the update rule that cuts costs even further: selecting only a fraction of the momentum matrix's rows to orthogonalize at each step. This update rule improves on Dion (another "compressed" version of Muon), in both speed and performance. Overall, Dion3 matches or improves on the loss achieved by Muon but reduces optimizer step time by up to 6x. Dion3 is available via the dion package (https://github.com/microsoft/dion) as a drop-in replacement for Muon.

cs.LG↗

arXiv2026

Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop

This document presents a series of open questions arising in matrix computations, i.e., the numerical solution of linear algebra problems. It is a result of working groups at the workshop Linear Systems and Eigenvalue Problems, which was organized at the Simons Institute for the Theory of Computing program on Complexity and Linear Algebra in Fall 2025. The complexity and numerical solution of linear algebra problems is a crosscutting area between theoretical computer science and numerical analysis. The value of the particular problem formulations here is that they were produced via discussions between researchers from both groups. The open questions are organized in five categories: iterative solvers for linear systems, eigenvalue computation, low-rank approximation, randomized sketching, and other areas including tensors, quantum systems, and matrix functions. (Updated to reflect the status of the open problems as of August 20, 2026.)

math.NA↗

arXiv2025

Dion2: A Simple Method to Shrink Matrix in Muon

The Muon optimizer enjoys strong empirical performance and theoretical grounding. However, the super-linear cost of its orthonormalization step introduces increasing overhead with scale. To alleviate this cost, several works have attempted to reduce the size of the matrix entering the orthonormalization step. We introduce Dion2, a much simpler method for shrinking the matrix involved in Muon's computation compared to prior approaches. At a high level, Dion2 selects a fraction of rows or columns at each iteration and orthonormalizes only those. This sampling procedure makes the update sparse, reducing both computation and communication costs which in turn improves the scalability of Muon.

cs.LG↗

arXiv2025

Customizing the Inductive Biases of Softmax Attention using Structured Matrices

The core component of attention is the scoring function, which transforms the inputs into low-dimensional queries and keys and takes the dot product of each pair. While the low-dimensional projection improves efficiency, it causes information loss for certain tasks that have intrinsically high-dimensional inputs. Additionally, attention uses the same scoring function for all input pairs, without imposing a distance-dependent compute bias for neighboring tokens in the sequence. In this work, we address these shortcomings by proposing new scoring functions based on computationally efficient structured matrices with high ranks, including Block Tensor-Train (BTT) and contiguous Multi-Level Low Rank (MLR) matrices. On in-context regression tasks with high-dimensional inputs, our proposed scoring functions outperform standard attention for any fixed compute budget. On language modeling, a task that exhibits locality patterns, our MLR-based attention method achieves improved scaling laws compared to both standard attention and variants of sliding window attention. Additionally, we show that both BTT and MLR fall under a broader family of efficient structured matrices capable of encoding either full-rank or distance-dependent compute biases, thereby addressing significant shortcomings of standard attention. Finally, we show that MLR attention has promising results for long-range time-series forecasting.

cs.LG↗

arXiv2025

Quasi-optimal hierarchically semi-separable matrix approximation

We present a randomized algorithm for producing a quasi-optimal hierarchically semi-separable (HSS) approximation to an $N\times N$ matrix $A$ using only matrix-vector products with $A$ and $A^T$. We prove that, using $O(k \log(N/k))$ matrix-vector products and ${O}(N k^2 \log(N/k))$ additional runtime, the algorithm returns an HSS matrix $B$ with rank-$k$ blocks whose expected Frobenius norm error $\mathbb{E}[\|A - B\|_F^2]$ is at most $O(\log(N/k))$ times worse than the best possible approximation error by an HSS rank-$k$ matrix. In fact, the algorithm we analyze in a simple modification of an empirically effective method proposed by [Levitt & Martinsson, SISC 2024]. As a stepping stone towards our main result, we prove two results that are of independent interest: a similar guarantee for a variant of the algorithm which accesses $A$'s entries directly, and explicit error bounds for near-optimal subspace approximation using projection-cost-preserving sketches. To the best of our knowledge, our analysis constitutes the first polynomial-time quasi-optimality result for HSS matrix approximation, both in the explicit access model and the matrix-vector product query model.

math.NA↗

arXiv2025

Query Efficient Structured Matrix Learning

We study the problem of learning a structured approximation (low-rank, sparse, banded, etc.) to an unknown matrix $A$ given access to matrix-vector product (matvec) queries of the form $x \rightarrow Ax$ and $x \rightarrow A^Tx$. This problem is of central importance to algorithms across scientific computing and machine learning, with applications to fast multiplication and inversion for structured matrices, building preconditioners for first-order optimization, and as a model for differential operator learning. Prior work focuses on obtaining query complexity upper and lower bounds for learning specific structured matrix families that commonly arise in applications. We initiate the study of the problem in greater generality, aiming to understand the query complexity of learning approximations from general matrix families. Our main result focuses on finding a near-optimal approximation to $A$ from any finite-sized family of matrices, $\mathcal{F}$. Standard results from matrix sketching show that $O(\log|\mathcal{F}|)$ matvec queries suffice in this setting. This bound can also be achieved, and is optimal, for vector-matrix-vector queries of the form $x,y\rightarrow x^TAy$, which have been widely studied in work on rank-$1$ matrix sensing. Surprisingly, we show that, in the matvec model, it is possible to obtain a nearly quadratic improvement in complexity, to $\tilde{O}(\sqrt{\log|\mathcal{F}|})$. Further, we prove that this bound is tight up to log-log factors. Via covering number arguments, our result extends to well-studied infinite families. As an example, we establish that a near-optimal approximation from any \emph{linear matrix family} of dimension $q$ can be learned with $\tilde{O}(\sqrt{q})$ matvec queries, improving on an $O(q)$ bound achievable via sketching techniques and vector-matrix-vector queries.

cs.DS↗

arXiv2025

The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm

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 optimizer 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 trained on one to ten billion tokens from the FineWeb dataset, outperforming recent alternatives across a range of learning rates.

cs.LG↗

arXiv2025

Compositional Reasoning with Transformers, RNNs, and Chain of Thought

It is well understood that different neural network architectures are suited to different tasks, but is there always a single best architecture for a given task? We compare the expressive power of transformers, RNNs, and transformers with chain of thought tokens on a simple and natural class of tasks we term Compositional Reasoning Questions (CRQ). This family captures multi-step problems with tree-like compositional structure, such as evaluating Boolean formulas. We prove that under standard hardness assumptions, \emph{none} of these three architectures is capable of solving CRQs unless some hyperparameter (depth, embedding dimension, and number of chain of thought tokens, respectively) grows with the size of the input. We then provide constructions for solving CRQs with each architecture. For transformers, our construction uses depth that is logarithmic in the problem size. For RNNs, logarithmic embedding dimension is necessary and sufficient, so long as the inputs are provided in a certain order. For transformers with chain of thought, our construction uses $n$ CoT tokens for input size $n$. These results show that, while CRQs are inherently hard, there are several different ways for language models to overcome this hardness. Even for a single class of problems, each architecture has strengths and weaknesses, and none is strictly better than the others.

cs.LG↗

arXiv2024

Nearly Optimal Approximation of Matrix Functions by the Lanczos Method

Approximating the action of a matrix function $f(\mathbf{A})$ on a vector $\mathbf{b}$ is an increasingly important primitive in machine learning, data science, and statistics, with applications such as sampling high dimensional Gaussians, Gaussian process regression and Bayesian inference, principle component analysis, and approximating Hessian spectral densities. Over the past decade, a number of algorithms enjoying strong theoretical guarantees have been proposed for this task. Many of the most successful belong to a family of algorithms called Krylov subspace methods. Remarkably, a classic Krylov subspace method, called the Lanczos method for matrix functions (Lanczos-FA), frequently outperforms newer methods in practice. Our main result is a theoretical justification for this finding: we show that, for a natural class of rational functions, Lanczos-FA matches the error of the best possible Krylov subspace method up to a multiplicative approximation factor. The approximation factor depends on the degree of $f(x)$'s denominator and the condition number of $\mathbf{A}$, but not on the number of iterations $k$. Our result provides a strong justification for the excellent performance of Lanczos-FA, especially on functions that are well approximated by rationals, such as the matrix square root.

math.NA↗

arXiv2024

On the Benefits of Rank in Attention Layers

Attention-based mechanisms are widely used in machine learning, most prominently in transformers. However, hyperparameters such as the rank of the attention matrices and the number of heads are scaled nearly the same way in all realizations of this architecture, without theoretical justification. In this work we show that there are dramatic trade-offs between the rank and number of heads of the attention mechanism. Specifically, we present a simple and natural target function that can be represented using a single full-rank attention head for any context length, but that cannot be approximated by low-rank attention unless the number of heads is exponential in the embedding dimension, even for short context lengths. Moreover, we prove that, for short context lengths, adding depth allows the target to be approximated by low-rank attention. For long contexts, we conjecture that full-rank attention is necessary. Finally, we present experiments with off-the-shelf transformers that validate our theoretical findings.

cs.LG↗

arXiv2024

Fixed-sparsity matrix approximation from matrix-vector products

We study the problem of approximating a matrix $\mathbf{A}$ with a matrix that has a fixed sparsity pattern (e.g., diagonal, banded, etc.), when $\mathbf{A}$ is accessed only by matrix-vector products. We describe a simple randomized algorithm that returns an approximation with the given sparsity pattern with Frobenius-norm error at most $(1+\varepsilon)$ times the best possible error. When each row of the desired sparsity pattern has at most $s$ nonzero entries, this algorithm requires $O(s/\varepsilon)$ non-adaptive matrix-vector products with $\mathbf{A}$. We also prove a matching lower-bound, showing that, for any sparsity pattern with $\Theta(s)$ nonzeros per row and column, any algorithm achieving $(1+\epsilon)$ approximation requires $\Omega(s/\varepsilon)$ matrix-vector products in the worst case. We thus resolve the matrix-vector product query complexity of the problem up to constant factors, even for the well-studied case of diagonal approximation, for which no previous lower bounds were known.

cs.DS↗

arXiv2022

A Quantitative Theory of Bottleneck Structures for Data Networks

The conventional view of the congestion control problem in data networks is based on the principle that a flow's performance is uniquely determined by the state of its bottleneck link, regardless of the topological properties of the network. However, recent work has shown that the behavior of congestion-controlled networks is better explained by models that account for the interactions between bottleneck links. These interactions are captured by a latent \textit{bottleneck structure}, a model describing the complex ripple effects that changes in one part of the network exert on the other parts. In this paper, we present a \textit{quantitative} theory of bottleneck structures (QTBS), a mathematical and engineering framework comprising a family of polynomial-time algorithms that can be used to reason about a wide variety of network optimization problems, including routing, capacity planning and flow control. QTBS can contribute to traffic engineering by making clear predictions about the relative performance of alternative flow routes, and by providing numerical recommendations for the optimal rate settings of traffic shapers. A particularly novel result in the domain of capacity planning indicates that previously established rules for the design of folded-Clos networks are suboptimal when flows are congestion controlled. We show that QTBS can be used to derive the optimal rules for this important class of topologies, and empirically demonstrate the correctness and efficacy of these results using the BBR and Cubic congestion-control algorithms.

cs.NI↗

arXiv2021

Spectral Top-Down Recovery of Latent Tree Models

Modeling the distribution of high dimensional data by a latent tree graphical model is a prevalent approach in multiple scientific domains. A common task is to infer the underlying tree structure, given only observations of its terminal nodes. Many algorithms for tree recovery are computationally intensive, which limits their applicability to trees of moderate size. For large trees, a common approach, termed divide-and-conquer, is to recover the tree structure in two steps. First, recover the structure separately of multiple, possibly random subsets of the terminal nodes. Second, merge the resulting subtrees to form a full tree. Here, we develop Spectral Top-Down Recovery (STDR), a deterministic divide-and-conquer approach to infer large latent tree models. Unlike previous methods, STDR partitions the terminal nodes in a non random way, based on the Fiedler vector of a suitable Laplacian matrix related to the observed nodes. We prove that under certain conditions, this partitioning is consistent with the tree structure. This, in turn, leads to a significantly simpler merging procedure of the small subtrees. We prove that STDR is statistically consistent and bound the number of samples required to accurately recover the tree with high probability. Using simulated data from several common tree models in phylogenetics, we demonstrate that STDR has a significant advantage in terms of runtime, with improved or similar accuracy.

stat.ML↗

arXiv2020

Spectral neighbor joining for reconstruction of latent tree models

A common assumption in multiple scientific applications is that the distribution of observed data can be modeled by a latent tree graphical model. An important example is phylogenetics, where the tree models the evolutionary lineages of a set of observed organisms. Given a set of independent realizations of the random variables at the leaves of the tree, a key challenge is to infer the underlying tree topology. In this work we develop Spectral Neighbor Joining (SNJ), a novel method to recover the structure of latent tree graphical models. Given a matrix that contains a measure of similarity between all pairs of observed variables, SNJ computes a spectral measure of cohesion between groups of observed variables. We prove that SNJ is consistent, and derive a sufficient condition for correct tree recovery from an estimated similarity matrix. Combining this condition with a concentration of measure result on the similarity matrix, we bound the number of samples required to recover the tree with high probability. We illustrate via extensive simulations that in comparison to several other reconstruction methods, SNJ requires fewer samples to accurately recover trees with a large number of leaves or long edges.

stat.ML↗

arXiv2019

Finding Syntactic Representations in Neural Stacks

Neural network architectures have been augmented with differentiable stacks in order to introduce a bias toward learning hierarchy-sensitive regularities. It has, however, proven difficult to assess the degree to which such a bias is effective, as the operation of the differentiable stack is not always interpretable. In this paper, we attempt to detect the presence of latent representations of hierarchical structure through an exploration of the unsupervised learning of constituency structure. Using a technique due to Shen et al. (2018a,b), we extract syntactic trees from the pushing behavior of stack RNNs trained on language modeling and classification objectives. We find that our models produce parses that reflect natural language syntactic constituencies, demonstrating that stack RNNs do indeed infer linguistically relevant hierarchical structure.

cs.CL↗

arXiv2018

Context-Free Transductions with Neural Stacks

This paper analyzes the behavior of stack-augmented recurrent neural network (RNN) models. Due to the architectural similarity between stack RNNs and pushdown transducers, we train stack RNN models on a number of tasks, including string reversal, context-free language modelling, and cumulative XOR evaluation. Examining the behavior of our networks, we show that stack-augmented RNNs can discover intuitive stack-based strategies for solving our tasks. However, stack RNNs are more difficult to train than classical architectures such as LSTMs. Rather than employ stack-based strategies, more complex networks often find approximate solutions by using the stack as unstructured memory.

cs.NE↗