Can a Large Language Model Learn Matrix Functions In Context?

TL;DR

This paper investigates the ability of Large Language Models to learn and perform matrix functions, especially those involving Singular Value Decomposition, demonstrating their potential for complex numerical tasks in in-context learning.

Contribution

The study shows that LLMs can effectively learn complex matrix functions and outperform traditional models on challenging tasks, highlighting their scalability and efficiency in high-dimensional computations.

Findings

LLMs perform well on complex matrix functions involving SVD.

They outperform classical models on top-k singular value tasks.

High accuracy maintained as matrix size increases.

Abstract

Large Language Models (LLMs) have demonstrated the ability to solve complex tasks through In-Context Learning (ICL), where models learn from a few input-output pairs without explicit fine-tuning. In this paper, we explore the capacity of LLMs to solve non-linear numerical computations, with specific emphasis on functions of the Singular Value Decomposition. Our experiments show that while LLMs perform comparably to traditional models such as Stochastic Gradient Descent (SGD) based Linear Regression and Neural Networks (NN) for simpler tasks, they outperform these models on more complex tasks, particularly in the case of top-k Singular Values. Furthermore, LLMs demonstrate strong scalability, maintaining high accuracy even as the matrix size increases. Additionally, we found that LLMs can achieve high accuracy with minimal prior examples, converging quickly and avoiding the overfitting seen in classical models. These results suggest that LLMs could provide an efficient alternative to classical methods for solving high-dimensional problems. Future work will focus on extending these findings to larger matrices and more complex matrix operations while exploring the effect of using different numerical representations in ICL.