Shape, Symmetries, and Structure: The Changing Role of Mathematics in Machine Learning Research

Over the past decade, the approach to machine learning research has transformed. Carefully designed, math-principled architectures yielded only marginal improvements, while compute-intensive, engineering-first methods—by scaling up training sets and model parameters—achieved new capabilities that existing theories failed to predict. Mathematics and statistics, which once primarily guided machine learning research, now struggle to directly explain the latest breakthroughs.

This shift has sparked speculation about a diminished role for mathematics in future machine learning research. However, this article argues that mathematics remains relevant, but its role is evolving. For instance, mathematics may shift from providing theoretical guarantees of model performance to post-hoc explanation of empirical phenomena in training and performance—similar to its role in physics. Likewise, the application of mathematical intuition may move from fine-grained handcrafted features and architectural details to higher-level design choices, such as matching architecture to task structure or data symmetries.

The shift toward scaling has broadened the range of mathematical fields applicable to machine learning. 'Pure' math fields such as topology, algebra, and geometry are joining traditional applied fields like probability, analysis, and linear algebra. These pure math fields, which have developed over the past century the ability to handle high abstraction and complexity, help mathematicians discover insights about spaces, algebraic objects, and combinatorial processes that initially seem beyond human intuition. These capabilities hold promise for addressing many major challenges in modern deep learning.

When analyzing a neural network with 7 billion parameters and 50 layers, the standard practice is to compute performance statistics, such as accuracy on evaluation benchmarks. But deep learning models are complex and multifaceted; two computer vision models with the same accuracy can differ greatly in 'secondary statistics' like out-of-distribution generalization, calibration, and adversarial robustness. All evidence suggests that building a complete scientific understanding of deep learning requires going beyond evaluation scores.

Unlike understanding humans, all parameters and computational processes of a model are accessible. By extracting hidden activations, one can directly trace the transformation from raw input to prediction. However, the world of hidden activations is not friendly: they are typically high-dimensional and not structured in a human-comprehensible way like input data. In weight space, one can move from a starting point in millions to billions of orthogonal directions. How can we make sense of these worlds?

Mathematical research involves not only solving problems but also understanding what questions to ask. Many mathematical objects are far from everyday experience, with limited initial intuition. Significant effort is needed to build tools that leverage existing intuition and yield tractable results. Take rotations as an example: familiar in 2 and 3 dimensions, but as dimensions increase, intuition becomes inaccessible. Different perspectives provided by pure mathematics are crucial for gaining a more comprehensive understanding.

For instance, an n-dimensional rotation can be represented by an n×n orthogonal matrix (determinant 1), with the set denoted SO(n). There are multiple complementary ways to understand SO(n): explore its linear algebraic structure, study each element's behavior as an operator, or use spatial intuition. This multi-perspective approach helps understand objects in high-dimensional spaces, similarly applicable to analyzing model weights and activations.