Brandeis Math Bio Seminar - Apr. 22cnd, 2026 - Daniel Lazarev (MIT)

Organizers: Thomas Fai, Jonathan Touboul, Yangyang Wang, Sreshta Venkatakrishnan, Nate Sutton (Brandeis). Title: Entropy-extremizing representations with applications to biology Abstract: Exploring alternative representations of mathematical and physical objects is central to progress in mathematics and science. From changes of variables to integral transformations, the discovery of effective representations often reveals hidden structure and enables the solution of otherwise difficult, or even intractable problems. In the era of large-scale and high-dimensional data, this challenge has taken on renewed importance. Modern machine learning methods seek representations that extract meaningful structure from increasingly complex datasets, yet the discovery of such representations typically remains heuristic and problem-specific. Both in mathematics and in data science, a general theory explaining why certain representations are optimal, and how to construct them systematically, remains largely absent. In this talk, I’ll outline a unifying framework based on entropy-extremizing representations, in which optimal representations arise as solutions to variational principles defined by entropy under structural constraints. Within this framework, entropy serves as a measure of information content relative to a specified mathematical structure, and extremizing it identifies canonical representations satisfying the desired constraints. We will briefly outline some theoretical results before moving on and focusing on applications to computational biology and machine learning. The first of these is GUIDE (Generic Unmixing by Independent Decomposition), a statistical method based on entropy minimization that uncovers latent, biologically meaningful structure in complex genetic architectures. Next, we will introduce w-values, which are new statistical measures derived from hyperspherical geometry for evaluating neural network weights with applications to model compression and pruning. Finally, we will present DiffEvol, a diffusion-based model of evolutionary dynamics that recasts evolution as a mutation-driven diffusion within a constrained subspace of genotype space. Together, these results demonstrate how entropy-extremizing principles can provide both a mathematical foundation and a practical methodology for discovering informative representations across mathematics, machine learning, and biology.