Entropy-Extremizing Representations
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Lazarev-927381574-PhD-math-2026-thesis.pdf
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Author(s)
Lazarev, Daniel
Advisor(s)
Berger, Bonnie
Neale, Benjamin M.
Cohn, Henry
Date Issued
May 2026
Publisher
Massachusetts Institute of Technology
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.
This thesis presents 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.
Here, I develop both the theoretical foundations and practical applications of this idea. First, I introduce a theory of universal entropy in structure spaces, showing that entropies associated with different mathematical contexts arise as instances of a common universal construction. Building on this perspective, I develop entropy-extremizing formulations of several mathematical problems, including representations of uniformly distributed subspaces and duality relations based on the generalized Stokes theorem.
The framework is then applied to computational biology and machine learning. I develop GUIDE (Generic Unmixing by Independent Decomposition), a statistical method based on entropy minimization that uncovers latent, biologically meaningful structure in complex genetic architectures. I further introduce w-values, a statistically grounded measure derived from hyperspherical geometry for evaluating neural network weights with applications to model compression and pruning. Finally, I 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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