About my research
Neuromanifolds: the geometry of model expressivity
Given a parametrization Ψ: Θ → 𝓕, its image is a neuromanifold.
The geometry induced by this parametrization shapes optimization dynamics.
- Neural networks
- Nonnegative tensor factorization
- The principal-minor map
I am working on questions at the intersection of algebraic geometry and machine learning. More concretely, I am interested in how to apply algebro-geometric tools to understand the geometry of function spaces of algebraic neural networks.
For a fixed neural network architecture, the network parameters map into an ambient space, and the image is a semi-algebraic set called a neuromanifold. The geometry and topology of the neuromanifold affect the training dynamics of neural networks. I am interested in how topology, geometry, and invariants can improve our understanding of black-box neural network training and potentially bring practical advantages.
Publications
Submitted
Algebraic geometry of rational neural networks
- Authors
- with Alexandros Grosdos and Elina Robeva
- Record
- Submitted, 2026.
Abstract
A geometric study of shallow rational neural networks and the function spaces they represent, with singularities providing structure that can be investigated through algebraic geometry.
Editorial summarySubmitted
Expressivity of Shallow Neural Networks Over Finite Fields
- Authors
- with Carol Wu, Shiwei Yang, Param Mody, and Yifei Chen
- Record
- Submitted, 2026.
Abstract
An investigation of the expressivity and geometry of shallow neural networks over finite fields—a discrete setting that also offers a model for networks with low-precision weights.
Editorial summaryPublished
Sign patterns of principal minors of real symmetric matrices
- Authors
- with Tobias Boege and Jesse Selover
- Record
- Published in Linear Algebra and its Applications, 738 (2026), 161–188.
Abstract
A study of the sign patterns that can occur among the principal minors of real symmetric matrices, viewed through the structure of a constrained real-algebraic problem.
Editorial summaryPublished
Geometry of Rank Constraints in Shallow Polynomial Neural Networks
- Authors
- with Param Mody
- Record
- Published at MOSS @ ICML, 2025.
Abstract
A connection between shallow polynomial neural networks, tensor decomposition, and the geometry of rank constraints arising from monomial activations.
Editorial summaryPublished
Likelihood Geometry of Determinantal Point Processes
- Authors
- with Hannah Friedman and Bernd Sturmfels
- Record
- Published in Algebraic Statistics, 15 (2024), no. 1, 15–25.
Abstract
An examination of the likelihood geometry of determinantal point processes through the algebraic structure of their statistical models.
Editorial summaryPublished
Chromatic Graph Homology for Brace Algebras
- Authors
- with Vladimir Baranovsky
- Record
- Published in New York Journal of Mathematics, 23 (2017), 1307–1319.
Abstract
A development of chromatic graph homology in the setting of brace algebras, bringing graph-theoretic constructions into conversation with algebraic operations.
Editorial summary
The neural network framework
- Step 1
Fix the architecture
Depth, width, and connectivity stay fixed.
- Step 2
Choose an activation function
Polynomial, rational, ReLU, or tropical.
- Step 3
Define the parameter map
w ↦ fw
Neuromanifold
equations · fibers · topology
- I
Polynomial networks
Monomial activations connect neural-network function spaces with tensor decomposition and simultaneous Waring rank.
- II
Rational networks
Rational activations lead toward Chow varieties and toward questions about learning the location and type of singularities from data.
- III
Finite fields
Finite-field networks offer a discrete setting for studying expressivity, geometry, and low-precision weights.
- IV
ReLU neural networks
Prony varieties and piecewise-linear geometry.
- V
Tropical neural networks
Tropical geometry and out-of-distribution generalization.