Guido Montúfar

My research is in mathematical machine learning. I am interested in the mathematical structure of neural networks and learning algorithms, including the geometry and expressivity of neural architectures, optimization and implicit bias, and methods for verification and reliable AI.

Our work draws on geometry, algebra, probability, optimization, and computation. I lead the Mathematical Machine Learning group at MPI MiS and work with students and researchers across Mathematics and Statistics & Data Science at UCLA.

Math Machine Learning Seminar · Prospective students and researchers · Complete publication list

Research

Structure and representations

We study how architecture and parameterization constrain the functions and representations realized by neural networks. Topics include expressivity, identifiability, symmetries, functional dimension, and algebraic and geometric structure of ReLU, maxout, and attention models.

geometry · expressivity · attention

Program: DFG SPP 2298 · Theoretical Foundations of Deep Learning

Learning dynamics and implicit bias

We investigate how optimization selects among the many solutions available to overparameterized models. Current questions concern feature learning, implicit regularization, large-step dynamics, optimization geometry, and the interaction between parameterization and learned representations.

optimization · implicit bias

Program: NSF MoDL · Occam's Razor in Deep and Physical Learning

Verification of AI systems

We develop mathematical and computational methods for certifying AI systems. This includes algebraic approaches to robustness verification, systematic stress-testing of verifiers, and function-level guarantees for reliable machine learning systems.

verification · reliable AI

Program: DARPA AIQ · Constraints for Provable Extrapolation in AI

Mathematics and AI

In an emerging direction, we study the mathematical structure of learning and reasoning in AI systems, the development of AI systems capable of creative mathematical reasoning, and the use of AI for mathematical search and discovery.

AI & mathematics

Program: NSF AIMing · Synergistic Advancement of AI and Mathematics

Selected research highlights

2026

Effective width collapse in wide ReLU networks

implicit biasfeature learning

In the mean-field regime, noisy SGD with weight decay can select solutions whose nonlinear structure is supported on finitely many feature directions, providing a mechanism by which very wide networks acquire effectively finite width.

paper · code

2026

Most ReLU networks admit identifiable parameters

identifiabilityReLU geometry

We study parameter identifiability in ReLU networks and show that, under broad conditions, generic network parameters are identifiable from the realized function up to the inherent symmetries of the parameterization.

paper

2026

Large step sizes, chaos, and fractal convergence

optimizationdynamics

Gradient descent beyond the classical small-step regime can exhibit complicated dynamical behavior, including chaotic trajectories and fractal boundaries between regions with different convergence behavior.

paper · code

2026

VeriStress: stress-testing neural network verifiers

verificationreliable AI

VeriStress constructs provably robust neural-network instances designed to systematically stress-test verification methods, separating ground-truth robustness from the ability of a verifier to certify it.

paper · workshop · code

2026

Algebraic structure in self-attention

attentionalgebraic geometry

Lightning self-attention satisfies nontrivial algebraic constraints. Identifying these invariants provides a route toward understanding the geometry of attention models and toward exact algebraic methods for verification.

paper · code

2026

Learning to optimize triangulations

AI & mathematicscombinatorics

TriSearch studies learned search over triangulations through bistellar flips, providing a test bed for machine learning on highly structured mathematical state spaces.

paper

All publications →

Mathematical Machine Learning Group

The group spans UCLA and the Max Planck Institute for Mathematics in the Sciences. Our work develops mathematical foundations for understanding machine learning systems, drawing on ideas from geometry, algebra, probability, optimization, statistics, and computation.

Postdocs

Patrícia Muñoz EwaldHedrick Math Fellow, UCLA
Moritz GrilloPostdoc, MPI MiS

PhD students

Cash BowmanUCLA Statistics & Data Science
Ethan YoungUCLA Statistics & Data Science
David TroxellUCLA Statistics & Data Science
Yiran WangUCLA Electrical & Computer Engineering
Khang NguyenUCLA Mathematics
Shuang LiangUCLA Statistics & Data Science
Ryan AndersonUCLA Statistics & Data Science
Hao DuanUCLA Statistics & Data Science

Undergraduate research

Sofia HuntUCLA
Stephanie LeiUCLA
Emma Ventura FontMPI MiS

MPI MiS group page → · Former group members & research visitors →

Opportunities

I regularly receive inquiries from prospective students and researchers. The following summarizes the main routes for working with our group.

PhD students: Prospective PhD students apply through the UCLA Mathematics PhD program or the UCLA Statistics & Data Science PhD program. Applications are reviewed by the respective departmental admissions committees. Students interested in mathematical machine learning are welcome to indicate relevant faculty and research interests in their applications.
Postdoctoral researchers: Postdoctoral opportunities arise through UCLA and MPI MiS. See UCLA Mathematics positions for 2026–27 and the Statistics and Data Theory RTG opportunities for current routes.
UCLA students: Research projects for UCLA undergraduate and graduate students depend on preparation, research fit, and current advising capacity. Courses and the Mathematical Machine Learning Seminar are useful ways to become familiar with our research.
Visitors and internships: For students outside UCLA, structured research programs or externally funded exchanges are generally the most practical route. For undergraduate research routes and summer programs, see the UCLA Mathematics research opportunities for undergraduates.
High-school students: For mathematically advanced school students, the UCLA Olga Radko Endowed Math Circle (ORMC) offers enrichment and advanced mathematical activities for K–12 students.

When contacting me about research: please indicate the specific research direction or recent paper that connects with your background, and briefly describe the mathematical question or connection that interests you. You are also very welcome to attend the Math Machine Learning Seminar, which is a good way to learn about the research interests of our group and the broader community around it.