Jiayi Li

Geometry · Learning dynamics · AI alignment

From the geometry of learning to the mathematics of AI alignment.

I am a postdoctoral researcher in the Section of Mathematics and Artificial Intelligence at MPI-CBG in Dresden and an ILIAD Fellow in applied mathematics for AI alignment. I received my PhD in Statistics from UCLA's Mathematical Machine Learning Group.

My research asks how the geometry of a learning system shapes what it can represent, which solutions training selects, and which features of the resulting model are identifiable. I use algebraic geometry, singular learning theory, and statistics to develop rigorous foundations for interpretability, reliability, and AI alignment.

Explore my research Curriculum vitae
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Research agenda

Structure, dynamics, and trustworthy claims

I study three connected layers of learning: the geometry of functions a model can represent, the dynamics that select among those functions, and the invariant properties on which interpretation and oversight can rely.

Selected work

Three views of learning through geometry

These projects move from the geometry of representable functions, through the dynamics that select solutions, to the identifiability needed for reliable interpretation.

Parameters map to a neuromanifold in function space Network parameters map to the neuromanifold of representable functions. Its dimension measures expressivity, while its learning degree measures algebraic training complexity. PARAMETER SPACE FUNCTION SPACE θ ∈ Θ φ 𝓜 = im(φ) dim 𝓜 measures expressivity learning degree measures training complexity

01 · Structure · Neuroalgebraic geometry

Geometry of Polynomial Neural Networks

What geometry is traced out by the functions a network can represent?

We describe the function spaces of polynomial networks as semialgebraic sets. Their dimension measures expressivity, while the learning degree captures an algebraic aspect of training complexity.

With Kaie Kubjas and Maximilian Wiesmann

Paper Research context
Training moves toward a solution with more degenerate local geometry A schematic projection of local loss geometry. The memorizing solution is drawn with a quadratic curve as a visual proxy, without implying that it is regular or nondegenerate. The lower LLC generalizing solution is drawn with a more pronounced degenerate parameter direction, representing greater near minimal parameter volume rather than ordinary curvature alone. SCHEMATIC PROJECTION OF LOCAL GEOMETRY late transition · training loss ≈ 0 MEMORIZATION λmem GENERALIZATION more degenerate · λgen < λmem lower LLC → greater near minimal volume → more posterior mass

02 · Dynamics · Singular learning theory

A Basin-Selection Perspective on Grokking via Singular Learning Theory

Why does a network generalize long after it has already fit the training data?

We interpret grokking as a transition between competing basins whose losses are nearly zero. The local learning coefficient ranks their statistical preference; optimization dynamics determine when training moves between them.

With Ben Cullen, Sergio Estan-Ruiz, and Riya Danait, from a project I mentored at LOGML 2025.

Paper Research context
Third and fourth cumulants identify a directed dynamical system For a vector autoregressive model at steady state, third and fourth cumulants supplement covariance information and allow the directed parameter matrix to be recovered when the noise is not Gaussian. VECTOR AUTOREGRESSION AT STEADY STATE xt+1 = A xt + εt x₁x₂x₃ A ENCODES THE GRAPH κ₂covariance κ₃non-Gaussian κ₄information Arecovered DAG + self-loops → generic identifiability parameters are rational functions of the cumulants

03 · Guarantees · Identifiability

Identifiability in Graphical Discrete Lyapunov Models

When can a directed mechanism be recovered from observations at steady state?

Third and fourth cumulants reveal parameters that covariance alone cannot. This establishes identifiability beyond the Gaussian setting and provides a foundation for structure learning in directed systems.

With Cecilie Olesen Recke, Sarah Lumpp, Nataliia Kushnerchuk, Janike Oldekop, Jane Ivy Coons, and Elina Robeva

Paper Research context

Now & next

Research in motion

Recent papers, talks, programs, and community building around the mathematics of learning.

I will participate in IPAM's Foundations of Interpretability workshop at UCLA, connecting mathematical perspectives on learning with questions in interpretability and AI safety.

I am an ILIAD Fellow in applied mathematics for AI alignment.

Beyond the papers

Mathematics is also a community.

I organize focused meetings and special sessions, including the inaugural Singular Learning Theory Days and sessions at the Joint Mathematics Meetings. From 2022 to 2025, I served as Editor in Chief of ACM XRDS.

I care deeply about mentoring, teaching across mathematical backgrounds, and creating spaces where new collaborations can take shape. Outside work, you will usually find me with my husband and cat, playing badminton, or trying to keep everyone near a computer properly stretched.

A little more about me
A capybara, Jiayi's lighthearted spiritual portrait

Spiritually, a slightly more faithful portrait.