Algebra ⟶ non-linear statistics
I build statistical models for discrete relational data that capture more complex behavior than traditional models, study them through the algebrogeometric lens, prove their interpretability in practice, and develop scalable model/data fit testing methodologies using a blend of combinatorial, algebraic, probabilistic, and Bayesian algorithms.
Randomness ⟶ non-linear algebra
I study randomized algorithm approaches to computational algebra problems whose expected runtimes are much lower then the well-known worst-case complexity bounds, develop probabilistic models to study average and extreme behavior of algebraic objects, and use machine learning to predict and improve behavior of algebraic computations.
Algebraic statistics for network models
Non-asymptotic goodness-of-fit tests based on Markov bases;
Existence and complexity of MLE;
Dynamic combinatorially-inspired data-oriented algorithms for model fitting;
Application to (large) sparse network data;
DARPA FA9550-12-1-0392 (2012-2013), AFOSR FA9550-14-1-0141 (2014-2017).
Randomness and learning for non-linear algebra
Stochastic non-linear algebra;
Solving systems of multivariate equations;
Fast randomized algorithms for large structured systems;
Randomized structures in algebra with applications;
Machine learning for computational algebra.
Simons Foundation, through a Collaboration Grant for Mathematicians 854770 (2021-2026).