About me

Hi! I am a PhD candidate at MIT advised by Philippe Rigollet. I obtained my B.Sc. (2016) in mathematics and electrical engineering from Universidad de los Andes.

I enjoy teaching at all levels and have taught at MIT Mathematics, IDSS, Mathroots, Universidad de los Andes, and Olimpiadas Colombianas de Matemáticas.

I worked at Quantil (2017) as a data scientist, and at Truveta (2022) as a research intern working on large language models.

Research interests

I am broadly interested in the theory and applications of statistics, optimization, and machine learning.

During my PhD, I worked on geometry and optimization aspects of different problems in machine learning:

• Non-convex dynamics of neural network training. There is a phase tansition in the learning between gradient flow regime and a “edge of stability” regime. We characterize a simple and generalizable class of functions with this property.
• Interpolation based on unbalanced optimal transport. The Eulerian and Lagrangian dynamic formulations of unbalanced optimal transport agree to second order. We construct an algorithm that interpolates measures of different masses using De Casteljau’s algorithm on the cone.
• Fisher-Rao geometry. Regularized linear programs are equivalent to gradient flows in the geometry of the barrier’s Hessian. We provide a unified framework that explains the differences and similarities of using entropy (Sinkhorn, mirror descent) versus Self-concordant (interior-point) algorithms.
• Matrix Factorization. Least squares costs for matrix factorization does not have spurious optima over the Grassmanian (modulo rotations). We provide a simple proof of convexity along orthogonal directions using principal angles.