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My research in the mathematics of data science lies at the intersection of stochastic and convex geometry, high-dimensional probability, and statistical learning theory.
An up-to-date list of my publications can be found on Google Scholar. My research is currently supported by the NSF.
Star Geometry + Regularizer Learning
Star Geometry + Regularizer Learning
In optimization-based approaches to inverse problems and statistical inference, it is common to add a regularizer to the objective to promote desired structural properties in the solution. A combination of prior domain information and computational considerations typically drives the choice of a suitable regularizer. Convex regularizers are attractive computationally, but are limited in the types of structure they can promote. Nonconvex regularizers are more flexible, and have strong empirical performance in some applications, but the associated optimization problems come with computational challenges. We study the class of regularizers defined by Minkowski functionals of star bodies, a class of compact sets that include both convex bodies and nonconvex sets. Some of the questions our work aims to answer include:
What is the optimal regularizer to employ for a given data source from this class?
What properties of a data source govern whether its optimal regularizer is convex?
Machine Learning + Random Tessellations
Machine Learning + Random Tessellations
The Mondrian process in machine learning is a recursive partition of space with random axis-aligned cuts used to build random forests that achieve minimax optimal rates and Laplace kernel approximations, and it can be viewed as a special case of the stable under iterated tessellation (STIT) in stochastic geometry. My work uses the theory of stationary random tessellations to study learning algorithms based on oblique variants of the Mondrian process where linear combinations of covariates are used to generate splits.
Spectrahedral Regression
Spectrahedral Regression
Convex regression is the problem of fitting a convex function to a data set consisting of input-output pairs. In joint work with Venkat Chandrasekaran, we present a new approach to this problem called spectrahedral regression. In this method, we fit a function that is given by the maximum eigenvalue of an affine matrix expression of the input, which we call a spectrahedral function. This method generalizes max-affine regression, in which a maximum of a fixed number of affine functions is fit to the data. In this work, we obtained bounds on how well spectrahedral functions can approximate arbitrary convex functions via statistical risk analysis. We also analyzed an alternating minimization algorithm for the non-convex optimization problem of fitting the best spectrahedral function to a given data set.
Determinantal Point Processes
Determinantal Point Processes
The most commonly studied class of point processes are Poisson processes, where all points are stochastically independent from each other. However, models exhibiting interaction are often needed. Determinantal point processes (DPPs) are a useful model exhibiting repulsion between particles that appear in random matrix theory and have found uses in machine learning for selecting diverse sets from a data collection. A useful property of DPPs is that conditional distributions of DPPs are also DPPs. This allows us to quantify the repulsiveness of a DPP, and in joint work with Jesper Møller, we showed that the repulsive effect of a point at a given location is to push out one other point with some probability, and the distribution of the point removed depends on the associated kernel of the DPP.
High Dimensional Random Convex Sets
High Dimensional Random Convex Sets
In joint work with Gilles Bonnet, we considered the model of i.i.d. points chosen uniformly from the unit sphere and study the asymptotic behavior of the facets as dimension and number of points tend to infinity. The geometry of high dimensional space imposes different regimes for the number of points and dimension with different asymptotic behavior of the facets. We obtain asymptotic formulas for approximation of the sphere and random spherical Delaunay tessellations in each case, illuminating the limiting shapes of these polytopes in high dimensions.
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