Research
My research in the mathematics of data science lies at the intersection of stochastic and convex geometry, high dimensional probability, and statistical learning theory.
Random tessellation forests and features
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. In joint work with Ngoc Tran, we utilize tools from stochastic geometry to resolve open questions on the generalization of cut directions in the Mondrian process. Our results generalize the use of random partitions for kernel approximation, show minimax rates of convergence for STIT random forests, and allow precise comparisons between the Mondrian forest and the Laplace kernel in density estimation.
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
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 convex set approximation
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.
