Geometry and Statistical Computing

As an optimization-based alternative to traditional Markov chain Monte Carlo approaches, variational inference (VI) is becoming increasingly popular for approximating intractable posterior distributions in large-scale Bayesian models due to its comparable efficacy and superior computational efficiency. Several recent works provide theoretical justifications of VI by proving its statistical optimality for parameter estimation under various settings. More recently, there is increasing interest in studying algorithmic properties of popular variational inference algorithms. In this talk, we present careful modifications to Gaussian VI and coordinate ascent VI algorithms, and demonstrate their improved convergence behavior across a variety of statistical applications.  

We introduce the \textit{Riemannian Proximal Sampler}, a method for sampling from densities defined on Riemannian manifolds. The performance of this sampler critically depends on two key oracles: the \textit{Manifold Brownian Increments (MBI)} oracle and the \textit{Riemannian Heat-kernel (RHK)} oracle. We establish high-accuracy sampling guarantees for the Riemannian Proximal Sampler, showing that generating samples with \(\varepsilon\)-accuracy requires \(\mathcal{O}(\log(1/\varepsilon))\) iterations in Kullback-Leibler divergence assuming access to exact oracles and \(\mathcal{O}(\log^2(1/\varepsilon))\) iterations in the total variation metric assuming access to sufficiently accurate inexact oracles. Furthermore, we present practical implementations of these oracles by leveraging heat-kernel truncation and Varadhan's asymptotics. In the latter case, we interpret the Riemannian Proximal Sampler as a discretization of the entropy-regularized Riemannian Proximal Point Method on the associated Wasserstein space. We provide preliminary numerical results that illustrate the effectiveness of the proposed methodology.
 

Phylogenetic trees represent the shared evolutionary history of organisms, and collections of these trees arise in modern biodiversity studies. The Billera-Holmes-Vogtmann (BHV) space is a non-Euclidean continuous metric space whose metric compares trees with respect to both their discrete structure and branch lengths. Remarkably, while the number of possible topologies grows super-exponentially in the number of leaves, the geometrical properties of BHV space admit a polynomial-time algorithm for distance computations. However, BHV distances are only defined between trees with identical leaf sets, limiting their use in practice. To address this, we propose an extension of BHV space to compare trees with overlapping, but not identical, leaf sets. Our proposal consists of "towers" of BHV spaces, where tower levels are traversed by adding or removing leaves with external branches of length zero adjacent to internal edges. We discuss the construction of the "Towering Tree Space", its geometric properties, distance computation, and statistical analysis using the towering metric. This is joint work with Amy Willis.