There are a variety of problems at the foundation of Data Science, for which the best problems require ideas and techniques from multiple mathematical disciplines. Currently, I am particularly interested in using ideas from mathematical optimization, geometry, and statistics. Below are some research areas I am actively exploring.
Riemannian Optimization
The central problem of interest of Riemannian optimization is to do the following task:
\[ \min_{x \in \mathcal{M}} f(x) \]
where \(\mathcal{M}\) is a Riemannian manifold, and \(f: \mathcal{M} \to \mathbb{R}\) is some cost function. The goal of the optimizer is to provide black-box iterative algorithms to solve this problem, with provable guarantees.
The central difficulty of Riemannian optimization is that the manifold \(\mathcal{M}\) are generally nonlinear spaces. Given first-order information of \(f\), i.e. an oracle such that \(x \mapsto (f(x), \nabla_{\mathcal{M}}f(x))\), one does not have access to exact exponential maps — the operation which turns directions into new iterates. Additionally, some schemes gradients from nearby points construct more accurate surrogates of \(f\). In the Riemannian setting, we do not even have access to exact parallel transports — the operation which moves first-order information to a shared space! Even worse, due to the intrinsic curvature of \(\mathcal{M}\), exact parallel transports also incur errors!
Submanifolds of Wasserstein Space
In Construction
Optimization in Wasserstein Space
In Construction