Some research projects (under construction)
Strict saddle optimization
Some nonconvex problems have a strict saddle geometry: strong convexity near minimizers and negative curvature at saddles. We study Riemannian trust-region methods in this setting, showing that they find approximate minimizers with an iteration count that scales only logarithmically with accuracy. We also design an inexact variant that chooses steps using the strict saddle structure, improving complexity guarantees over the general nonconvex case.
Nonlinear matrix recovery
Reconstruct structured matrices from a small set of affine measurements by enforcing that columns satisfy a nonlinear geometric structure (clusters, algebraic varieties). We cast the problem as a optimization on a smooth manifold and solve it with Riemannian trust-region or alternating minimization.
Point cloud registration on algebraic varieties
Align two point clouds of the same object that is well approximated by an algebraic variety. We first estimate the algebraic variety that best approximates the data. Then, we compute the transformation by enforcing that the surfaces of both objects overlap. This handles noisy samples and preserves the shape of the data.