Georgia Institute of Technology College of Sciences

We will prove lower bounds for energy functionals of mappings of the real, complex and quaternionic projective spaces with their canonical Riemannian metrics.  For real and complex projective spaces, these results are sharp, and we will characterize the family of energy-minimizing mappings which occur in these results.  For complex projective spaces, these results extend to all Kahler metrics.  We will discuss the connections between these results and several theorems and questions in systolic geometry.

Wild-type zebrafish are named for their dark and light stripes, but mutant zebrafish feature variable skin patterns, including spots and labyrinth curves. All of these patterns form as the fish grow due to the interactions of tens of thousands of pigment cells in the skin. This leads to the question: how do cell interactions change to create mutant patterns? The longterm biological motivation for my work is to shed light on this question — I strive to help link genes, cell behavior, and visible animal characteristics. Toward this goal, I build agent-based models to describe cell behavior in growing fish body and fin-shaped domains. However, my models are stochastic and have many parameters, and comparing simulated patterns, alternative models, and fish images is often a qualitative process. This, in turn, drives my mathematical goal: I am interested in developing methods for quantifying variable cell-based patterns and linking computational and analytically tractable models. In this talk, I will overview our agent-based models for body and fin pattern formation, share how topological data analysis can be used to quantify cell-based patterns and models, and discuss ongoing work on relating agent-based and continuum models for zebrafish patterns.

Given an affine space of matrices L and a matrix Θ ∈ L, consider the problem of computing the closest rank deficient matrix to Θ on L with respect to the Frobenius norm. This is a nonconvex problem with several applications in control theory, computer algebra, and computer vision. We introduce a novel semidefinite programming (SDP) relaxation, and prove that it always gives the global minimizer of the nonconvex problem in the low noise regime, i.e., when Θ is close to be rank deficient. Our SDP is the first convex relaxation for this problem with provable guarantees. We evaluate the performance of our SDP relaxation in examples from system identification, approximate GCD, triangulation, and camera resectioning. Our relaxation reliably obtains the global minimizer under non-adversarial noise, and its noise tolerance is significantly better than state of the art methods.

In the past decade, the revival of deep neural networks has led to dramatic success in numerous applications ranging from computer vision to natural language processing to scientific discovery and beyond. Nevertheless, the practice of deep networks has been shrouded with mystery as our theoretical understanding of the success of deep learning remains elusive.

In this talk, we will exploit low-dimensional modeling to help understand and improve deep learning performance. We will first provide a geometric analysis for understanding neural collapse, an intriguing empirical phenomenon that persists across different neural network architectures and a variety of standard datasets. We will utilize our understanding of neural collapse to improve training efficiency. We will then exploit principled methods for dealing with sparsity and sparse corruptions to address the challenges of overfitting for modern deep networks in the presence of training data corruptions. We will introduce a principled approach for robustly training deep networks with noisy labels and robustly recovering natural images by deep image prior.