Georgia Institute of Technology College of Sciences

Generative Adversarial Networks (GANs) are powerful tools for creating new content, but they face challenges such as sensitivity to starting conditions and mode collapse. To address these issues, we propose a deep generative model that utilizes the Gromov-Monge embedding (GME). It helps identify the low-dimensional structure of the underlying measure of the data and then map it, while preserving its geometry, into a measure in a low-dimensional latent space, which is then optimally transported to the reference measure. We guarantee the preservation of the underlying geometry by the GME and c-cyclical monotonicity of the generative map, where c is an intrinsic embedding cost employed by the GME. The latter property is a first step in guaranteeing better robustness to initialization of parameters and mode collapse. Numerical experiments demonstrate the effectiveness of our approach in generating high-quality images, avoiding mode collapse, and exhibiting robustness to different starting conditions.

Galois groups embody the structure of algebraic equations arising in both enumerative geometry and various scientific applications where such equations must be solved. I will describe a line of work that aims to elucidate the role of Galois groups in applications where data taken from multiple images are used to reconstruct a 3D scene. From this perspective, I will revisit two well-known solutions to camera pose estimation problems, which originate from classical photogrammetry and are still heavily used within modern 3D reconstruction systems. I will then discuss some less-classical problems, for which the insight we gleaned from computing Galois groups led to significant practical improvements over previous solutions. A key ingredient was the use of numerical homotopy continuation methods to (heuristically) compute monodromy permutations. Time-permitting, I will explain how such methods may also be used to automatically recover certain symmetries underlying enumerative problems. 

Given graphs G and H and a positive integer q, an (H,q)-coloring of G is an edge-coloring in which each copy of H receives at least q colors. Erdős and Shelah raised the question of determining the minimum number of colors, f(G,H,q), which are required for an (H,q)-coloring of G. Determining f(K_n,K_p,2) for all n and p is equivalent to determining the classical multicolor Ramsey numbers. Recently, Mubayi and Joos introduced the use of a new method for proving upper bounds on these generalized Ramsey numbers; by finding a “conflict-free" matching in an appropriate auxiliary hypergraph, they determined the values of f(K_{n,n},C_4,3) and f(K_n,K_4,5). In this talk, we will show how to generalize their approach to give bounds on the generalized Ramsey numbers for several families of graphs. This is joint work with Deepak Bal, Patrick Bennett, and Shira Zerbib.

We present new results concerning characterizations of the spaces $C^{1,\alpha}$ and “$LI_{\alpha+1}$” for $0<\alpha<1$.  The space $LI_{\alpha +1}$ is the space of Lipschitz functions with $\alpha$-order fractional derivative having bounded mean oscillation.  These characterizations involve geometric square functions which measure how well the graph of a function is approximated by a hyperplane at every point and scale.  We will also discuss applications of these results to higher-order rectifiability.

In this talk, we introduce and survey continuous-time deep learning approaches based on neural ordinary differential equations (neural ODEs) arising in supervised learning, generative modeling, and numerical solution of high-dimensional optimal control problems. We will highlight theoretical advantages and numerical benefits of neural ODEs in deep learning and their use to solve otherwise intractable PDE problems.