Georgia Institute of Technology College of Sciences

We present a mixed atomic matrix norm that, when used as regularization in optimization problems, promotes low-rank matrices with sparse factors. We show that in convex lifted formulations of sparse phase retrieval and sparse principal component analysis (PCA), this norm provides near-optimal sample complexity and error rate guarantees. Since statistically optimal sparse PCA is widely believed to be NP-hard, this leaves open questions about how practical it is to compute and optimize this atomic norm. Motivated by convex duality analysis, we present a heuristic algorithm in the case of sparse phase retrieval and show that it empirically matches existing state-of-the-art algorithms.

One interesting question in low-dimensional topology is to understand the structure of mapping class group of a given manifold. In dimension 2, this topic is very well studied. The structure of this group is known for various 3-manifolds as well (ref- Hatcher's famous work on Smale's conjecture). But virtually nothing is known in dimension 4. In this talk I will try to motivate why this problem in dimension 4 is interesting and how it is different from dimension 2 and 3. I will demonstrate some "exotic" phenomena and if time permits, I will talk a few words on my upcoming work with Jianfeng Lin. 

Fix a graph $F$. A classical problem in extremal graph theory asks about how many induced copies of $F$ can a graph with edge density $\rho$ have? The only case in which we know the asymptotic solution is when $F$ is a complete graph, and it was solved completely only recently by Reiher using the flag algebra machinery. We will consider the other cases and show some results when $F$ is a complete bipartite graph or a complete graph minus one edge. Many interesting related open problems will also be introduced. Joint work with Dhruv Mubayi and Christian Reiher.

In this talk, we develop algorithms for numerical computation, based on ideas from competitive games and statistical inference. 

In the first part, we propose competitive gradient descent (CGD) as a natural generalization of gradient descent to saddle point problems and general sum games. Whereas gradient descent minimizes a local linear approximation at each step, CGD uses the Nash equilibrium of a local bilinear approximation. Explicitly accounting for agent-interaction significantly improves the convergence properties, as demonstrated in applications to GANs, reinforcement learning, and computer graphics.

In the second part, we show that the conditional near-independence properties of smooth Gaussian processes imply the near-sparsity of Cholesky factors of their dense covariance matrices. We use this insight to derive simple, fast solvers with state-of-the-art complexity vs. accuracy guarantees for general elliptic differential- and integral equations. Our methods come with rigorous error estimates, are easy to parallelize, and show good performance in practice.