Georgia Institute of Technology College of Sciences

Neural networks with a large number of parameters admit a mean-field description, which has recently served as a theoretical explanation for the favorable training properties of "overparameterized" models. In this regime, gradient descent obeys a deterministic partial differential equation (PDE) that converges to a globally optimal solution for networks with a single hidden layer under appropriate assumptions. In this talk, we propose a non-local mass transport dynamics that leads to a modified PDE with the same minimizer. We implement this non-local dynamics as a stochastic neuronal birth-death process and we prove that it accelerates the rate of convergence in the mean-field limit. We subsequently realize this PDE with two classes of numerical schemes that converge to the mean-field equation, each of which can easily be implemented for neural networks with finite numbers of parameters. We illustrate our algorithms with two models to provide intuition for the mechanism through which convergence is accelerated. Joint work with G. Rotskoff (NYU), S. Jelassi (Princeton) and E. Vanden-Eijnden (NYU).

We prove sparse bounds for the spherical maximal operator of Magyar,Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint esti-mate. The new method of proof is inspired by ones by Bourgain and Ionescu, is veryefficient, and has not been used in the proof of sparse bounds before. The Hardy-Littlewood Circle method is used to decompose the multiplier into major and minor arccomponents. The efficiency arises as one only needs a single estimate on each elementof the decomposition.

In the design of complex engineering systems like aircraft/rotorcraft/spacecraft, computer experiments offer a cheaper alternative to physical experiments due to high-fidelity(HF) models. However, such models are still not cheap enough for application to Global Optimization(GO) and Uncertainty Quantification(UQ) to find the best possible design alternative. In such cases, surrogate models of HF models become necessary. The construction of surrogate models requires an offline database of the system response generated by running the expensive model several times. In general, the training sample size and distribution for a given problem is unknown apriori and can be over/under predicted, which leads to wastage of resources and poor decision-making. An adaptive model building approach eliminates this problem by sequentially sampling points based on information gained in the previous step. However, an approach that works for highly non-stationary response is still lacking in the literature. Here, we use Gaussian Process(GP) models as surrogate model. We employ a novel process-convolution approach to generate parameterized non-stationary.

GPs that offer control on the process smoothness. We show that our approach outperforms existing methods, particularly for responses that have localized non-smoothness. This leads to better performance in terms of GO, UQ and mean-squared-prediction-errors for a given budget of HF function calls.

Matroids are basic combinatorial objects arising from graphs and vector configurations. Given a vector configuration, I will introduce a “matroid Schubert variety” which shares various similarities with classical Schubert varieties. I will discuss how the Hodge theory of such matroid Schubert varieties can be used to prove a purely combinatorial conjecture, the “top-heavy” conjecture of Dowling-Wilson. I will also report an on-going project joint with Tom Braden, June Huh, Jacob Matherne, Nick Proudfoot on the cohomology theory of non-realizable matroids.