# Research Projects

#### Invnet emulator

Generative Adversarial Networks (GANs), while widely successful in modeling complex data distributions, have not yet been sufficiently leveraged in scientific computing and design. Reasons for this include the lack of flexibility of GANs to represent discrete-valued image data, as well as the lack of control over physical properties of generated samples. We propose a new conditional generative modeling approach (InvNet) that efficiently enables modeling discrete-valued images, while allowing control over their parameterized geometric and statistical properties. We evaluate our approach on several synthetic and real world problems, navigating manifolds of geometric shapes with desired sizes; generation of binary two-phase materials; and the (challenging) problem of generating multi-orientation polycrystalline microstructures.

#### Robustifying Machine Learning under Semantic Constraints

We explore the space of adversarial examples in terms of semantically valid images. Our approach relies on the use of generative models to simulate the semantic transformations of images.

#### Physics-aware Deep Generative Models for Creating Synthetic Microstructures

We integrate classical engineering approaches (i.e., physics models) with machine learning models such as generative models to transform and accelerate such design exploration process.

#### Reconstruction from Periodic Nonlinearities, with applications in HDR imaging

Our aim is a reliable estimation of a signal or image from its periodic nonlinearities, with a focus on a periodic nonlinear observation model named modulo sensor encountered in high-dynamic range (HDR) imaging.

#### Solving inverse problems using GAN priors

We leverage the ability of Generative Adversarial Networks (GANs) of learning the real data distribution by using the Generator function as a prior on natural images.

#### Provable Algorithms for training ReLU networks

We prove linear convergence of both gradient descent and a new scheme called alternating minimization for training ReLU based 2-layer networks.

#### Sparse image super resolution

We consider the problem of super-resolution for sub-diffraction imaging using our CoPRAM algorithm for sparse phase retrieval.

#### Phase retrieval of structured signals

We consider the problem of recovering a signal from magnitude-only measurements under a structured sparsity prior. The problem is of interest in the fields of nano- and bioimaging systems, astronomical imaging and speech processing.

#### Fast Low-Rank Estimation for Ill-Conditioned Matrices

In this paper, we study the general problem of optimizing a convex function $F(L)$ over the set of $p\times p$ matrices, subject to rank constraints on $L$.

#### Fast and Provable Algorithms for Learning Two-Layer Polynomial Neural Networks

We study the problem of (provably) learning the weights of a two-layer neural network with quadratic activations.