maradan
Scholar : Twitter : GitHub.
I am an Applied Mathematics PhD candidate at MIT. My advisor is Phillipe Rigollet. I am interested in developing theory and principled methodologies for deep machine learning, often in the context of image generation and imaging or spatial inverse problems. Previously, I was fortunate to work under the supervision of Dr. Paul Hand at Northeastern University and Dr. Lenka Zdeborová at EPFL.
* indicates equal contribution.
M. Daniels.
We investigate stochastic interpolation, a recently introduced framework for high dimensional sampling which bears many similarities to diffusion modeling. Like diffusion, stochastic interpolation involves sample generation through the simulation of a deterministic or stochastic evolution equation. We investigate these dynamics and prove bounds on contractivity of the system, uncovering some interesting comparisons to optimal transport theory.
M. Daniels*, Cédric Gerbelot*, Florent Krzakala, Lenka Zdeborová. Published in NeurIPS 2022.
We study signal recovery in a multi-layer model with convolutional matrices, which is a simple model for a convolutional neural network. We prove state evolution equations for Approximate Message Passing, an algorithm that can be used to compute posterior statistics in high dimensional Bayesian models. These equations provide a tractable method to predict signal recovery error in a large-size limit.
M. Daniels, Tyler Maunu, Paul Hand. Published in NeurIPS 2021.
We propose a new method to solve a regularized form of the Optimal Transport problem. The goal is to learn a transportation plan between a given source and target probability distribution so that the cost to execute that plan is minimized. We prove global optimization guarantees for a fast, large-scale learning algorithm to solve this problem and we demonstrate strong empirical performance.
Jung Yeon Park*, Niklas Smedemark-Marguilies*, M. Daniels, Rose Yu, Jan-Willem van de Meent, Paul Hand. Presented at NeurIPS 2020 Deep Inverse Workshop.
Generative priors for imaging inverse problems allow one to model images from a dataset by generating samples from the dataset. We demonstrate a simple method to improve recovery performance by modifying these priors after training, but this performance boost comes at the cost of no longer being able to generate samples using the prior.
Muhammad Asim*, M. Daniels*, Oscar Leong, Paul Hand, and Ali Ahmed. Published in ICML 2020.
In an imaging inverse problem, one must recover missing information about a target image using prior assumptions on the image structure. We show that Invertible Neural Networks can be used to vastly outperform classical approaches when one has access to a dataset of known images.
