My research lies at the intersection of applied harmonic analysis, inverse problems, and machine learning, with the goal of enhancing our theoretical understanding while addressing practical challenges. I am especially interested in how harmonic analysis can contribute to a deeper understanding of inverse problems, where the aim is to reconstruct hidden information from indirect or incomplete measurements, as well as of modern machine learning methods, such as deep neural networks, whose impressive performance often lacks a full mathematical explanation. Currently, my research focuses on two main topics: phase retrieval and mathematical foundation of neural networks. Phase retrieval arises in many real-world applications where one aims to recover missing phase information from magnitude-only measurements. I study the mathematical properties of these problems, focusing on the stability and uniqueness of possible reconstructions. Concerning mathematical foundation of neural networks, I use tools from functional analysis to study the function spaces associated with neural networks. My goal is to characterize these spaces and identify suitable norms that reflect the behavior and expressivity of neural networks. This research direction contributes to building a theoretical framework that helps explain and guide the practical success of deep learning methods.
Key Words
Applied Harmonic Analysis • Group Representation Theory • Phase Retrieval Problems
Radon Transforms • Functional Analysis of Neural Networks