The generalized phase retrieval problem over compact groups aims to recover a set of matrices, representing an unknown signal, from their associated Gram matrices, leveraging prior structural knowledge about the signal. This framework generalizes the classical phase retrieval problem, which reconstructs a signal from the magnitudes of its Fourier transform, to a richer setting involving non-abelian compact group.

The fundamental matrix of a pair of pinhole cameras lies at the core of systems that reconstruct 3D scenes from 2D images. However, for more than two cameras, the relations between the various fundamental matrices of camera pairs are not yet completely understood. In joint work with Viktor Korotynskiy, Anton Leykin, and Tomas Pajdla, we characterize all polynomial constraints that hold for an arbitrary triple of fundamental matrices.

We consider the problem of stably separating the orbits of the action of a group of isometries on Euclidean space. We will present two different constructions of maps that can separate the orbits of such an action and discuss when these can also be bilipschitz in an appropriate metric. Both of these constructions can be viewed as generalizations of phase retrieval which we will use as a prototypical example. Time permitting we will also discuss some recent results on optimal bilipschitz embeddings and approximations of such embeddings.

We study maximum likelihood estimation (MLE) in the generalized group orbit recovery model, where each observation is generated by applying a random group action and a known, fixed linear operator to an unknown signal, followed by additive noise. This model is motivated by single-particle cryo-electron microscopy (cryo-EM) and can be viewed primarily as a structured continuous Gaussian mixture model.

The Euclidean distance geometry (EDG) problem concerns the reconstruction of point configurations in R^n from prior partial knowledge of pairwise inter-point distances, often accompanied by assumptions on the prior being noisy, incomplete or unlabeled. Instances of this problem appear across diverse domains, including dimensionality reduction techniques in machine learning, predicting molecular conformations in computational chemistry, and sensor network localization for acoustic vision.

Synchronization is crucial for the success of many data-intensive applications. This problem involves estimating global states from relative measurements between states. While many studies have explored synchronization in different contexts using pairwise measurements, relying solely on pairwise measurements often fails to capture the full complexity of the system.

Frames for a Hilbert space allow for a linear and stable reconstruction of a vector from linear measurements. In many real-world applications, sensors are set up such that any measurement above and below a certain threshold would be clipped as the signal gets saturated. We study the recovery of a vector from such measurements which is called declipping or saturation recovery. Phase retrieval is the problem of recovering a vector where only the intensity of each linear measurement of a signal is available and the phase information is lost.

Consider a rooted d-regular tree with \ell layers, where each vertex is colored either blue or green. Starting from the root, the color propagates down the tree so that each child inherits its parent’s color but flips with probability 30%. Now, suppose you only observe the colors of the leaves—can you infer the color of the root?