Abstract: 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. Our main result shows that for a suitable class of semi-algebraic priors, the generalized phase retrieval problem not only admits a unique solution (up to inherent group symmetries), but also satisfies a bi-Lipschitz property.  This implies robustness to both noise and model mismatch, an essential requirement for practical use, especially when measurements are severely corrupted by noise. 

Abstract: 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. Unlike most constraints in previous works, our constraints hold independently of the relative scaling of the fundamental matrices, which is unknown in practice. We also provide a partial characterization for essential matrix triples arising from calibrated cameras.

Abstract: 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.

Abstract: 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. In practice, signal estimation is often performed by marginalizing over the group using a uniform distribution—an assumption that typically does not hold and renders the MLE misspecified. This raises a fundamental question: how does the misspecified MLE perform? We address this question from several angles. First, we show that in the absence of projection, the misspecified population log-likelihood has desired optimization landscape that leads to correct signal recovery. In contrast, when projections are present, the global optimizers of the misspecified likelihood deviate from the true signal, with the magnitude of the bias depending on the noise level. To address this issue, we propose a joint estimation approach tailored to the cryo-EM setting, which parameterizes the unknown distribution of the group elements and estimates both the signal and distribution parameters simultaneously.

Abstract: 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. In this work, we focus on a specific instance of the synchronization problem within the context of structure from motion (SfM) in computer vision, where each state represents the orientation and location of a camera. We exploit the higher-order interactions encoded in trifocal tensors and introduce the block trifocal tensor. We carefully study the mathematical properties of the block trifocal tensors and use these theoretical insights to develop an effective synchronization framework based on tensor decomposition. Experimental comparisons with state-of-the-art global synchronization methods on real datasets demonstrate the potential of this algorithm for significantly improving location estimation accuracy. To our knowledge, this is the first global SfM synchronization algorithm that directly operates on higher-order measurements. This is joint work with Joe Kileel (UT Austin) and Gilad Lerman (UMN).

Abstract: 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. This talk provides a concise (inexhaustive) overview of the typical priors that arise in EDG problems. We will then turn to a specific instance motivated by Cryo-Electron Microscopy (cryo-EM), where recovering the 3-dimensional structure of proteins can be reformulated as an EDG problem with partially labeled distances. For this problem, we will outline a generic recovery result and present a recovery algorithm that is polynomial-time in fixed dimension. The algorithm achieves exact recovery when distances are noiseless and is robust to small levels of noise.

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