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.

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).

TBA

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.