Abstract : We discuss a class of estimation problems that aim for unknown group elements or a signal affected by group actions. Three prominent examples of such problems are synchronization over groups, multireference alignment, and the recovery problem in single-particle cryo-EM. The talks will cover computational and theoretical aspects, including the sample complexity of the problems, constructing group invariant operators, sparsity, recovery strategies, machine learning-based methods, group-robust metrics, data modeling, autocorrelation analysis, and its acceleration techniques, manifold optimization in cryo-EM, synchronization analysis, and more. This mini-symposium is divided into three sections and will host senior and junior researchers as its speakers.
Organizer(s) : Yuehaw Khoo, Nir Sharon, Amit Singer
[05667] Heterogeneity analysis by Covariance Estimation
Format : Talk at Waseda University
Author(s) :
Marc Aurèle Tiberius Gilles (Princeton University)
Abstract : While theoretically attractive, heterogeneity analysis by 3D covariance estimation has long been plagued by its exorbitant computational and storage cost. We overcome this challenge by designing a new low-rank estimator computed by SVD. This allows us to compute up to 50 principal components of the covariance in real datasets in a few minutes at medium resolutions, and under 30 minutes for high resolutions, a 10x speed-up over other linear-subspace methods, and 100x over nonlinear methods.
[05122] Vector bundles for alignment and dimensionality reduction
Format : Online Talk on Zoom
Author(s) :
Jose Perea (Northeastern University)
Luis Scoccola (Northeastern University)
Abstract : Vector bundles have rich structure and arise naturally when trying to solve dimensionality reduction and synchronization problems in data science. I will show in this talk how the classical machinery (e.g., classifying maps, characteristic classes, etc) can be adapted to the world of algorithms and noisy data, as well as the insights one can gain.
[04867] Group-robust metrics
Format : Online Talk on Zoom
Author(s) :
William Leeb (University of Minnesota, Twin Cities)
Abstract : This talk will describe a family of metrics between functions. These metrics are provably robust to a large class of perturbations of the inputs, including the group of integral-preserving reparameterizations; they are also robust to additive noise, and can be evaluated rapidly. Their theoretical properties will be illustrated by numerical experiments.
[04814] The sample complexity of multireference alignment and cryo-EM
Format : Talk at Waseda University
Author(s) :
Tamir Bendory (Tel Aviv University)
Abstract : The problem of multi-reference alignment (MRA) involves retrieving a signal from multiple copies that have been corrupted by noise and transformed by a random group element. MRA is of particular interest in the context of single-particle cryo-electron microscopy (cryo-EM), a prominent technique used to reconstruct biological molecular structures. During this talk, I will examine the sample complexity of both the MRA and cryo-EM models using tools from representation theory, sparse coding, and generative models.
[05662] Simultaneous denoising in low-SNR regimes using data-driven priors
Format : Online Talk on Zoom
Author(s) :
Joakim Andén (KTH Royal Institute of Technology)
Abstract : The application of DNNs to the denoising problem has resulted in highly performant denoisers for a wide range of applications from photographic image restoration to medical imaging. However, these images are typically subjected to relatively low degree of noise compared to applications such as cryogenic electron microscopy (cryo-EM), where noise power dominates the clean signal to the point where traditional denoising methods fail. In this talk, we will study the related problem of multireference alignment (MRA). Here, a single clean signal is observed at random shifts and subjected to additive noise and the goal is to recover the original signal. We should how a transformer architecture can be used to encode a signal prior that is used to aggregate the information from the entire set of observations to yield a superior denoising method.
[04753] Autocorrelation analysis for cryo-EM with sparsity constraints
Format : Talk at Waseda University
Author(s) :
Tamir Bendory (Tel Aviv University)
Yuehaw Khoo (The University of Chicago)
Joe Kileel (UT Austin)
Oscar Mickelin (Princeton University)
Amit Singer (Princeton University)
Abstract : This work presents new results for the method of moments applied to cryo-electron microscopy. We prove that autocorrelations of noisy tomographic projection images can reconstruct molecular structures that are modeled as sparse sums of Gaussians. This significantly reduces the sample complexity of the problem, compared to previous results. Additionally, we detail a practical ab initio reconstruction algorithm using tools adapted from crystallographic phase retrieval.
[03118] Optimal Spectral Methods for Synchronization Problems
Format : Talk at Waseda University
Author(s) :
Anderson Ye Zhang (University of Pennsylvania)
Abstract : We study the performance of spectral methods for synchronization problems with additive Gaussian noises and incomplete data. Spectral methods refer to algorithms that use the leading eigenvectors of the data matrix followed by a normalization step. (1) For phase synchronization and orthogonal group synchronization, we prove that they achieve the minimax lower bound of the problem with a matching leading constant under a squared $\ell_2$ loss. This shows that the spectral method has the same performance as more sophisticated procedures including MLE, generalized power method, and SDP when consistent parameter estimation is possible. (2) For permutation synchronization, we propose a novel spectral method that overcomes a crucial limitation of existing ones and has improved numerical performance. We further show the proposed method is statistically optimal with an exponentially small error that matches the minimax rate.
[05181] Orthogonal Matrix Retrieval with Spatial Consensus for 3D Unknown-View Tomography
Format : Talk at Waseda University
Author(s) :
Shuai Huang (Emory University)
Mona Zehni (University of Illinois at Urbana-Champaign)
Ivan Dokmanic (University of Basel)
Zhizhen Zhao (University of Illinois at Urbana Champaign)
Abstract : A line of work starting with Kam (1980) employs the method of moments (MoM) with rotation-invariant features to reconstruct a 3D density map from its 2D tomographic projections at unknown, random orientations, assuming that the orientations are uniformly distributed. This line of work includes the recent orthogonal matrix retrieval approaches. In this talk, we extend the previous approaches and propose to jointly recover the density map and the orthogonal matrices by imposing spatial nonnegativity constraint.
[05647] Applications of Riemannian geometry in protein dynamics and single particle cryo-EM
Author(s) :
Ozan Öktem (KTH Royal Institute of Technology)
Willem Diepeveen (University of Cambridge)
Jan Lellmann (University of Lübeck)
Carola-Bibiane Schönlieb (University of Cambridge)
Carlos Esteve-Yague (University of Cambridge)
Abstract : We will show how to construct a Riemannian geometry that is modeled after an energy landscape relevant for protein conformations. A key difficulty is to construct a smooth Riemannian structure based on an energy landscape. Another is to have computationally feasible means for computing geodesics and related mappings. We address these challenges by a local approximation technique. The resulting method can accurately predict molecular dynamics trajectories of two proteins within seconds on a laptop.
