arXiv subjects

Amit Singer

Publications and source records attributed to Amit Singer.

At least 19 recordsLinked to original sources

arXiv2026

Fast rigid rotational alignment of voxelized 3-D density maps in sliced Wasserstein distance

We develop a fast method for the rigid alignment of voxelized three-dimensional density functions over $\mathsf{SO}(3)$ using sliced optimal transport. Our approach efficiently transforms each density map into a tensor of one-dimensional quantile functions, taken from central slices over the unit sphere, such that the Euclidean distance of the transformed volumes approximates the sliced 2-Wasserstein distance. Our formulation is therefore compatible with existing, fast, voxel-based Euclidean alignment algorithms. We provide theoretical analysis and numerical experiments demonstrating that the sliced 2-Wasserstein distance is a robust metric for comparing three-dimensional shapes and for computing their rigid alignment.

math.NA↗

arXiv2026

Diagonally-Weighted Generalized Method of Moments Estimation for Gaussian Mixture Modeling

With direct computation of moments through order $L$ in $d$ dimensions, standard method of moments (MM) incurs $Ω(d^L)$ computation and storage costs, and standard generalized method of moments (GMM) additionally requires estimating and inverting an often ill-conditioned covariance matrix of size $Ω(d^L)$-by-$Ω(d^L)$ to obtain the optimal weighting matrix. Pereira, Kileel, and Kolda [arXiv:2202.06930 (2022)] avoid the tensor computation cost of MM through implicit moment computation. Despite this progress, incorporating statistically principled weighting while preserving computational efficiency remains an important challenge, particularly at low signal-to-noise ratio (SNR), where sampling variances can differ substantially across moment orders. We introduce diagonally-weighted GMM (DGMM), which assigns one adaptive weight to each moment order through a structured approximation of the optimal weighting matrix while avoiding the covariance inversion, thereby balancing statistical efficiency, computational complexity, and numerical stability. For general parametric models, we prove that DGMM is consistent, asymptotically normal, and near-efficient under explicit structural conditions. For heteroscedastic low-rank Gaussian mixtures, we design a computationally efficient and numerically stable algorithm that extends the implicit moment computation to the DGMM objective, gradients, and weights and incorporates Nyström approximations of the kernel sums. In low-SNR experiments with $d=100$ and $L=4$, where standard MM and GMM are infeasible, controlled weighting comparisons show that DGMM achieves consistently small parameter errors across tested initializations, including the challenging starts where alternative weightings struggle, with shorter median runtimes and fewer iterations. The code and data will be available upon publication at https://github.com/liu-lzhang/dgmm.

cs.LG↗

arXiv2026

Image Moment Estimation for Crowded Datasets

Single-particle cryogenic electron microscopy (cryo-EM) is an imaging technique capable of reconstructing high-resolution three-dimensional (3D) structure of biological macromolecules from a set of noisy projection images taken at random viewing directions. The mathematical model for cryo-EM images typically assumes that each image contains exactly one particle, which is centered at the origin. However, many experimental datasets contain particle images which include multiple particles: one central particle and a few crowding particles. This deviation from the model leads to inaccurate estimation of image statistics, such as the low-order image moments. To account for this, we propose a method to estimate image moments while correcting for the effect of crowding particles. This method can be run directly from a set of cryo-EM micrographs with picked particles, and we demonstrate its efficacy on a wide array of synthetic data.

eess.IV↗

arXiv2026

Orientation Estimation of Cryo-EM Images for Molecules with Arbitrary Symmetry by Semidefinite Programming

This paper extends the non-unique games framework introduced in Bandeira et al. (2020) to address symmetric structures. Precisely, we consider the problem \[ \begin{aligned} \underset{g_1,\ldots,g_n\in \mathcal{G}}{\operatorname{min}}\,\, \frac{1}{|S|}\sum_{i,j=1}^n \sum_{s\in S} f_{ij}(g_i s g_j^{-1}), \end{aligned} \] where $\mathcal{G}$ is a compact group, $S\subset \mathcal{G}$ is a finite subgroup of symmetry elements, and $f_{ij}:\mathcal{G}\rightarrow \mathbb{R}$ is a loss function. A notable example is common lines based orientation estimation of cryo-EM images for molecules with point group symmetry $S$, where $\mathcal{G}$ is the rotation group $\operatorname{SO}(3)$. With a semidefinite programming relaxation, we turn the problem into a convex program that is solved with the alternating direction method of multipliers. Unlike existing approaches which are tailored to specific symmetry groups, ours provides a unified computational framework for addressing all point group symmetries. We demonstrate the applicability of the proposed algorithm on the cryo-EM orientation estimation problem on both simulated and experimental datasets. In particular, our algorithm is effective in generating high-quality ab initio models from only a few class averages computed from experimental images.

cs.CV↗

arXiv2026

Mathematics of Data Science

This book is about the mathematical foundations of data science. 1. Introduction 2. Curses, Blessings, and Surprises in High Dimensions 3. Singular Value Decomposition and Principal Component Analysis 4. Linear Regression and Regularization 5. Graphs, Networks, and Clustering 6. Nonlinear Dimension Reduction and Diffusion Maps 7. Linear Dimension Reduction via Random Projections 8. Optimization for Data Science 9. Classification 10. A Mathematical Introduction to Deep Learning 11. Large Sample Limit of Graph Laplacians 12. Community 13. Concentration of Measure and Gaussian Analysis 14. Matrix Concentration Inequalities 15. Compressive Sensing and Sparsity 16. Low-Rank Matrix Recovery

cs.LG↗

arXiv2026

Two Datasets Are Better Than One: Method of Double Moments for 3-D Reconstruction in Cryo-EM

Cryo-electron microscopy (cryo-EM) is a powerful imaging technique for reconstructing three-dimensional molecular structures from noisy tomographic projection images of randomly oriented particles. We introduce a new data fusion framework, termed the method of double moments (MoDM), which reconstructs molecular structures from two instances of the second-order moment of projection images obtained under distinct orientation distributions: one uniform, the other non-uniform and unknown. We prove that these moments generically uniquely determine the underlying structure, up to a global rotation and reflection, and we develop a convex-relaxation-based algorithm that achieves accurate recovery using only second-order statistics. Our results demonstrate the advantage of collecting and modeling multiple datasets under different experimental conditions, illustrating that leveraging dataset diversity can substantially enhance reconstruction quality in computational imaging tasks.

cs.CV↗

arXiv2026

Robust Moment-Based Estimation via Spectral Gradient Reweighting

Moment-based estimation is a theoretically attractive approach to parametric inference, especially when likelihood-based estimation is unavailable, misspecified, or computationally inconvenient. However, the moment equations involve sample averages, which makes moment-based estimation sensitive to outliers. We propose the SGR-GMM algorithm, a robust generalized method of moments (GMM) procedure that uses a spectral gradient reweighting (SGR) primitive to soft-reweight the per-observation gradients during the moment-matching optimization. Our analysis has three layers. First, for a fixed center, the SGR primitive is formulated as an entropy-regularized spectral game between a sample-weight player and a density-matrix player, which is analyzed using classical multiplicative-weights and matrix-multiplicative-weights regret bounds. Second, we establish explicit convergence radius and finite termination bound for the fixed-center updates in the SGR primitive. Third, we prove a local finite-sample parameter estimation error bound with explicit dependence on the contamination fraction, inlier gradient stability, local GMM identification strength, and optimization accuracy. We further specialize the SGR-GMM algorithm to obtain a robust diagonally-weighted GMM (DGMM) estimator for estimating heteroscedastic low-rank Gaussian mixtures observed under additive Gaussian noise and strong contamination. In the numerical experiments, the SGR primitive produces nearly-oracle gradient estimation and the robust DGMM specialization substantially improves over non-robust moment baselines. The code and data are available at https://github.com/liu-lzhang/sgr-gmm.

math.ST↗

arXiv2026

Expectation-maximization for structure determination directly from cryo-EM micrographs

A single-particle cryo-electron microscopy (cryo-EM) measurement, called a micrograph, consists of multiple two-dimensional tomographic projections of a three-dimensional (3-D) molecular structure at unknown locations, taken under unknown viewing directions. All existing cryo-EM algorithmic pipelines first locate and extract the projection images, and then reconstruct the structure from the extracted images. However, if the molecular structure is small, the signal-to-noise ratio (SNR) of the data is very low, making it challenging to accurately detect projection images within the micrograph. Consequently, all standard techniques fail in low-SNR regimes. To recover molecular structures from measurements of low SNR, and in particular small molecular structures, we devise an approximate expectation-maximization algorithm to estimate the 3-D structure directly from the micrograph, bypassing the need to locate the projection images. We corroborate our computational scheme with numerical experiments and present successful structure recoveries from simulated noisy measurements.

eess.IV↗

arXiv2026

Bayesian Perspective for Orientation Determination in Cryo-EM with Application to Structural Heterogeneity Analysis

Accurate orientation estimation is a crucial component of 3D molecular structure reconstruction, both in single-particle cryo-electron microscopy (cryo-EM) and in the increasingly popular field of cryo-electron tomography (cryo-ET). The dominant approach, which involves searching for the orientation that maximizes cross-correlation relative to given templates, is sub-optimal, particularly under low signal-to-noise conditions. In this work, we propose a Bayesian framework for more accurate and flexible orientation estimation, with the minimum mean square error (MMSE) estimator serving as a key example. Through simulations, we demonstrate that the MMSE estimator consistently outperforms the cross-correlation-based method, especially in challenging low signal-to-noise scenarios, and we provide a theoretical framework that supports these improvements. When incorporated into iterative refinement algorithms in the 3D reconstruction pipeline, the MMSE estimator markedly improves reconstruction accuracy, reduces model bias, and enhances robustness to the ``Einstein from Noise'' artifact. Crucially, we demonstrate that orientation estimation accuracy has a decisive effect on downstream structural heterogeneity analysis. In particular, integrating the MMSE-based pose estimator into frameworks for continuous heterogeneity recovery yields accuracy improvements approaching those obtained with ground-truth poses, establishing MMSE-based pose estimation as a key enabler of high-fidelity conformational landscape reconstruction. These findings indicate that the proposed Bayesian framework could substantially advance cryo-EM and cryo-ET by enhancing the accuracy, robustness, and reliability of 3D molecular structure reconstruction, thereby facilitating deeper insights into complex biological systems.

stat.AP↗

arXiv2025

Subspace method of moments for ab initio 3-D single-particle cryo-EM reconstruction

Cryo-electron microscopy (cryo-EM) is a widely used technique for recovering the 3-D structure of biological molecules from a large number of experimentally generated noisy 2-D tomographic projection images of the 3-D structure, taken from unknown viewing angles. Through computationally intensive algorithms, these observed images are processed to reconstruct the 3-D structures. Many popular computational methods rely on estimating the unknown angles as part of the reconstruction process, which becomes particularly challenging at low signal-to-noise ratios. The method of moments (MoM) offers an alternative approach that circumvents the estimation of viewing orientations of individual projection images by instead estimating the underlying distribution of the viewing angles, and is robust to noise given sufficiently many images. However, the method of moments typically entails computing higher-order moments of the projection images, incurring significant computational and memory costs. To mitigate this, we propose a new approach called the subspace method of moments (SubspaceMoM), which compresses the first three moments using data-driven low-rank tensor techniques as well as expansion into a suitable function basis. The compressed moments can be efficiently computed from the set of projection images using numerical quadrature and can be employed to jointly reconstruct the 3-D structure and the distribution of viewing orientations. We illustrate the practical applicability of SubspaceMoM through numerical experiments using up to the third-order moment on synthetic datasets with a simplified cryo-EM image formation model, which significantly improves the reconstruction resolution compared to previous MoM approaches.

math.NA↗

arXiv2025

SO(3)-invariant PCA with application to molecular data

Principal component analysis (PCA) is a fundamental technique for dimensionality reduction and denoising; however, its application to three-dimensional data with arbitrary orientations -- common in structural biology -- presents significant challenges. A naive approach requires augmenting the dataset with many rotated copies of each sample, incurring prohibitive computational costs. In this paper, we extend PCA to 3D volumetric datasets with unknown orientations by developing an efficient and principled framework for SO(3)-invariant PCA that implicitly accounts for all rotations without explicit data augmentation. By exploiting underlying algebraic structure, we demonstrate that the computation involves only the square root of the total number of covariance entries, resulting in a substantial reduction in complexity. We validate the method on real-world molecular datasets, demonstrating its effectiveness and opening up new possibilities for large-scale, high-dimensional reconstruction problems.

eess.SP↗

arXiv2025

Misspecified Maximum Likelihood Estimation for Non-Uniform Group Orbit Recovery

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.

math.ST↗

arXiv2025

Manifold learning in metric spaces

Laplacian-based methods are popular for the dimensionality reduction of data lying in $\mathbb{R}^N$. Several theoretical results for these algorithms depend on the fact that the Euclidean distance locally approximates the geodesic distance on the underlying submanifold which the data are assumed to lie on. However, for some applications, other metrics, such as the Wasserstein distance, may provide a more appropriate notion of distance than the Euclidean distance. We provide a framework that generalizes the problem of manifold learning to metric spaces and study when a metric satisfies sufficient conditions for the pointwise convergence of the graph Laplacian.

cs.LG↗

arXiv2025

Fast rigid alignment of heterogeneous images in sliced Wasserstein distance

Many applications of computer vision rely on the alignment of similar but non-identical images. We present a fast algorithm for aligning heterogeneous images based on optimal transport. Our approach combines the speed of fast Fourier methods with the robustness of sliced probability metrics and allows us to efficiently compute the alignment between two $L \times L$ images using the sliced 2-Wasserstein distance in $O(L^2 \log L)$ operations. We show that our method is robust to translations, rotations and deformations in the images.

cs.CV↗

arXiv2025

Fast expansion into harmonics on the ball

We devise fast and provably accurate algorithms to transform between an $N\times N \times N$ Cartesian voxel representation of a three-dimensional function and its expansion into the {ball harmonics}, that is, the eigenbasis of the Dirichlet Laplacian on the unit ball in $\mathbb{R}^3$. Given $\varepsilon > 0$, our algorithms achieve relative $\ell^1$ - $\ell^\infty$ accuracy $\varepsilon$ in time $O(N^3 (\log N)^2 + N^3 |\log \varepsilon|^2)$, while the naïve direct application of the expansion operators has time complexity $O(N^6)$. We illustrate our methods on numerical examples.

math.NA↗

arXiv2024

Alignment of Density Maps in Wasserstein Distance

In this paper we propose an algorithm for aligning three-dimensional objects when represented as density maps, motivated by applications in cryogenic electron microscopy. The algorithm is based on minimizing the 1-Wasserstein distance between the density maps after a rigid transformation. The induced loss function enjoys a more benign landscape than its Euclidean counterpart and Bayesian optimization is employed for computation. Numerical experiments show improved accuracy and efficiency over existing algorithms on the alignment of real protein molecules. In the context of aligning heterogeneous pairs, we illustrate a potential need for new distance functions.

eess.IV↗

arXiv2024

Moment-based metrics for molecules computable from cryo-EM images

Single particle cryogenic electron microscopy (cryo-EM) is an imaging technique capable of recovering the high-resolution 3-D structure of biological macromolecules from many noisy and randomly oriented projection images. One notable approach to 3-D reconstruction, known as Kam's method, relies on the moments of the 2-D images. Inspired by Kam's method, we introduce a rotationally invariant metric between two molecular structures, which does not require 3-D alignment. Further, we introduce a metric between a stack of projection images and a molecular structure, which is invariant to rotations and reflections and does not require performing 3-D reconstruction. Additionally, the latter metric does not assume a uniform distribution of viewing angles. We demonstrate uses of the new metrics on synthetic and experimental datasets, highlighting their ability to measure structural similarity.

q-bio.BM↗

arXiv2023

Autocorrelation analysis for cryo-EM with sparsity constraints: Improved sample complexity and projection-based algorithms

The number of noisy images required for molecular reconstruction in single-particle cryo-electron microscopy (cryo-EM) is governed by the autocorrelations of the observed, randomly-oriented, noisy projection images. In this work, we consider the effect of imposing sparsity priors on the molecule. We use techniques from signal processing, optimization, and applied algebraic geometry to obtain new theoretical and computational contributions for this challenging non-linear inverse problem with sparsity constraints. We prove that molecular structures modeled as sums of Gaussians are uniquely determined by the second-order autocorrelation of their projection images, implying that the sample complexity is proportional to the square of the variance of the noise. This theory improves upon the non-sparse case, where the third-order autocorrelation is required for uniformly-oriented particle images and the sample complexity scales with the cube of the noise variance. Furthermore, we build a computational framework to reconstruct molecular structures which are sparse in the wavelet basis. This method combines the sparse representation for the molecule with projection-based techniques used for phase retrieval in X-ray crystallography.

math.NA↗