arXiv subjects

Shay Kreymer

Publications and source records attributed to Shay Kreymer.

11 recordsLinked to original sources

arXiv2026

Provable orbit recovery over SO(3) from the non-uniform second moment

We study the recovery of an unknown three-dimensional band-limited signal from multiple noisy observations that are randomly rotated by latent elements of SO(3), where the rotations are drawn from an unknown, non-uniform distribution. Because the rotations are unobserved, only the signal orbit under the rotation group can be recovered. We show that the signal orbit and the rotation distribution are jointly identifiable from the first and second moments. This yields an improved high-noise sample complexity that scales quadratically with the noise variance, rather than cubically as in the uniform-rotation case. We further develop a provable, computationally efficient reconstruction algorithm that recovers the 3-D signal by successively solving a sequence of well-conditioned linear systems. The algorithm is validated through extensive numerical experiments. Our results provide a principled and tractable framework for high-noise 3-D orbit recovery, with potential relevance to cryo-electron microscopy and cryo-electron tomography modeling, where molecules are observed in unknown orientations.

eess.SP↗

arXiv2026

A Fast Approximate Maximum Likelihood Estimator for Low SNR Multi-Reference Alignment

Motivated by single-particle cryo-electron microscopy, multi-reference alignment (MRA) models the task of recovering an unknown signal from multiple noisy observations corrupted by random rotations. The standard approach for computing the maximum likelihood estimator (MLE) is the expectation-maximization (EM) algorithm; however, it often becomes computationally prohibitive, particularly in low signal-to-noise ratio (SNR) settings. We introduce a fast approximate MLE for MRA over the special orthogonal groups $\mathrm{SO}(2)$ and $\mathrm{SO}(3)$ in the low-SNR regime. A low-SNR Taylor expansion of the likelihood reveals a closed-form, non-iterative approximate MLE. We show that this approach yields a consistent estimator in the low-SNR limit and requires substantially lower computational complexity than both EM and invariant-based alternatives. Numerical experiments generated from molecular volumes show that the proposed method provides a favorable accuracy-runtime trade-off, especially in challenging low-SNR regimes, and can serve as an effective initialization for EM.

eess.SP↗

arXiv2025

Solving ill-conditioned polynomial equations using score-based priors with application to multi-target detection

Recovering signals from low-order moments is a fundamental yet notoriously difficult task in inverse problems. This recovery process often reduces to solving ill-conditioned systems of polynomial equations. In this work, we propose a new framework that integrates score-based diffusion priors with moment-based estimators to regularize and solve these nonlinear inverse problems. This introduces a new role for generative models: stabilizing polynomial recovery from noisy statistical features. As a concrete application, we study the multi-target detection (MTD) model in the high-noise regime. We demonstrate two main results: (i) diffusion priors substantially improve recovery from third-order moments, and (ii) they make the super-resolution MTD problem, otherwise ill-posed, feasible. Numerical experiments on MNIST data confirm consistent gains in reconstruction accuracy across SNR levels. Our results suggest a promising new direction for combining generative priors with nonlinear polynomial inverse problems.

eess.SP↗

arXiv2025

Orbit recovery for spherical functions

Orbit recovery is a central problem in both mathematics and applied sciences, with important applications to structural biology. This paper focuses on recovering generic orbits of functions on ${\mathbb R}^{n}$ and the sphere $S^{n-1}$ under the rotation action of $SO(n)$. Specifically, we demonstrate that invariants of degree three (called the bispectrum) suffice to recover generic orbits of functions in finite-dimensional approximations of $L^2({\mathbb R}^n)$ obtained by band-limiting the spherical component and discretizing the radial direction. In particular, our main result explicitly bounds the number of samples in the radial direction required for recovery from the degree three invariants. From an application perspective, the most important case is $SO(3)$, which arises in many scientific fields, and in particular, plays a central role in leading structural biology applications such as cryo-electron tomography and cryo-electron microscopy. Our result for $SO(3)$ states that considering three spherical shells (i.e., samples in the radial direction) is sufficient to recover generic orbits, which verifies an implicit conjecture made in a paper of Bandeira et al. Our proof technique provides an explicit, computationally efficient algorithm to recover the signal by successively solving systems of linear equations. We implemented this algorithm and demonstrated its effectiveness on three protein structures.

math.NA↗

arXiv2025

A note on the sample complexity of multi-target detection

This work studies the sample complexity of the multi-target detection (MTD) problem, which involves recovering a signal from a noisy measurement containing multiple instances of a target signal in unknown locations, each transformed by a random group element. This problem is primarily motivated by single-particle cryo-electron microscopy (cryo-EM), a groundbreaking technology for determining the structures of biological molecules. We establish upper and lower bounds for various MTD models in the high-noise regime as a function of the group, the distribution over the group, and the arrangement of signal occurrences within the measurement. The lower bounds are established through a reduction to the related multi-reference alignment problem, while the upper bounds are derived from explicit recovery algorithms utilizing autocorrelation analysis. These findings provide fundamental insights into estimation limits in noisy environments and lay the groundwork for extending this analysis to more complex applications, such as cryo-EM.

eess.SP↗

arXiv2023

Score-based diffusion priors for multi-target detection

Multi-target detection (MTD) is the problem of estimating an image from a large, noisy measurement that contains randomly translated and rotated copies of the image. Motivated by the single-particle cryo-electron microscopy technology, we design data-driven diffusion priors for the MTD problem, derived from score-based stochastic differential equations models. We then integrate the prior into the approximate expectation-maximization algorithm. In particular, our method alternates between an expectation step that approximates the expected log-likelihood and a maximization step that balances the approximated log-likelihood with the learned log-prior. We show on two datasets that adding the data-driven prior substantially reduces the estimation error, in particular in high noise regimes.

eess.SP↗

arXiv2023

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↗

arXiv2022

An approximate expectation-maximization for two-dimensional multi-target detection

We consider the two-dimensional multi-target detection (MTD) problem of estimating a target image from a noisy measurement that contains multiple copies of the image, each randomly rotated and translated. The MTD model serves as a mathematical abstraction of the structure reconstruction problem in single-particle cryo-electron microscopy, the chief motivation of this study. We focus on high noise regimes, where accurate detection of image occurrences within a measurement is impossible. To estimate the image, we develop an expectation-maximization framework that aims to maximize an approximation of the likelihood function. We demonstrate image recovery in highly noisy environments, and show that our framework outperforms the previously studied autocorrelation analysis in a wide range of parameters.

eess.SP↗

arXiv2022

Two-dimensional multi-target detection: an autocorrelation analysis approach

We consider the two-dimensional multi-target detection problem of recovering a target image from a noisy measurement that contains multiple copies of the image, each randomly rotated and translated. Motivated by the structure reconstruction problem in single-particle cryo-electron microscopy, we focus on the high noise regime, where the noise hampers accurate detection of the image occurrences. We develop an autocorrelation analysis framework to estimate the image directly from a measurement with an arbitrary spacing distribution of image occurrences, bypassing the estimation of individual locations and rotations. We conduct extensive numerical experiments, and demonstrate image recovery in highly noisy environments. The code to reproduce all numerical experiments is publicly available at https://github.com/krshay/MTD-2D.

eess.SP↗

arXiv2021

Generalized autocorrelation analysis for multi-target detection

We study the multi-target detection problem of recovering a target signal from a noisy measurement that contains multiple copies of the signal at unknown locations. Motivated by the structure reconstruction problem in cryo-electron microscopy, we focus on the high noise regime, where noise hampers accurate detection of signal occurrences. Previous works proposed an autocorrelation analysis framework to estimate the signal directly from the measurement, without detecting signal occurrences. Specifically, autocorrelation analysis entails finding a signal that best matches the observable autocorrelations by minimizing a least squares objective. This paper extends this line of research by developing a generalized autocorrelation analysis framework that replaces the least squares by a weighted least squares. The optimal weights can be computed directly from the data and guarantee favorable statistical properties. We demonstrate signal recovery from highly noisy measurements, and show that the proposed framework outperforms autocorrelation analysis in a wide range of parameters.

eess.SP↗

arXiv2021

Signal recovery from a few linear measurements of its high-order spectra

The $q$-th order spectrum is a polynomial of degree $q$ in the entries of a signal $x\in\mathbb{C}^N$, which is invariant under circular shifts of the signal. For $q\geq 3$, this polynomial determines the signal uniquely, up to a circular shift, and is called a high-order spectrum. The high-order spectra, and in particular the bispectrum ($q=3$) and the trispectrum ($q=4$), play a prominent role in various statistical signal processing and imaging applications, such as phase retrieval and single-particle reconstruction. However, the dimension of the $q$-th order spectrum is $N^{q-1}$, far exceeding the dimension of $x$, leading to increased computational load and storage requirements. In this work, we show that it is unnecessary to store and process the full high-order spectra: a signal can be characterized uniquely, up to symmetries, from only $N+1$ linear measurements of its high-order spectra. The proof relies on tools from algebraic geometry and is corroborated by numerical experiments.

cs.IT↗