Tamir Bendory

Amnon Balanov, Tamir Bendory, Dan Edidin

We study estimation in the low signal-to-noise ratio (SNR) regime for a broad class of Gaussian latent-variable models, including Gaussian mixtures and orbit recovery problems. We show that, in this regime, the generalized method-of-moments (GMoM) matches the first-order asymptotic efficiency of maximum likelihood. In particular, if the moment features are chosen up to the minimal local order required for identification and are weighted optimally, then the resulting GMoM estimator has the same leading asymptotic covariance as the maximum-likelihood estimator. Our analysis shows that, in low SNR, this equivalence is governed by a layered local geometry: different directions become informative at different moment orders, partitioning the space into layers with distinct SNR scalings. We prove that the observed Fisher information and the GMoM information operator admit matching layerwise expansions across these layers. As a consequence, in the low-SNR regime, GMoM provides a statistically efficient alternative to maximum likelihood, while preserving the computational advantages of moment-based estimation.

Amnon Balanov, Josh Katz, Tamir Bendory et al.

Motivated by structural biology applications, we study the projected multi-reference alignment (MRA) model, in which an unknown signal is observed through noisy samples, each generated by applying a random cyclic shift followed by a fixed projection. The projection merges reflection-symmetric index pairs, thereby discarding orientation information. The goal is to recover the dihedral orbit of the signal. We prove that in the high-noise regime, the first three moments of the projected observations determine a generic dihedral orbit. The main mechanism is a reduction, at the moment level, from projected MRA to the reflection-invariant phase-coupling structure of dihedral MRA. In Fourier-cosine coordinates adapted to the projection, the first moment determines the mean component, the second moment determines the Fourier magnitudes, and selected third moments yield the cosine phase-coupling relations appearing in the dihedral bispectrum. These relations lead to a constructive recovery scheme from moments up to order three. We complement the population theory with finite-sample experiments comparing expectation--maximization (EM), direct moment optimization, and direct Fourier-cosine moment optimization. The results show that, in the high-noise regime, both EM and direct moment optimization are consistent with the predicted third-moment sample-complexity scaling $n \gtrsim σ^6$, where $n$ is the number of observations and $σ^2$ is the noise variance.

Noam Janco, Tamir Bendory

The group synchronization problem involves estimating a collection of group elements from noisy measurements of their pairwise ratios. This task is a key component in many computational problems, including the molecular reconstruction problem in single-particle cryo-electron microscopy (cryo-EM). The standard methods to estimate the group elements are based on iteratively applying linear and non-linear operators, and are not necessarily optimal. Motivated by the structural similarity to deep neural networks, we adopt the concept of algorithm unrolling, where training data is used to optimize the algorithm. We design unrolled algorithms for several group synchronization instances, including synchronization over the group of 3-D rotations: the synchronization problem in cryo-EM. We also apply a similar approach to the multi-reference alignment problem. We show by numerical experiments that the unrolling strategy outperforms existing synchronization algorithms in a wide variety of scenarios.

Kweku Abraham, Amnon Balanov, Tamir Bendory et al.

Motivated by single-particle cryo-electron microscopy, we study the sample complexity of the multi-target detection (MTD) problem, in which an unknown signal appears multiple times at unknown locations within a long, noisy observation. We propose a patching scheme that reduces MTD to a non-i.i.d. multi-reference alignment (MRA) model. In the one-dimensional setting, the latent group elements form a Markov chain, and we show that the convergence rate of any estimator matches that of the corresponding i.i.d. MRA model, up to a logarithmic factor in the number of patches. Moreover, for estimators based on empirical averaging, such as the method of moments, the convergence rates are identical in both settings. We further establish an analogous result in two dimensions, where the latent structure arises from an exponentially mixing random field generated by a hard-core placement model. As a consequence, if the signal in the corresponding i.i.d. MRA model is determined by moments up to order $n_{\min}$, then in the low-SNR regime the number of patches required to estimate the signal in the MTD model scales as $σ^{2n_{\min}}$, where $σ^2$ denotes the noise variance.

Roy R. Lederman, David Silva-Sánchez, Ziling Chen et al.

Lloyd's k-means algorithm is one of the most widely used clustering methods. We prove that in high-dimensional, high-noise settings, the algorithm exhibits catastrophic failure: with high probability, essentially every partition of the data is a fixed point. Consequently, Lloyd's algorithm simply returns its initial partition - even when the underlying clusters are trivially recoverable by other methods. In contrast, we prove that Hartigan's k-means algorithm does not exhibit this pathology. Our results show the stark difference between these algorithms and offer a theoretical explanation for the empirical difficulties often observed with k-means in high dimensions.

Amnon Balanov, Tamir Bendory, Wasim Huleihel

Confirmation bias, the tendency to interpret information in a way that aligns with one's preconceptions, can profoundly impact scientific research, leading to conclusions that reflect the researcher's hypotheses even when the observational data do not support them. This issue is especially critical in scientific fields involving highly noisy observations, such as cryo-electron microscopy. This study investigates confirmation bias in Gaussian mixture models. We consider the following experiment: A team of scientists assumes they are analyzing data drawn from a Gaussian mixture model with known signals (hypotheses) as centroids. However, in reality, the observations consist entirely of noise without any informative structure. The researchers use a single iteration of the K-means or expectation-maximization algorithms, two popular algorithms to estimate the centroids. Despite the observations being pure noise, we show that these algorithms yield biased estimates that resemble the initial hypotheses, contradicting the unbiased expectation that averaging these noise observations would converge to zero. Namely, the algorithms generate estimates that mirror the postulated model, although the hypotheses (the presumed centroids of the Gaussian mixture) are not evident in the observations. Specifically, among other results, we prove a positive correlation between the estimates produced by the algorithms and the corresponding hypotheses. We also derive explicit closed-form expressions of the estimates for a finite and infinite number of hypotheses. This study underscores the risks of confirmation bias in low signal-to-noise environments, provides insights into potential pitfalls in scientific methodologies, and highlights the importance of prudent data interpretation.

Amnon Balanov, Tamir Bendory, Dan Edidin

Let $f:\mathbb{R}^n\to\mathbb{R}$ be an unknown object, and suppose the observations are tomographic projections of randomly rotated copies of $f$ of the form $Y = P(R\cdot f)$, where $R$ is Haar-uniform in $\mathrm{SO}(n)$ and $P$ is the projection onto an $m$-dimensional subspace, so that $Y:\mathbb{R}^m\to\mathbb{R}$. We prove that, whenever $d\le m$, the $d$-th order moment of the projected data determines the full $d$-th order Haar-orbit moment of $f$, independently of the ambient dimension $n$. We further provide an explicit algorithmic procedure for recovering the latter from the former. As a consequence, any identifiability result for the unprojected model based on $d$-th order group-invariant moment extends directly to the tomographic setting at the same moment order. In particular, for $n=3$, $m=2$, and $d=2$, our result recovers a classical result in the cryo-EM literature: the covariance of the 2D projection images determines the second order rotationally invariant moment of the underlying 3D object.

Michael Fraiman, Paulina Hoyos, Tamir Bendory et al.

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.

Elad Romanov, Tamir Bendory, Or Ordentlich

This paper studies the sample complexity of learning the $k$ unknown centers of a balanced Gaussian mixture model (GMM) in $\mathbb{R}^d$ with spherical covariance matrix $σ^2\mathbf{I}$. In particular, we are interested in the following question: what is the maximal noise level $σ^2$, for which the sample complexity is essentially the same as when estimating the centers from labeled measurements? To that end, we restrict attention to a Bayesian formulation of the problem, where the centers are uniformly distributed on the sphere $\sqrt{d}\mathcal{S}^{d-1}$. Our main results characterize the exact noise threshold $σ^2$ below which the GMM learning problem, in the large system limit $d,k\to\infty$, is as easy as learning from labeled observations, and above which it is substantially harder. The threshold occurs at $\frac{\log k}{d} = \frac12\log\left( 1+\frac{1}{σ^2} \right)$, which is the capacity of the additive white Gaussian noise (AWGN) channel. Thinking of the set of $k$ centers as a code, this noise threshold can be interpreted as the largest noise level for which the error probability of the code over the AWGN channel is small. Previous works on the GMM learning problem have identified the minimum distance between the centers as a key parameter in determining the statistical difficulty of learning the corresponding GMM. While our results are only proved for GMMs whose centers are uniformly distributed over the sphere, they hint that perhaps it is the decoding error probability associated with the center constellation as a channel code that determines the statistical difficulty of learning the corresponding GMM, rather than just the minimum distance.

Noam Janco, Tamir Bendory

The multi-reference alignment (MRA) problem entails estimating an image from multiple noisy and rotated copies of itself. If the noise level is low, one can reconstruct the image by estimating the missing rotations, aligning the images, and averaging out the noise. While accurate rotation estimation is impossible if the noise level is high, the rotations can still be approximated, and thus can provide indispensable information. In particular, learning the approximation error can be harnessed for efficient image estimation. In this paper, we propose a new computational framework, called Synch-EM, that consists of angular synchronization followed by expectation-maximization (EM). The synchronization step results in a concentrated distribution of rotations; this distribution is learned and then incorporated into the EM as a Bayesian prior. The learned distribution also dramatically reduces the search space, and thus the computational load, of the EM iterations. We show by extensive numerical experiments that the proposed framework can significantly accelerate EM for MRA in high noise levels, occasionally by a few orders of magnitude, without degrading the reconstruction quality.

Tamir Bendory, Ti-Yen Lan, Nicholas F. Marshall et al.

We consider the multi-target detection problem of estimating a two-dimensional target image from a large noisy measurement image that contains many randomly rotated and translated copies of the target image. Motivated by single-particle cryo-electron microscopy, we focus on the low signal-to-noise regime, where it is difficult to estimate the locations and orientations of the target images in the measurement. Our approach uses autocorrelation analysis to estimate rotationally and translationally invariant features of the target image. We demonstrate that, regardless of the level of noise, our technique can be used to recover the target image when the measurement is sufficiently large.

Elad Romanov, Tamir Bendory, Or Ordentlich

Multi-reference alignment entails estimating a signal in $\mathbb{R}^L$ from its circularly-shifted and noisy copies. This problem has been studied thoroughly in recent years, focusing on the finite-dimensional setting (fixed $L$). Motivated by single-particle cryo-electron microscopy, we analyze the sample complexity of the problem in the high-dimensional regime $L\to\infty$. Our analysis uncovers a phase transition phenomenon governed by the parameter $α= L/(σ^2\log L)$, where $σ^2$ is the variance of the noise. When $α>2$, the impact of the unknown circular shifts on the sample complexity is minor. Namely, the number of measurements required to achieve a desired accuracy $\varepsilon$ approaches $σ^2/\varepsilon$ for small $\varepsilon$; this is the sample complexity of estimating a signal in additive white Gaussian noise, which does not involve shifts. In sharp contrast, when $α\leq 2$, the problem is significantly harder and the sample complexity grows substantially quicker with $σ^2$.

Ye Zhou, Amit Moscovich, Tamir Bendory et al.

Single-particle cryo-Electron Microscopy (EM) has become a popular technique for determining the structure of challenging biomolecules that are inaccessible to other technologies. Recent advances in automation, both in data collection and data processing, have significantly lowered the barrier for non-expert users to successfully execute the structure determination workflow. Many critical data processing steps, however, still require expert user intervention in order to converge to the correct high-resolution structure. In particular, strategies to identify homogeneous populations of particles rely heavily on subjective criteria that are not always consistent or reproducible among different users. Here, we explore the use of unsupervised strategies for particle sorting that are compatible with the autonomous operation of the image processing pipeline. More specifically, we show that particles can be successfully sorted based on a simple statistical model for the distribution of scores assigned during refinement. This represents an important step towards the development of automated workflows for protein structure determination using single-particle cryo-EM.

Nicholas F. Marshall, Ti-Yen Lan, Tamir Bendory et al.

We introduce a framework for recovering an image from its rotationally and translationally invariant features based on autocorrelation analysis. This work is an instance of the multi-target detection statistical model, which is mainly used to study the mathematical and computational properties of single-particle reconstruction using cryo-electron microscopy (cryo-EM) at low signal-to-noise ratios. We demonstrate with synthetic numerical experiments that an image can be reconstructed from rotationally and translationally invariant features and show that the reconstruction is robust to noise. These results constitute an important step towards the goal of structure determination of small biomolecules using cryo-EM.

Chao Ma, Tamir Bendory, Nicolas Boumal et al.

Motivated by the task of 2-D classification in single particle reconstruction by cryo-electron microscopy (cryo-EM), we consider the problem of heterogeneous multireference alignment of images. In this problem, the goal is to estimate a (typically small) set of target images from a (typically large) collection of observations. Each observation is a rotated, noisy version of one of the target images. For each individual observation, neither the rotation nor which target image has been rotated are known. As the noise level in cryo-EM data is high, clustering the observations and estimating individual rotations is challenging. We propose a framework to estimate the target images directly from the observations, completely bypassing the need to cluster or register the images. The framework consists of two steps. First, we estimate rotation-invariant features of the images, such as the bispectrum. These features can be estimated to any desired accuracy, at any noise level, provided sufficiently many observations are collected. Then, we estimate the images from the invariant features. Numerical experiments on synthetic cryo-EM datasets demonstrate the effectiveness of the method. Ultimately, we outline future developments required to apply this method to experimental data.

Tamar Rott, Dorin Shriki, Tamir Bendory

Image registration is a challenging task in the world of medical imaging. Particularly, accurate edge registration plays a central role in a variety of clinical conditions. The Modality Independent Neighbourhood Descriptor (MIND) demonstrates state of the art alignment, based on the image self-similarity. However, this method appears to be less accurate regarding edge registration. In this work, we propose a new registration method, incorporating gradient intensity and MIND self-similarity metric. Experimental results show the superiority of this method in edge registration tasks, while preserving the original MIND performance for other image features and textures.

Tamir Bendory, Shai Dekel, Arie Feuer

In this work we consider the problem of recovering non-uniform splines from their projection onto spaces of algebraic polynomials. We show that under a certain Chebyshev-type separation condition on its knots, a spline whose inner-products with a polynomial basis and boundary conditions are known, can be recovered using Total Variation norm minimization. The proof of the uniqueness of the solution uses the method of `dual' interpolating polynomials and is based on \cite{SR}, where the theory was developed for trigonometric polynomials. We also show results for the multivariate case.