An Ultra-Fast MLE 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, expectation-maximization (EM), often becomes computationally prohibitive, particularly in low signal-to-noise ratio (SNR) settings. We introduce an alternative, ultra-fast algorithm for MRA over the special orthogonal group $\mathrm{SO}(2)$. By performing a Taylor expansion of the log-likelihood in the low-SNR regime, we estimate the signal by sequentially computing data-driven averages of observations. Our method requires only one pass over the data, dramatically reducing computational cost compared to EM. Numerical experiments show that the proposed approach achieves high accuracy in low-SNR environments and provides an excellent initialization for subsequent EM refinement.

Orbit recovery under the rigid motions group

We study the orbit recovery problem under the rigid-motion group SE(n), where the objective is to reconstruct an unknown signal from multiple noisy observations subjected to unknown rotations and translations. This problem is fundamental in signal processing, computer vision, and structural biology. Our main theoretical contribution is bounding the sample complexity of this problem. We show that if the d-th order moment under the rotation group SO(n) uniquely determines the signal orbit, then orbit recovery under SE(n) is achievable with $N\gtrsim σ^{2d+4}$ samples as the noise variance $σ^2 \to \infty$. The key technical insight is that the d-th order SO(n) moments can be explicitly recovered from (d+2)-order SE(n) autocorrelations, enabling us to transfer known results from the rotation-only setting to the rigid-motion case. We further harness this result to derive a matching bound to the sample complexity of the multi-target detection model that serves as an abstract framework for electron-microscopy-based technologies in structural biology, such as single-particle cryo-electron microscopy (cryo-EM) and cryo-electron tomography (cryo-ET). Beyond theory, we present a provable computational pipeline for rigid-motion orbit recovery in three dimensions. Starting from rigid-motion autocorrelations, we extract the SO(3) moments and demonstrate successful reconstruction of a 3-D macromolecular structure. Importantly, this algorithmic approach is valid at any noise level, suggesting that even very small macromolecules, long believed to be inaccessible using structural biology electron-microscopy-based technologies, may, in principle, be reconstructed given sufficient data.

Expectation-maximization for multi-reference alignment: Two pitfalls and one remedy

We study the multi-reference alignment model, which involves recovering a signal from noisy observations that have been randomly transformed by an unknown group action, a fundamental challenge in statistical signal processing, computational imaging, and structural biology. While much of the theoretical literature has focused on the asymptotic sample complexity of this model, the practical performance of reconstruction algorithms, particularly of the omnipresent expectation maximization (EM) algorithm, remains poorly understood. In this work, we present a detailed investigation of EM in the challenging low signal-to-noise ratio (SNR) regime. We identify and characterize two failure modes that emerge in this setting. The first, called Einstein from Noise, reveals a strong sensitivity to initialization, with reconstructions resembling the input template regardless of the true underlying signal. The second phenomenon, referred to as the Ghost of Newton, involves EM initially converging towards the correct solution but later diverging, leading to a loss of reconstruction fidelity. We provide theoretical insights and support our findings through numerical experiments. Finally, we introduce a simple, yet effective modification to EM based on mini-batching, which mitigates the above artifacts. Supported by both theory and experiments, this mini-batching approach processes small data subsets per iteration, reducing initialization bias and computational cost, while maintaining accuracy comparable to full-batch EM.

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.

Confirmation Bias in Gaussian Mixture Models

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.

Einstein from Noise: Statistical Analysis

``Einstein from noise" (EfN) is a prominent example of the model bias phenomenon: systematic errors in the statistical model that lead to erroneous but consistent estimates. In the EfN experiment, one falsely believes that a set of observations contains noisy, shifted copies of a template signal (e.g., an Einstein image), whereas in reality, it contains only pure noise observations. To estimate the signal, the observations are first aligned with the template using cross-correlation, and then averaged. Although the observations contain nothing but noise, it was recognized early on that this process produces a signal that resembles the template signal! This pitfall was at the heart of a central scientific controversy about validation techniques in structural biology. This paper provides a comprehensive statistical analysis of the EfN phenomenon above. We show that the Fourier phases of the EfN estimator (namely, the average of the aligned noise observations) converge to the Fourier phases of the template signal, explaining the observed structural similarity. Additionally, we prove that the convergence rate is inversely proportional to the number of noise observations and, in the high-dimensional regime, to the Fourier magnitudes of the template signal. Moreover, in the high-dimensional regime, the Fourier magnitudes converge to a scaled version of the template signal's Fourier magnitudes. This work not only deepens the theoretical understanding of the EfN phenomenon but also highlights potential pitfalls in template matching techniques and emphasizes the need for careful interpretation of noisy observations across disciplines in engineering, statistics, physics, and biology.