The sample complexity of sparse multi-reference alignment and single-particle cryo-electron microscopy

Multi-reference alignment (MRA) is the problem of recovering a signal from its multiple noisy copies, each acted upon by a random group element. MRA is mainly motivated by single-particle cryo-electron microscopy (cryo-EM) that has recently joined X-ray crystallography as one of the two leading technologies to reconstruct biological molecular structures. Previous papers have shown that in the high noise regime, the sample complexity of MRA and cryo-EM is $n=\omega(\sigma^{2d})$, where $n$ is the number of observations, $\sigma^2$ is the variance of the noise, and $d$ is the lowest-order moment of the observations that uniquely determines the signal. In particular, it was shown that in many cases, $d=3$ for generic signals, and thus the sample complexity is $n=\omega(\sigma^6)$. In this paper, we analyze the second moment of the MRA and cryo-EM models. First, we show that in both models the second moment determines the signal up to a set of unitary matrices, whose dimension is governed by the decomposition of the space of signals into irreducible representations of the group. Second, we derive sparsity conditions under which a signal can be recovered from the second moment, implying sample complexity of $n=\omega(\sigma^4)$. Notably, we show that the sample complexity of cryo-EM is $n=\omega(\sigma^4)$ if at most one third of the coefficients representing the molecular structure are non-zero; this bound is near-optimal. The analysis is based on tools from representation theory and algebraic geometry. We also derive bounds on recovering a sparse signal from its power spectrum, which is the main computational problem of X-ray crystallography

Multi-Reference Alignment for sparse signals, Uniform Uncertainty Principles and the Beltway Problem

Motivated by cutting-edge applications like cryo-electron microscopy (cryo-EM), the Multi-Reference Alignment (MRA) model entails the learning of an unknown signal from repeated measurements of its images under the latent action of a group of isometries and additive noise of magnitude $\sigma$. Despite significant interest, a clear picture for understanding rates of estimation in this model has emerged only recently, particularly in the high-noise regime $\sigma \gg 1$ that is highly relevant in applications. Recent investigations have revealed a remarkable asymptotic sample complexity of order $\sigma^6$ for certain signals whose Fourier transforms have full support, in stark contrast to the traditional $\sigma^2$ that arise in regular models. Often prohibitively large in practice, these results have prompted the investigation of variations around the MRA model where better sample complexity may be achieved. In this paper, we show that \emph{sparse} signals exhibit an intermediate $\sigma^4$ sample complexity even in the classical MRA model. Our results explore and exploit connections of the MRA estimation problem with two classical topics in applied mathematics: the \textit{beltway problem} from combinatorial optimization, and \textit{uniform uncertainty principles} from harmonic analysis

Maximum likelihood for high-noise group orbit estimation and single-particle cryo-EM

Motivated by applications to single-particle cryo-electron microscopy (cryo-EM), we study several problems of function estimation in a low SNR regime, where samples are observed under random rotations of the function domain. In a general framework of group orbit estimation with linear projection, we describe a stratification of the Fisher information eigenvalues according to a sequence of transcendence degrees in the invariant algebra, and relate critical points of the log-likelihood landscape to a sequence of method-of-moments optimization problems. This extends previous results for a discrete rotation group without projection. We then compute these transcendence degrees and the forms of these moment optimization problems for several examples of function estimation under $SO(2)$ and $SO(3)$ rotations, including a simplified model of cryo-EM as introduced by Bandeira, Blum-Smith, Kileel, Perry, Weed, and Wein. For several of these examples, we affirmatively resolve numerical conjectures that $3^\text{rd}$-order moments are sufficient to locally identify a generic signal up to its rotational orbit. For low-dimensional approximations of the electric potential maps of two small protein molecules, we empirically verify that the noise-scalings of the Fisher information eigenvalues conform with these theoretical predictions over a range of SNR, in a model of $SO(3)$ rotations without projection

Efficient Estimation of Signals via Non-Convex Approaches

This dissertation aims to highlight the importance of methodological development and the need for tailored algorithms in non-convex statistical problems. Specifically, we study three non-convex estimation problems with novel ideas and techniques in both statistical methodologies and algorithmic designs. Chapter 2 discusses my work with Zhou Fan on estimation of a piecewise-constant image, or a gradient-sparse signal on a general graph, from noisy linear measurements. In this work, we propose and study an iterative algorithm to minimize a penalized least-squares objective, with a penalty given by the ``$\ell_0$-norm\u27\u27 of the signal\u27s discrete graph gradient. The method uses a non-convex variant of proximal gradient descent, applying the alpha-expansion procedure to approximate the proximal mapping in each iteration, and using a geometric decay of the penalty parameter across iterations to ensure convergence. Under a cut-restricted isometry property for the measurement design, we prove global recovery guarantees for the estimated signal. For standard Gaussian designs, the required number of measurements is independent of the graph structure, and improves upon worst-case guarantees for total-variation (TV) compressed sensing on the 1-D line and 2-D lattice graphs by polynomial and logarithmic factors, respectively. The method empirically yields lower mean-squared recovery error compared with TV regularization in regimes of moderate undersampling and moderate to high signal-to-noise, for several examples of changepoint signals and gradient-sparse phantom images. Chapter 3 discusses my work with Zhou Fan and Sahand Negahban on tree-projected gradient descent for estimating gradient-sparse parameters. We consider estimating a gradient-sparse parameter $\boldsymbol{\theta}^*\in\mathbb{R}^p$, having strong gradient-sparsity $s^*:=\|\nabla_G \boldsymbol{\theta}^*\|_0$ on an underlying graph $G$. Given observations$Z_1,\ldots,Z_n$ and a smooth, convex loss function $\mathcal{L}$ for which our parameter of interest $\boldsymbol{\theta}^*$ minimizes the population risk \mathbb{E}[\mathcal{L}(\btheta;Z_1,\ldots,Z_n)], we propose to estimate $\boldsymbol{\theta}^*$ by a projected gradient descent algorithm that iteratively and approximately projects gradient steps onto spaces of vectors having small gradient-sparsity over low-degree spanning trees of $G$. We show that, under suitable restricted strong convexity and smoothness assumptions for the loss, the resulting estimator achieves the squared-error risk $\frac{s^*}{n} \log (1+\frac{p}{s^*})$ up to a multiplicative constant that is independent of $G$. In contrast, previous polynomial-time algorithms have only been shown to achieve this guarantee in more specialized settings, or under additional assumptions for $G$ and/or the sparsity pattern of $\nabla_G \boldsymbol{\theta}^*$. As applications of our general framework, we apply our results to the examples of linear models and generalized linear models with random design. Chapter 4 discusses my joint work with Zhou Fan, Roy R. Lederman, Yi Sun, and Tianhao Wang on maximum likelihood for high-noise group orbit estimation. Motivated by applications to single-particle cryo-electron microscopy (cryo-EM), we study several problems of function estimation in a low SNR regime, where samples are observed under random rotations of the function domain. In a general framework of group orbit estimation with linear projection, we describe a stratification of the Fisher information eigenvalues according to a sequence of transcendence degrees in the invariant algebra, and relate critical points of the log-likelihood landscape to a sequence of method-of-moments optimization problems. This extends previous results for a discrete rotation group without projection. We then compute these transcendence degrees and the forms of these moment optimization problems for several examples of function estimation under $\mathsf{SO}(2)$ and $\mathsf{SO}(3)$ rotations. For several of these examples, we affirmatively resolve numerical conjectures that $3^\text{rd}$-order moments are sufficient to locally identify a generic signal up to its rotational orbit, and also confirm the existence of spurious local optima for the landscape of the population log-likelihood. For low-dimensional approximations of the electric potential maps of two small protein molecules, we empirically verify that the noise-scalings of the Fisher information eigenvalues conform with these theoretical predictions over a range of SNR, in a model of $\mathsf{SO}(3)$ rotations without projection