In the multireference alignment model, a signal is observed by the action of a random circular translation and the addition of Gaussian noise. The goal is to recover the signal's orbit by accessing multiple independent observations. Of particular interest is the sample complexity, i.e., the number of observations/samples needed in terms of the signal-to-noise ratio (the signal energy divided by the noise variance) in order to drive the mean-square error (MSE) to zero. Previous work showed that if the translations are drawn from the uniform distribution, then, in the low SNR regime, the sample complexity of the problem scales as ω(1/ SNR 3 ). In this work, using a generalization of the Chapman-Robbins bound for orbits and expansions of the χ 2 divergence at low SNR, we show that in the same regime the sample complexity for any aperiodic translation distribution scales as ω(1/ SNR 2 ). This rate is achieved by a simple spectral algorithm. We propose two additional algorithms based on nonconvex optimization and expectation-maximization. We also draw a connection between the multireference alignment problem and the spiked covariance model.

Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray crystallography and NMR spectroscopy as a high-resolution structural method for biological macromolecules. In a cryo-EM experiment, the microscope produces images called micrographs. Projections of the molecule of interest are embedded in the micrographs at unknown locations, and under unknown viewing directions. Standard imaging techniques rst locate these projections (detection) and then reconstruct the 3-D structure from them. Unfortunately, high noise levels hinder detection. When reliable detection is rendered impossible, the standard techniques fail. This is a problem especially for small molecules, which can be particularly hard to detect. In this paper, we propose a radically dierent approach: we contend that the structure could, in principle, be reconstructed directly from the micrographs, without intermediate detection. As a result, even small molecules should be within reach for cryo-EM. To support this claim, we setup a simplied mathematical model and demonstrate how our autocorrelation analysis technique allows to go directly from the micrographs to the sought signals. This involves only one pass over the micrographs, which is desirable for large experiments. We show numerical results and discuss challenges that lay ahead to turn this proof-of-concept into a competitive alternative to state-of-the-art algorithms.

No abstract

We consider the multi-target detection problem of recovering a set of signals that appear multiple times at unknown locations in a noisy measurement. In the low noise regime, one can estimate the signals by first detecting occurrences, then clustering and averaging them. In the high noise regime, however, neither detection nor clustering can be performed reliably, so that strategies along these lines are destined to fail. Notwithstanding, using autocorrelation analysis, we show that the impossibility to detect and cluster signal occurrences in the presence of high noise does not necessarily preclude signal estimation. Specifically, to estimate the signals, we derive simple relations between the autocorrelations of the observation and those of the signals. These autocorrelations can be estimated accurately at any noise level given a sufficiently long measurement. To recover the signals from the observed autocorrelations, we solve a set of polynomial equations through nonlinear least-squares. We provide analysis regarding well-posedness of the task, and demonstrate numerically the effectiveness of the method in a variety of settings.The main goal of this work is to provide theoretical and numerical support for a recently proposed framework to image 3-D structures of biological macromolecules using cryo-electron microscopy in extreme noise levels. arXiv:1903.06022v2 [cs.IT] 3 Jun 2019In words: the starting positions of any two occurrences must be separated by at least 2L − 1 positions, so that their end points are necessarily separated by at least L − 1 signal-free (but still noisy) entries in the data. Furthermore, we require that the last signal occurrence in y is also followed by at least L − 1 signal-free entries. This property ensures that correlating y with versions of itself shifted by at most L−1 entries does not involve correlating distinct signal occurrences. Once s is determined, for each position i such that s[i] = 1, one of the signals x k is selected independently at random, and accordingly we set s k [i] = 1. As a result, the only properties of s 1 , . . . , s K that affect the autocorrelations of y (for shifts up to L − 1) are the total number of occurrences of the distinct signals: their individual and relative locations do not intervene. We detail this in Section 3.The Poisson model. If the separation condition is violated, more knowledge about the location distribution is necessary to disentangle the autocorrelations of y. To that effect, we analyze a Poisson generative model. Specifically, for each position i, the number s[i] of signal occurrences starting at that position is drawn independently from a Poisson distribution with parameter γ/L, that is, s[i] i.i.d.

No abstract