We consider the problem of estimating the covariance matrix of a random signal observed through unknown translations (modeled by cyclic shifts) and corrupted by noise. Solving this problem allows to discover low-rank structures masked by the existence of translations (which act as nuisance parameters), with direct application to Principal Components Analysis (PCA). We assume that the underlying signal is of length @math and follows a standard factor model with mean zero and @math normally-distributed factors. To recover the covariance matrix in this case, we propose to employ the second- and fourth-order shift-invariant moments of the signal known as the @math and the @math . We prove that they are sufficient for recovering the covariance matrix (under a certain technical condition) when @math . Correspondingly, we provide a polynomial-time procedure for estimating the covariance matrix from many (translated and noisy) observations, where no explicit knowledge of @math is required, and prove the procedure's statistical consistency. While our results establish that covariance estimation is possible from the power spectrum and the trispectrum for low-rank covariance matrices, we prove that this is not the case for full-rank covariance matrices. We conduct numerical experiments that corroborate our theoretical findings, and demonstrate the favorable performance of our algorithms in various settings, including in high levels of noise.
Last, we mention @cite_21 , where the authors considered the problem of MRFA in the restricted case of @math . A key observation in @cite_21 is that if the distribution of the shifts is uniform, then the bispectrum vanishes entirely. Therefore, it was proposed to employ the fourth-order shift-invariant moment known as the trispectrum @cite_4 to estimate the signal parameters ( @math and @math ), and algorithms were presented for consistently estimating @math and @math as @math . The trispectrum is analogous to the power spectrum and the bispectrum, in the sense that it consists of certain quadruple correlations in the Fourier domain of the signal, which are invariant to cyclic shifts. As in MRA, much focus was put on the sample complexity of the algorithms in the low-SNR regime, since in the high-SNR regime the observations can be accurately aligned using correlations (a fact which is only true for the case @math ).
@cite_21 @cite_4
2013913072 2761720500 2726175563 2946901449
Abstract This paper is concerned with the development of techniques to detect and analyse non-linearities. The methods developed are based on the concepts of higher-order spectra (HOS), in particular the bispectrum and trispectrum. The study of HOS has been dominated by work on the bispectrum. The bispectrum can be viewed as a decomposition of the third moment (skewness) of a signal over frequency and proves useful for analysing systems with asymmetric non-linearities. In studying symmetric non-linearities, the trispectrum is a more powerful tool, as it represents a decomposition of kurtosis over frequency. Techniques are presented that enable the estimation and display of bispectra and trispectra. HOS are studied in detail with particular attention being paid to normalisation methods. Two traditional methods, the bicoherence and skewness function, are studied and these are extended to their fourth-order equivalents, the tricoherence and kurtosis functions. Under certain conditions, notably narrowband signals, the above normalisation methods are shown to fail, and so a new technique based on pre-whitening the signal in the time domain is developed. Examples of these functions are given both for an amplitude modulated process and the Duffing oscillator. Multireference alignment (MRA) is the problem of estimating a signal from many noisy and cyclically shifted copies of itself. In this paper, we consider an extension called heterogeneous MRA, where @math signals must be estimated, and each observation comes from one of those signals, unknown to us. This is a simplified model for the heterogeneity problem notably arising in cryo-electron microscopy. We propose an algorithm which estimates the @math signals without estimating either the shifts or the classes of the observations. It requires only one pass over the data and is based on low-order moments that are invariant under cyclic shifts. Given sufficiently many measurements, one can estimate these invariant features averaged over the @math signals. We then design a smooth, non-convex optimization problem to compute a set of signals which are consistent with the estimated averaged features. We find that, in many cases, the proposed approach estimates the set of signals accurately despite non-convexity, and conjecture the number of signals @math that can be resolved as a function of the signal length @math is @math . The growing role of data-driven approaches to scientific discovery has unveiled a large class of models that involve latent transformations with a rigid algebraic constraint. Among them, multi-reference alignment (MRA) is a simple model that captures fundamental aspects of the statistical and algorithmic challenges arising from this new paradigm. In this model, an unknown signal is subject to two types of corruption: a latent cyclic shift and the more traditional additive white noise. The goal is to recover the signal at a certain precision from independent samples. While at high signal-to-noise ratio (SNR), the number of observations needed to recover a generic signal is proportional to 1 SNR, we show that it rises to 1 SNR^3 in the more realistic low SNR regime. We propose an algorithm that achieves this optimal dependence on the SNR. Furthermore, we extend our results to cover a heterogeneous MRA model where the samples come from a mixture of signals, as is often the case in applications such as Cryo-Electron Microscopy, where molecules may have different conformations. We provide the first known procedure that provably achieves signal recovery in the low SNR regime for heterogeneous MRA. In recent years, there is a growing need for processing methods aimed at extracting useful information from large datasets. In many cases the challenge is to discover a low-dimensional structure in the data, often concealed by the existence of nuisance parameters and noise. Motivated by such challenges, we consider the problem of estimating a signal from its scaled, cyclically-shifted and noisy observations. We focus on the particularly challenging regime of low signal-to-noise ratio (SNR), where different observations cannot be shift-aligned. We show that an accurate estimation of the signal from its noisy observations is possible, and derive a procedure which is proved to consistently estimate the signal. The asymptotic sample complexity (the number of observations required to recover the signal) of the procedure is @math . Additionally, we propose a procedure which is experimentally shown to improve the sample complexity by a factor equal to the signal's length. Finally, we present numerical experiments which demonstrate the performance of our algorithms, and corroborate our theoretical findings.