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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 $1/operatorname{SNR}^4$. Additionally, we propose a procedure which is experimentally shown to improve the sample complexity by a factor equal to the signals length. Finally, we present numerical experiments which demonstrate the performance of our algorithms, and corroborate our theoretical findings.
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 existenc
A striking result of [Acharya et al. 2017] showed that to estimate symmetric properties of discrete distributions, plugging in the distribution that maximizes the likelihood of observed multiset of frequencies, also known as the profile maximum likel
Two existing approaches to functional principal components analysis (FPCA) are due to Rice and Silverman (1991) and Silverman (1996), both based on maximizing variance but introducing penalization in different ways. In this article we propose an alte
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In this work we study the fundamental limits of approximate recovery in the context of group testing. One of the most well-known, theoretically optimal, and easy to implement testing procedures is the non-adaptive Bernoulli group testing problem, whe