We consider the problem of recovering a continuoustime bandlimited signal from the discrete-time signal obtained from sampling it every Ts seconds and reducing the result modulo ∆, for some ∆ > 0. For ∆ = ∞ the celebrated Shannon-Nyquist sampling theorem guarantees that perfect recovery is possible provided that the sampling rate 1/Ts exceeds the socalled Nyquist rate. Recent work by Bhandari et al. has shown that for any ∆ > 0 perfect reconstruction is still possible if the sampling rate exceeds the Nyquist rate by a factor of πe. In this letter we improve upon this result and show that for finite energy signals, perfect recovery is possible for any ∆ > 0 and any sampling rate above the Nyquist rate. Thus, modulo folding does not degrade the signal, provided that the sampling rate exceeds the Nyquist rate. This claim is proved by establishing a connection between the recovery problem of a discrete-time signal from its modulo reduced version and the problem of predicting the next sample of a discrete-time signal from its past, and leveraging the fact that for a bandlimited signal the prediction error can be made arbitrarily small.

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We consider the problem of recovering n i.i.d samples from a zero mean multivariate Gaussian distribution with an unknown covariance matrix, from their modulo wrapped measurements, i.e., measurement where each coordinate is reduced modulo ∆, for some ∆ > 0. For this setup, which is motivated by quantization and analog-to-digital conversion, we develop a low-complexity iterative decoding algorithm. We show that if a benchmark informed decoder that knows the covariance matrix can recover each sample with small error probability, and n is large enough, the performance of the proposed blind recovery algorithm closely follows that of the informed one. We complement the analysis with numeric results that show that the algorithm performs well even in non-asymptotic conditions.

Multi-reference alignment entails estimating a signal in \BbbR L from its circularly shifted and noisy copies. This problem has been studied thoroughly in recent years, focusing on the finite-dimensional setting (fixed L). Motivated by single-particle cryo-electron microscopy, we analyze the sample complexity of the problem in the high-dimensional regime L \rightar \infty . Our analysis uncovers a phase transition phenomenon governed by the parameter \alpha = L/(\sigma 2 log L), where \sigma 2 is the variance of the noise. When \alpha > 2, the impact of the unknown circular shifts on the sample complexity is minor. Namely, the number of measurements required to achieve a desired accuracy \varepsi approaches \sigma 2 /\varepsi for small \varepsi ; this is the sample complexity of estimating a signal in additive white Gaussian noise, which does not involve shifts. In sharp contrast, when \alpha \leq 2, the problem is significantly harder, and the sample complexity grows substantially more quickly with \sigma 2 .

In Group Synchronization, one attempts to find a collection of unknown group elements from noisy measurements of their pairwise differences. Several important problems in vision and data analysis reduce to group synchronization over various compact groups. Spectral Group Synchronization is a commonly used, robust algorithm for solving group synchronization problems, which relies on diagonalization of a block matrix whose blocks are matrix representations of the measured pairwise differences. Assuming uniformly distributed measurement errors, we present a rigorous analysis of the accuracy and noise sensitivity of spectral group synchronization algorithms over any compact group, up to the rounding error. We identify a Baik-Ben Arous-Péché type phase transition in the noise level, beyond which spectral group synchronization necessarily fails. Below the phase transition, spectral group synchronization succeeds in recovering the unknown group elements, but its performance deteriorates with the noise level. We provide asymptotically exact formulas for the accuracy of spectral group synchronization below the phase transition, up to the rounding error. We also provide a consistent risk estimate, allowing practitioners to estimate the method's accuracy from available measurements. *