We consider the problem of recovering a continuous-time bandlimited signal from the discrete-time signal obtained from sampling it every $T_s$ seconds and reducing the result modulo Δ, for some Δ>0. For Δ=∞ the celebrated Shannon-Nyquist sampling theorem guarantees that perfect recovery is possible provided that the sampling rate 1/ $T_s$ exceeds the so-called Nyquist rate. Recent work by Bhandari et al. has shown that for any Δ>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 Δ>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.

This paper studies the related problems of prediction, covariance estimation, and principal component analysis for the spiked covariance model with heteroscedastic noise. We consider an estimator of the principal components based on whitening the noise, and we derive optimal singular value and eigenvalue shrinkers for use with these estimated principal components. Underlying these methods are new asymptotic results for the high-dimensional spiked model with heteroscedastic noise, and consistent estimators for the relevant population parameters. We extend previous analysis on out-of-sample prediction to the setting of predictors with whitening. We demonstrate certain advantages of noise whitening. Specifically, we show that in a certain asymptotic regime, optimal singular value shrinkage with whitening converges to the best linear predictor, whereas without whitening it converges to a suboptimal linear predictor. We prove that for generic signals, whitening improves estimation of the principal components, and increases a natural signal-to-noise ratio of the observations. We also show that for rank one signals, our estimated principal components achieve the asymptotic minimax rate.

Multi-reference alignment entails estimating a signal in $\mathbb{R}^L$ from its circularly shifted and noisy copies. This problem has been studied thoroughly in recent years, focusing on the finit...

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., measurements where each coordinate is reduced modulo Δ, for some Δ > 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 numerical results that show that the algorithm performs well even in non-asymptotic conditions.

Motivated by applications in unsourced random access, this paper develops a novel scheme for the problem of compressed sensing of binary signals. In this problem, the goal is to design a sensing matrix A and a recovery algorithm, such that the sparse binary vector $\mathbf{x}$ can be recovered reliably from the measurements $\mathbf{y}=A\mathbf{x}+\sigma \mathbf{z}$, where $\mathbf{z}$ is additive white Gaussian noise. We propose to design A as a parity check matrix of a low-density parity-check code (LDPC) and to recover $\mathbf{x}$ from the measurements $\mathbf{y}$ using a Markov chain Monte Carlo algorithm, which runs relatively fast due to the sparse structure of A. The performance of our scheme is comparable to state-of-the-art schemes, which use dense sensing matrices, while enjoying the advantages of using a sparse sensing matrix.

We study principal components regression (PCR) in an asymptotic high-dimensional regression setting, where the number of data points is proportional to the dimension. We derive exact limiting formulas for the estimation and prediction risks, which depend in a complicated manner on the eigenvalues of the population covariance, the alignment between the population PCs and the true signal and the number of selected PCs. A key challenge in the high-dimensional setting stems from the fact that the sample covariance is an inconsistent estimate of its population counterpart, so that sample PCs may fail to fully capture potential latent low-dimensional structure in the data. We demonstrate this point through several case studies, including that of a spiked covariance model. To calculate the asymptotic prediction risk, we leverage tools from random matrix theory, which to our knowledge have not seen much use to date in the statistics literature: multiresolvent traces and their associated eigenvector overlap measures.