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Motivated by recent advances in serverless cloud computing, in particular the "function as a service" (FaaS) model, we consider the problem of minimizing a convex function in a massively parallel fashion, where communication between workers is limited. Focusing on the case of a twice-differentiable objective subject to an L2 penalty, we propose a scheme where the central node (server) effectively runs a Newton method, offloading its high per-iteration cost -- stemming from the need to invert the Hessian -- to the workers. In our solution, workers produce independently coarse but low-bias estimates of the inverse Hessian, using an adaptive sketching scheme. The server then averages the descent directions produced by the workers, yielding a good approximation for the exact Newton step. The main component of our adaptive sketching scheme is a low-complexity procedure for selecting the sketching dimension, an issue that was left largely unaddressed in the existing literature on Hessian sketching for distributed optimization. Our solution is based on ideas from asymptotic random matrix theory, specifically the Marchenko-Pastur law. For Gaussian sketching matrices, we derive non asymptotic guarantees for our algorithm which are essentially dimension-free. Lastly, when the objective is self-concordant, we provide convergence guarantees for the approximate Newton's method with noisy Hessians, which may be of independent interest beyond the setting considered in this paper.

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: multi-resolvent traces and their associated eigenvector overlap measures.

The randomized singular value decomposition (R-SVD) is a popular sketching-based algorithm for efficiently computing the partial SVD of a large matrix. When the matrix is low-rank, the R-SVD produces its partial SVD exactly; but when the rank is large, it only yields an approximation. Motivated by applications in data science and principal component analysis (PCA), we analyze the R-SVD under a low-rank signal plus noise measurement model; specifically, when its input is a spiked random matrix. The singular values produced by the R-SVD are shown to exhibit a BBP-like phase transition: when the SNR exceeds a certain detectability threshold, that depends on the dimension reduction factor, the largest singular value is an outlier; below the threshold, no outlier emerges from the bulk of singular values. We further compute asymptotic formulas for the overlap between the ground truth signal singular vectors and the approximations produced by the R-SVD. Dimensionality reduction has the adverse affect of amplifying the noise in a highly nonlinear manner. Our results demonstrate the statistical advantage -- in both signal detection and estimation -- of the R-SVD over more naive sketched PCA variants; the advantage is especially dramatic when the sketching dimension is small. Our analysis is asymptotically exact, and substantially more fine-grained than existing operator-norm error bounds for the R-SVD, which largely fail to give meaningful error estimates in the moderate SNR regime. It applies for a broad family of sketching matrices previously considered in the literature, including Gaussian i.i.d. sketches, random projections, and the sub-sampled Hadamard transform, among others. Lastly, we derive an optimal singular value shrinker for singular values and vectors obtained through the R-SVD, which may be useful for applications in matrix denoising.

We study the problem of estimating a large, low-rank matrix corrupted by additive noise of unknown covariance, assuming one has access to additional side information in the form of noise-only measurements. We study the Whiten-Shrink-reColor (WSC) workflow, where a "noise covariance whitening" transformation is applied to the observations, followed by appropriate singular value shrinkage and a "noise covariance re-coloring" transformation. We show that under the mean square error loss, a unique, asymptotically optimal shrinkage nonlinearity exists for the WSC denoising workflow, and calculate it in closed form. To this end, we calculate the asymptotic eigenvector rotation of the random spiked F-matrix ensemble, a result which may be of independent interest. With sufficiently many pure-noise measurements, our optimally-tuned WSC denoising workflow outperforms, in mean square error, matrix denoising algorithms based on optimal singular value shrinkage which do not make similar use of noise-only side information; numerical experiments show that our procedure's relative performance is particularly strong in challenging statistical settings with high dimensionality and large degree of heteroscedasticity.

Tyler's and Maronna's M-estimators, as well as their regularized variants, are popular robust methods to estimate the scatter or covariance matrix of a multivariate distribution. In this work, we study the non-asymptotic behavior of these estimators, for data sampled from a distribution that satisfies one of the following properties: 1) independent sub-Gaussian entries, up to a linear transformation; 2) log-concave distributions; 3) distributions satisfying a convex concentration property. Our main contribution is the derivation of tight non-asymptotic concentration bounds of these M-estimators around a suitably scaled version of the data sample covariance matrix. Prior to our work, non-asymptotic bounds were derived only for Elliptical and Gaussian distributions. Our proof uses a variety of tools from non asymptotic random matrix theory and high dimensional geometry. Finally, we illustrate the utility of our results on two examples of practical interest: sparse covariance and sparse precision matrix estimation.

This paper studies the sample complexity of learning the $k$ unknown centers of a balanced Gaussian mixture model (GMM) in $\mathbb{R}^d$ with spherical covariance matrix $\sigma^2\mathbf{I}$. In particular, we are interested in the following question: what is the maximal noise level $\sigma^2$, for which the sample complexity is essentially the same as when estimating the centers from labeled measurements? To that end, we restrict attention to a Bayesian formulation of the problem, where the centers are uniformly distributed on the sphere $\sqrt{d}\mathcal{S}^{d-1}$. Our main results characterize the exact noise threshold $\sigma^2$ below which the GMM learning problem, in the large system limit $d,k\to\infty$, is as easy as learning from labeled observations, and above which it is substantially harder. The threshold occurs at $\frac{\log k}{d} = \frac12\log\left( 1+\frac{1}{\sigma^2} \right)$, which is the capacity of the additive white Gaussian noise (AWGN) channel. Thinking of the set of $k$ centers as a code, this noise threshold can be interpreted as the largest noise level for which the error probability of the code over the AWGN channel is small. Previous works on the GMM learning problem have identified the minimum distance between the centers as a key parameter in determining the statistical difficulty of learning the corresponding GMM. While our results are only proved for GMMs whose centers are uniformly distributed over the sphere, they hint that perhaps it is the decoding error probability associated with the center constellation as a channel code that determines the statistical difficulty of learning the corresponding GMM, rather than just the minimum distance., Comment: COLT 2022

Consider the rank-1 spiked model: $\bf{X}=\sqrt{\nu}\xi \bf{u}+ \bf{Z}$, where $\nu$ is the spike intensity, $\bf{u}\in\mathbb{S}^{k-1}$ is an unknown direction and $\xi\sim \mathcal{N}(0,1),\bf{Z}\sim \mathcal{N}(\bf{0},\bf{I})$. Motivated by recent advances in analog-to-digital conversion, we study the problem of recovering $\bf{u}\in \mathbb{S}^{k-1}$ from $n$ i.i.d. modulo-reduced measurements $\bf{Y}=[\bf{X}]\mod \Delta$, focusing on the high-dimensional regime ($k\gg 1$). We develop and analyze an algorithm that, for most directions $\bf{u}$ and $\nu=\mathrm{poly}(k)$, estimates $\bf{u}$ to high accuracy using $n=\mathrm{poly}(k)$ measurements, provided that $\Delta\gtrsim \sqrt{\log k}$. Up to constants, our algorithm accurately estimates $\bf{u}$ at the smallest possible $\Delta$ that allows (in an information-theoretic sense) to recover $\bf{X}$ from $\bf{Y}$. A key step in our analysis involves estimating the probability that a line segment of length $\approx\sqrt{\nu}$ in a random direction $\bf{u}$ passes near a point in the lattice $\Delta \mathbb{Z}^k$. Numerical experiments show that the developed algorithm performs well even in a non-asymptotic setting., Comment: AISTATS, 2022

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., Comment: Accepted to Entropy Special Issue on "Information-Theoretic Aspects of Non-Orthogonal and Massive Access for Future Wireless Networks"

We derive a formula for optimal hard thresholding of the singular value decomposition in the presence of correlated additive noise; although it nominally involves unobservables, we show how to apply it even where the noise covariance structure is not a-priori known or is not independently estimable. The proposed method, which we call ScreeNOT, is a mathematically solid alternative to Cattell's ever-popular but vague Scree Plot heuristic from 1966. ScreeNOT has a surprising oracle property: it typically achieves exactly, in large finite samples, the lowest possible MSE for matrix recovery, on each given problem instance - i.e. the specific threshold it selects gives exactly the smallest achievable MSE loss among all possible threshold choices for that noisy dataset and that unknown underlying true low rank model. The method is computationally efficient and robust against perturbations of the underlying covariance structure. Our results depend on the assumption that the singular values of the noise have a limiting empirical distribution of compact support; this model, which is standard in random matrix theory, is satisfied by many models exhibiting either cross-row correlation structure or cross-column correlation structure, and also by many situations where there is inter-element correlation structure. Simulations demonstrate the effectiveness of the method even at moderate matrix sizes. The paper is supplemented by ready-to-use software packages implementing the proposed algorithm: package ScreeNOT in Python (via PyPI) and R (via CRAN).

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 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 $L\to\infty$. 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 $\varepsilon$ approaches $\sigma^2/\varepsilon$ for small $\varepsilon$; 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 quicker with $\sigma^2$.

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 $\Delta$, for some $\Delta>0$. For $\Delta=\infty$ 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 $\Delta>0$ perfect reconstruction is still possible if the sampling rate exceeds the Nyquist rate by a factor of $\pi e$. In this letter we improve upon this result and show that for finite energy signals, perfect recovery is possible for any $\Delta>0$ 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.

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 $\Delta$, for some $\Delta>0$. 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.

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.

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-P\'ech\'e 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.

In matrix recovery from random linear measurements, one is interested in recovering an unknown $M$-by-$N$ matrix $X_0$ from $n