Search

We consider the problem of localizing change points in high-dimensional linear regression. We propose an Approximate Message Passing (AMP) algorithm for estimating both the signals and the change point locations. Assuming Gaussian covariates, we give an exact asymptotic characterization of its estimation performance in the limit where the number of samples grows proportionally to the signal dimension. Our algorithm can be tailored to exploit any prior information on the signal, noise, and change points. It also enables uncertainty quantification in the form of an efficiently computable approximate posterior distribution, whose asymptotic form we characterize exactly. We validate our theory via numerical experiments, and demonstrate the favorable performance of our estimators on both synthetic data and images., Comment: 24 pages, 8 figures

We consider communication over the Gaussian multiple-access channel in the regime where the number of users grows linearly with the codelength. In this regime, schemes based on sparse superposition coding can achieve a near-optimal tradeoff between spectral efficiency and signal-to-noise ratio. However, these schemes are feasible only for small values of user payload. This paper investigates efficient schemes for larger user payloads, focusing on coded CDMA schemes where each user's information is encoded via a linear code before being modulated with a signature sequence. We propose an efficient approximate message passing (AMP) decoder that can be tailored to the structure of the linear code, and provide an exact asymptotic characterization of its performance. Based on this result, we consider a decoder that integrates AMP and belief propagation and characterize its tradeoff between spectral efficiency and signal-to-noise ratio, for a given target error rate. Simulation results show that the decoder achieves state-of-the-art performance at finite lengths, with a coded CDMA scheme defined using LDPC codes and a spatially coupled matrix of signature sequences., Comment: 23 pages, 8 figures. A shorter version of this paper to appear in the Proceedings of IEEE ISIT 2024

In the pooled data problem, the goal is to identify the categories associated with a large collection of items via a sequence of pooled tests. Each pooled test reveals the number of items of each category within the pool. We study an approximate message passing (AMP) algorithm for estimating the categories and rigorously characterize its performance, in both the noiseless and noisy settings. For the noiseless setting, we show that the AMP algorithm is equivalent to one recently proposed by El Alaoui et al. Our results provide a rigorous version of their performance guarantees, previously obtained via non-rigorous techniques. For the case of pooled data with two categories, known as quantitative group testing (QGT), we use the AMP guarantees to compute precise limiting values of the false positive rate and the false negative rate. Though the pooled data problem and QGT are both instances of estimation in a linear model, existing AMP theory cannot be directly applied since the design matrices are binary valued. The key technical ingredient in our analysis is a rigorous asymptotic characterization of AMP for generalized linear models defined via generalized white noise design matrices. This result, established using a recent universality result of Wang et al., is of independent interest. Our theoretical results are validated by numerical simulations. For comparison, we propose estimators based on convex relaxation and iterative thresholding, without providing theoretical guarantees. The simulations indicate that AMP outperforms the convex estimator for noiseless pooled data and QGT, but the convex estimator performs slightly better for noisy pooled data with three categories when the number of observations is small., Comment: 62 pages, 11 figures

We consider the problem of signal estimation in a generalized linear model (GLM). GLMs include many canonical problems in statistical estimation, such as linear regression, phase retrieval, and 1-bit compressed sensing. Recent work has precisely characterized the asymptotic minimum mean-squared error (MMSE) for GLMs with i.i.d. Gaussian sensing matrices. However, in many models there is a significant gap between the MMSE and the performance of the best known feasible estimators. In this work, we address this issue by considering GLMs defined via spatially coupled sensing matrices. We propose an efficient approximate message passing (AMP) algorithm for estimation and prove that with a simple choice of spatially coupled design, the MSE of a carefully tuned AMP estimator approaches the asymptotic MMSE in the high-dimensional limit. To prove the result, we first rigorously characterize the asymptotic performance of AMP for a GLM with a generic spatially coupled design. This characterization is in terms of a deterministic recursion (`state evolution') that depends on the parameters defining the spatial coupling. Then, using a simple spatially coupled design and a judicious choice of functions for the AMP algorithm, we analyze the fixed points of the resulting state evolution and show that it achieves the asymptotic MMSE. Numerical results for phase retrieval and rectified linear regression show that spatially coupled designs can yield substantially lower MSE than i.i.d. Gaussian designs at finite dimensions when used with AMP algorithms., Comment: 41 pages, 4 figures. A shorter version of this paper appeared in the proceedings of the 2023 IEEE International Symposium on Information Theory

We consider the problem of parameter estimation in a high-dimensional generalized linear model. Spectral methods obtained via the principal eigenvector of a suitable data-dependent matrix provide a simple yet surprisingly effective solution. However, despite their wide use, a rigorous performance characterization, as well as a principled way to preprocess the data, are available only for unstructured (i.i.d.\ Gaussian and Haar orthogonal) designs. In contrast, real-world data matrices are highly structured and exhibit non-trivial correlations. To address the problem, we consider correlated Gaussian designs capturing the anisotropic nature of the features via a covariance matrix $\Sigma$. Our main result is a precise asymptotic characterization of the performance of spectral estimators. This allows us to identify the optimal preprocessing that minimizes the number of samples needed for parameter estimation. Surprisingly, such preprocessing is universal across a broad set of designs, which partly addresses a conjecture on optimal spectral estimators for rotationally invariant models. Our principled approach vastly improves upon previous heuristic methods, including for designs common in computational imaging and genetics. The proposed methodology, based on approximate message passing, is broadly applicable and opens the way to the precise characterization of spiked matrices and of the corresponding spectral methods in a variety of settings.

We study the problem of regression in a generalized linear model (GLM) with multiple signals and latent variables. This model, which we call a matrix GLM, covers many widely studied problems in statistical learning, including mixed linear regression, max-affine regression, and mixture-of-experts. In mixed linear regression, each observation comes from one of $L$ signal vectors (regressors), but we do not know which one; in max-affine regression, each observation comes from the maximum of $L$ affine functions, each defined via a different signal vector. The goal in all these problems is to estimate the signals, and possibly some of the latent variables, from the observations. We propose a novel approximate message passing (AMP) algorithm for estimation in a matrix GLM and rigorously characterize its performance in the high-dimensional limit. This characterization is in terms of a state evolution recursion, which allows us to precisely compute performance measures such as the asymptotic mean-squared error. The state evolution characterization can be used to tailor the AMP algorithm to take advantage of any structural information known about the signals. Using state evolution, we derive an optimal choice of AMP `denoising' functions that minimizes the estimation error in each iteration. The theoretical results are validated by numerical simulations for mixed linear regression, max-affine regression, and mixture-of-experts. For max-affine regression, we propose an algorithm that combines AMP with expectation-maximization to estimate intercepts of the model along with the signals. The numerical results show that AMP significantly outperforms other estimators for mixed linear regression and max-affine regression in most parameter regimes., Comment: 44 pages. To appear in the Journal of Machine Learning Research. A shorter version of this paper appeared in the proceedings of AISTATS 2023

We consider the problem of mixed sparse linear regression with two components, where two real $k$-sparse signals $\beta_1, \beta_2$ are to be recovered from $n$ unlabelled noisy linear measurements. The sparsity is allowed to be sublinear in the dimension, and additive noise is assumed to be independent Gaussian with variance $\sigma^2$. Prior work has shown that the problem suffers from a $\frac{k}{SNR^2}$-to-$\frac{k^2}{SNR^2}$ statistical-to-computational gap, resembling other computationally challenging high-dimensional inference problems such as Sparse PCA and Robust Sparse Mean Estimation; here $SNR$ is the signal-to-noise ratio. We establish the existence of a more extensive computational barrier for this problem through the method of low-degree polynomials, but show that the problem is computationally hard only in a very narrow symmetric parameter regime. We identify a smooth information-computation tradeoff between the sample complexity $n$ and runtime for any randomized algorithm in this hard regime. Via a simple reduction, this provides novel rigorous evidence for the existence of a computational barrier to solving exact support recovery in sparse phase retrieval with sample complexity $n = \tilde{o}(k^2)$. Our second contribution is to analyze a simple thresholding algorithm which, outside of the narrow regime where the problem is hard, solves the associated mixed regression detection problem in $O(np)$ time with square-root the number of samples and matches the sample complexity required for (non-mixed) sparse linear regression; this allows the recovery problem to be subsequently solved by state-of-the-art techniques from the dense case. As a special case of our results, we show that this simple algorithm is order-optimal among a large family of algorithms in solving exact signed support recovery in sparse linear regression., Comment: To appear in Conference on Learning Theory (COLT) 2023

In a mixed generalized linear model, the objective is to learn multiple signals from unlabeled observations: each sample comes from exactly one signal, but it is not known which one. We consider the prototypical problem of estimating two statistically independent signals in a mixed generalized linear model with Gaussian covariates. Spectral methods are a popular class of estimators which output the top two eigenvectors of a suitable data-dependent matrix. However, despite the wide applicability, their design is still obtained via heuristic considerations, and the number of samples $n$ needed to guarantee recovery is super-linear in the signal dimension $d$. In this paper, we develop exact asymptotics on spectral methods in the challenging proportional regime in which $n, d$ grow large and their ratio converges to a finite constant. By doing so, we are able to optimize the design of the spectral method, and combine it with a simple linear estimator, in order to minimize the estimation error. Our characterization exploits a mix of tools from random matrices, free probability and the theory of approximate message passing algorithms. Numerical simulations for mixed linear regression and phase retrieval demonstrate the advantage enabled by our analysis over existing designs of spectral methods.

We consider the problem of recovering an $n_1 \times n_2$ low-rank matrix with $k$-sparse singular vectors from a small number of linear measurements (sketch). We propose a sketching scheme and an algorithm that can recover the singular vectors with high probability, with a sample complexity and running time that both depend only on $k$ and not on the ambient dimensions $n_1$ and $n_2$. Our sketching operator, based on a scheme for compressed sensing by Li et al. and Bakshi et al., uses a combination of a sparse parity check matrix and a partial DFT matrix. Our main contribution is the design and analysis of a two-stage iterative algorithm which recovers the singular vectors by exploiting the simultaneously sparse and low-rank structure of the matrix. We derive a nonasymptotic bound on the probability of exact recovery, which holds for any $n_1\times n_2 $ sparse, low-rank matrix. We also show how the scheme can be adapted to tackle matrices that are approximately sparse and low-rank. The theoretical results are validated by numerical simulations and comparisons with existing schemes that use convex programming for recovery., Comment: 46 pages, 14 figures. A shorter version appeared at the 2022 IEEE International Symposium on Information Theory

We consider the problem of signal estimation in generalized linear models defined via rotationally invariant design matrices. Since these matrices can have an arbitrary spectral distribution, this model is well suited for capturing complex correlation structures which often arise in applications. We propose a novel family of approximate message passing (AMP) algorithms for signal estimation, and rigorously characterize their performance in the high-dimensional limit via a state evolution recursion. Our rotationally invariant AMP has complexity of the same order as the existing AMP derived under the restrictive assumption of a Gaussian design; our algorithm also recovers this existing AMP as a special case. Numerical results showcase a performance close to Vector AMP (which is conjectured to be Bayes-optimal in some settings), but obtained with a much lower complexity, as the proposed algorithm does not require a computationally expensive singular value decomposition., Comment: 35 pages, 8 figures, to appear in International Conference on Machine Learning (ICML), 2022

We study the problem of estimating a rank-$1$ signal in the presence of rotationally invariant noise-a class of perturbations more general than Gaussian noise. Principal Component Analysis (PCA) provides a natural estimator, and sharp results on its performance have been obtained in the high-dimensional regime. Recently, an Approximate Message Passing (AMP) algorithm has been proposed as an alternative estimator with the potential to improve the accuracy of PCA. However, the existing analysis of AMP requires an initialization that is both correlated with the signal and independent of the noise, which is often unrealistic in practice. In this work, we combine the two methods, and propose to initialize AMP with PCA. Our main result is a rigorous asymptotic characterization of the performance of this estimator. Both the AMP algorithm and its analysis differ from those previously derived in the Gaussian setting: at every iteration, our AMP algorithm requires a specific term to account for PCA initialization, while in the Gaussian case, PCA initialization affects only the first iteration of AMP. The proof is based on a two-phase artificial AMP that first approximates the PCA estimator and then mimics the true AMP. Our numerical simulations show an excellent agreement between AMP results and theoretical predictions, and suggest an interesting open direction on achieving Bayes-optimal performance., Comment: 72 pages, 2 figures, appeared in Neural Information Processing Systems (NeurIPS), 2021

Over the last decade or so, Approximate Message Passing (AMP) algorithms have become extremely popular in various structured high-dimensional statistical problems. The fact that the origins of these techniques can be traced back to notions of belief propagation in the statistical physics literature lends a certain mystique to the area for many statisticians. Our goal in this work is to present the main ideas of AMP from a statistical perspective, to illustrate the power and flexibility of the AMP framework. Along the way, we strengthen and unify many of the results in the existing literature., Comment: 99 pages, 2 figures

This paper considers the Gaussian multiple-access channel (MAC) in the asymptotic regime where the number of users grows linearly with the code length. We propose efficient coding schemes based on random linear models with approximate message passing (AMP) decoding and derive the asymptotic error rate achieved for a given user density, user payload (in bits), and user energy. The tradeoff between energy-per-bit and achievable user density (for a fixed user payload and target error rate) is studied, and it is demonstrated that in the large system limit, a spatially coupled coding scheme with AMP decoding achieves near-optimal tradeoffs for a wide range of user densities. Furthermore, in the regime where the user payload is large, we also study the tradeoff between energy-per-bit and spectral efficiency and discuss methods to reduce decoding complexity., Comment: 15 pages, 4 figures. To appear in IEEE Journal on Selected Areas in Information Theory

We consider the problem of estimating a signal from measurements obtained via a generalized linear model. We focus on estimators based on approximate message passing (AMP), a family of iterative algorithms with many appealing features: the performance of AMP in the high-dimensional limit can be succinctly characterized under suitable model assumptions; AMP can also be tailored to the empirical distribution of the signal entries, and for a wide class of estimation problems, AMP is conjectured to be optimal among all polynomial-time algorithms. However, a major issue of AMP is that in many models (such as phase retrieval), it requires an initialization correlated with the ground-truth signal and independent from the measurement matrix. Assuming that such an initialization is available is typically not realistic. In this paper, we solve this problem by proposing an AMP algorithm initialized with a spectral estimator. With such an initialization, the standard AMP analysis fails since the spectral estimator depends in a complicated way on the design matrix. Our main contribution is a rigorous characterization of the performance of AMP with spectral initialization in the high-dimensional limit. The key technical idea is to define and analyze a two-phase artificial AMP algorithm that first produces the spectral estimator, and then closely approximates the iterates of the true AMP. We also provide numerical results that demonstrate the validity of the proposed approach., Comment: 38 pages, 5 figures, AISTATS 2021

We study the problem of recovering an unknown signal $\boldsymbol x$ given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator $\hat{\boldsymbol x}^{\rm L}$ and a spectral estimator $\hat{\boldsymbol x}^{\rm s}$. The former is a data-dependent linear combination of the columns of the measurement matrix, and its analysis is quite simple. The latter is the principal eigenvector of a data-dependent matrix, and a recent line of work has studied its performance. In this paper, we show how to optimally combine $\hat{\boldsymbol x}^{\rm L}$ and $\hat{\boldsymbol x}^{\rm s}$. At the heart of our analysis is the exact characterization of the joint empirical distribution of $(\boldsymbol x, \hat{\boldsymbol x}^{\rm L}, \hat{\boldsymbol x}^{\rm s})$ in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of $\hat{\boldsymbol x}^{\rm L}$ and $\hat{\boldsymbol x}^{\rm s}$, given the limiting distribution of the signal $\boldsymbol x$. When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form $\theta\hat{\boldsymbol x}^{\rm L}+\hat{\boldsymbol x}^{\rm s}$ and we derive the optimal combination coefficient. In order to establish the limiting distribution of $(\boldsymbol x, \hat{\boldsymbol x}^{\rm L}, \hat{\boldsymbol x}^{\rm s})$, we design and analyze an Approximate Message Passing (AMP) algorithm whose iterates give $\hat{\boldsymbol x}^{\rm L}$ and approach $\hat{\boldsymbol x}^{\rm s}$. Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately., Comment: 49 pages, 6 figures

This paper studies a generalization of sparse superposition codes (SPARCs) for communication over the complex additive white Gaussian noise (AWGN) channel. In a SPARC, the codebook is defined in terms of a design matrix, and each codeword is a generated by multiplying the design matrix with a sparse message vector. In the standard SPARC construction, information is encoded in the locations of the non-zero entries of the message vector. In this paper we generalize the construction and consider modulated SPARCs, where information in encoded in both the locations and the values of the non-zero entries of the message vector. We focus on the case where the non-zero entries take values from a phase-shift keying (PSK) constellation. We propose a computationally efficient approximate message passing (AMP) decoder, and obtain analytical bounds on the state evolution parameters which predict the error performance of the decoder. Using these bounds we show that PSK-modulated SPARCs are asymptotically capacity achieving for the complex AWGN channel, with either spatial coupling or power allocation. We also provide numerical simulation results to demonstrate the error performance at finite code lengths. These results show that introducing modulation to the SPARC design can significantly reduce decoding complexity without sacrificing error performance., Comment: 20 pages, 6 figures. To appear in IEEE Transactions on Information Theory

Sparse superposition codes, also called sparse regression codes (SPARCs), are a class of codes for efficient communication over the AWGN channel at rates approaching the channel capacity. In a standard SPARC, codewords are sparse linear combinations of columns of an i.i.d. Gaussian design matrix, while in a spatially coupled SPARC the design matrix has a block-wise structure, where the variance of the Gaussian entries can be varied across blocks. A well-designed spatial coupling structure can significantly enhance the error performance of iterative decoding algorithms such as Approximate Message Passing (AMP). In this paper, we obtain a non-asymptotic bound on the probability of error of spatially coupled SPARCs with AMP decoding. Applying this bound to a simple band-diagonal design matrix, we prove that spatially coupled SPARCs with AMP decoding achieve the capacity of the AWGN channel. The bound also highlights how the decay of error probability depends on each design parameter of the spatially coupled SPARC. An attractive feature of AMP decoding is that its asymptotic mean squared error (MSE) can be predicted via a deterministic recursion called state evolution. Our result provides the first proof that the MSE concentrates on the state evolution prediction for spatially coupled designs. Combined with the state evolution prediction, this result implies that spatially coupled SPARCs with the proposed band-diagonal design are capacity-achieving. Using the proof technique used to establish the main result, we also obtain a concentration inequality for the MSE of AMP applied to compressed sensing with spatially coupled design matrices. Finally we provide numerical simulation results that demonstrate the finite length error performance of spatially coupled SPARCs. The performance is compared with coded modulation schemes that use LDPC codes from the DVB-S2 standard., Comment: To appear in IEEE Transactions on Information Theory. This version contains proofs of two technical lemmas that were omitted in the journal version

Developing computationally-efficient codes that approach the Shannon-theoretic limits for communication and compression has long been one of the major goals of information and coding theory. There have been significant advances towards this goal in the last couple of decades, with the emergence of turbo codes, sparse-graph codes, and polar codes. These codes are designed primarily for discrete-alphabet channels and sources. For Gaussian channels and sources, where the alphabet is inherently continuous, Sparse Superposition Codes or Sparse Regression Codes (SPARCs) are a promising class of codes for achieving the Shannon limits. This survey provides a unified and comprehensive overview of sparse regression codes, covering theory, algorithms, and practical implementation aspects. The first part of the monograph focuses on SPARCs for AWGN channel coding, and the second part on SPARCs for lossy compression (with squared error distortion criterion). In the third part, SPARCs are used to construct codes for Gaussian multi-terminal channel and source coding models such as broadcast channels, multiple-access channels, and source and channel coding with side information. The survey concludes with a discussion of open problems and directions for future work., Comment: Published in Foundations and Trends in Communications and Information Theory, 2019

Motivated by the sequence reconstruction problem from traces in DNA-based storage, we consider the problem of designing codes for the deletion channel when multiple observations (or traces) are available to the decoder. We propose simple binary and non-binary codes based on Varshamov-Tenengolts (VT) codes. The proposed codes split the codeword in blocks and employ a VT code in each block. The availability of multiple traces helps the decoder to identify deletion-free copies of a block, and to avoid mis-synchronization while decoding. The encoding complexity of the proposed scheme is linear in the codeword length; the decoding complexity is linear in the codeword length, and quadratic in the number of deletions and the number of traces. The proposed scheme offers an explicit low-complexity technique for correcting deletions using multiple traces., Comment: This paper will be presented at ISIT 2019

This paper considers the problem of approximating a Boolean function $f$ using another Boolean function from a specified class. Two classes of approximating functions are considered: $k$-juntas, and linear Boolean functions. The $n$ input bits of the function are assumed to be independently drawn from a distribution that may be biased. The quality of approximation is measured by the mismatch probability between $f$ and the approximating function $g$. For each class, the optimal approximation and the associated mismatch probability is characterized in terms of the biased Fourier expansion of $f$. The technique used to analyze the mismatch probability also yields an expression for the noise sensitivity of $f$ in terms of the biased Fourier coefficients, under a general i.i.d. input perturbation model., Comment: 5 pages, 2 figures, To appear in IEEE ISIT 2018

We consider the design and analysis of spatially coupled sparse regression codes (SC-SPARCs), which were recently introduced by Barbier et al. for efficient communication over the additive white Gaussian noise channel. SC-SPARCs can be efficiently decoded using an Approximate Message Passing (AMP) decoder, whose performance in each iteration can be predicted via a set of equations called state evolution. In this paper, we give an asymptotic characterization of the state evolution equations for SC-SPARCs. For any given base matrix (that defines the coupling structure of the SC-SPARC) and rate, this characterization can be used to predict whether or not AMP decoding will succeed in the large system limit. We then consider a simple base matrix defined by two parameters $(\omega, \Lambda)$, and show that AMP decoding succeeds in the large system limit for all rates $R < \mathcal{C}$. The asymptotic result also indicates how the parameters of the base matrix affect the decoding progression. Simulation results are presented to evaluate the performance of SC-SPARCs defined with the proposed base matrix., Comment: 8 pages, 6 figures. A shorter version of this paper to appear in ISIT 2018

Sparse superposition codes, or sparse regression codes (SPARCs), are a recent class of codes for reliable communication over the AWGN channel at rates approaching the channel capacity. Approximate message passing (AMP) decoding, a computationally efficient technique for decoding SPARCs, has been proven to be asymptotically capacity-achieving for the AWGN channel. In this paper, we refine the asymptotic result by deriving a large deviations bound on the probability of AMP decoding error. This bound gives insight into the error performance of the AMP decoder for large but finite problem sizes, giving an error exponent as well as guidance on how the code parameters should be chosen at finite block lengths. For an appropriate choice of code parameters, we show that for any fixed rate less than the channel capacity, the decoding error probability decays exponentially in $n/(\log n)^{2T}$, where $T$, the number of AMP iterations required for successful decoding, is bounded in terms of the gap from capacity., Comment: 27 pages. IEEE Transactions on Information Theory

Consider the problem of estimating a low-rank matrix when its entries are perturbed by Gaussian noise. If the empirical distribution of the entries of the spikes is known, optimal estimators that exploit this knowledge can substantially outperform simple spectral approaches. Recent work characterizes the asymptotic accuracy of Bayes-optimal estimators in the high-dimensional limit. In this paper we present a practical algorithm that can achieve Bayes-optimal accuracy above the spectral threshold. A bold conjecture from statistical physics posits that no polynomial-time algorithm achieves optimal error below the same threshold (unless the best estimator is trivial). Our approach uses Approximate Message Passing (AMP) in conjunction with a spectral initialization. AMP algorithms have proved successful in a variety of statistical estimation tasks, and are amenable to exact asymptotic analysis via state evolution. Unfortunately, state evolution is uninformative when the algorithm is initialized near an unstable fixed point, as often happens in low-rank matrix estimation. We develop a new analysis of AMP that allows for spectral initializations. Our main theorem is general and applies beyond matrix estimation. However, we use it to derive detailed predictions for the problem of estimating a rank-one matrix in noise. Special cases of this problem are closely related---via universality arguments---to the network community detection problem for two asymmetric communities. For general rank-one models, we show that AMP can be used to construct confidence intervals and control false discovery rate. We provide illustrations of the general methodology by considering the cases of sparse low-rank matrices and of block-constant low-rank matrices with symmetric blocks (we refer to the latter as to the `Gaussian Block Model')., Comment: 76 pages, 6 pdf figures; Version 4 expands the introductory material and the applications to statistical inference

Varshamov-Tenengolts (VT) codes are a class of codes which can correct a single deletion or insertion with a linear-time decoder. This paper addresses the problem of efficient encoding of non-binary VT codes, defined over an alphabet of size $q >2$. We propose a simple linear-time encoding method to systematically map binary message sequences onto VT codewords. The method provides a new lower bound on the size of $q$-ary VT codes of length $n$., Comment: This paper will appear in the proceedings of ISIT 2018

The problem of estimating a high-dimensional sparse vector $\boldsymbol{\theta} \in \mathbb{R}^n$ from an observation in i.i.d. Gaussian noise is considered. The performance is measured using squared-error loss. An empirical Bayes shrinkage estimator, derived using a Bernoulli-Gaussian prior, is analyzed and compared with the well-known soft-thresholding estimator. We obtain concentration inequalities for the Stein's unbiased risk estimate and the loss function of both estimators. The results show that for large $n$, both the risk estimate and the loss function concentrate on deterministic values close to the true risk. Depending on the underlying $\boldsymbol{\theta}$, either the proposed empirical Bayes (eBayes) estimator or soft-thresholding may have smaller loss. We consider a hybrid estimator that attempts to pick the better of the soft-thresholding estimator and the eBayes estimator by comparing their risk estimates. It is shown that: i) the loss of the hybrid estimator concentrates on the minimum of the losses of the two competing estimators, and ii) the risk of the hybrid estimator is within order $\frac{1}{\sqrt{n}}$ of the minimum of the two risks. Simulation results are provided to support the theoretical results. Finally, we use the eBayes and hybrid estimators as denoisers in the approximate message passing (AMP) algorithm for compressed sensing, and show that their performance is superior to the soft-thresholding denoiser in a wide range of settings., Comment: 35 pages, to appear in Information and Inference: A Journal of the IMA

In statistical inference problems, we wish to obtain lower bounds on the minimax risk, that is to bound the performance of any possible estimator. A standard technique to obtain risk lower bounds involves the use of Fano's inequality. In an information-theoretic setting, it is known that Fano's inequality typically does not give a sharp converse result (error lower bound) for channel coding problems. Moreover, recent work has shown that an argument based on binary hypothesis testing gives tighter results. We adapt this technique to the statistical setting, and argue that Fano's inequality can always be replaced by this approach to obtain tighter lower bounds that can be easily computed and are asymptotically sharp. We illustrate our technique in three applications: density estimation, active learning of a binary classifier, and compressed sensing, obtaining tighter risk lower bounds in each case., Comment: In the latest version, the value of $\lambda$ in the statements of Lemma 4.1 and Proposition 4.2 is restricted to the interval $(0,1]$. This is the correct condition, rather than $\lambda>0$ stated in the journal version below

Consider two remote nodes (encoder and decoder), each with a binary sequence. The encoder's sequence $X$ differs from the decoder's sequence $Y$ by a small number of edits (deletions and insertions). The goal is to construct a message $M$, to be sent via a one-way error free link, such that the decoder can reconstruct $X$ using $M$ and $Y$. In this paper, we devise a coding scheme for this one-way synchronization model. The scheme is based on multiple layers of Varshamov-Tenengolts (VT) codes combined with off-the-shelf linear error-correcting codes, and uses a list decoder. We bound the expected list size of the decoder under certain assumptions, and validate its performance via numerical simulations. We also consider an alternative decoder that uses only the constraints from the VT codes (i.e., does not require a linear code), and has a smaller redundancy at the expense of a slightly larger average list size., Comment: This paper is accepted for publication in the IEEE Transactions on Information Theory

Sparse superposition codes are a recent class of codes introduced by Barron and Joseph for efficient communication over the AWGN channel. With an appropriate power allocation, these codes have been shown to be asymptotically capacity-achieving with computationally feasible decoding. However, a direct implementation of the capacity-achieving construction does not give good finite length error performance. In this paper, we consider sparse superposition codes with approximate message passing (AMP) decoding, and describe a variety of techniques to improve their finite length performance. These include an iterative algorithm for SPARC power allocation, guidelines for choosing codebook parameters, and estimating a critical decoding parameter online instead of pre-computation. We also show how partial outer codes can be used in conjunction with AMP decoding to obtain a steep waterfall in the error performance curves. We compare the error performance of AMP-decoded sparse superposition codes with coded modulation using LDPC codes from the WiMAX standard., Comment: 13 pages, 16 figures. To appear in IEEE Transactions on Communications

This paper considers insertion and deletion channels with the additional assumption that the channel input sequence is implicitly divided into segments such that at most one edit can occur within a segment. No segment markers are available in the received sequence. We propose code constructions for the segmented deletion, segmented insertion, and segmented insertion-deletion channels based on subsets of Varshamov-Tenengolts codes chosen with pre-determined prefixes and/or suffixes. The proposed codes, constructed for any finite alphabet, are zero-error and can be decoded segment-by-segment. We also derive an upper bound on the rate of any zero-error code for the segmented edit channel, in terms of the segment length. This upper bound shows that the rate scaling of the proposed codes as the segment length increases is the same as that of the maximal code., Comment: Appeared in IEEE Transactions on Information Theory

Approximate message passing (AMP) refers to a class of efficient algorithms for statistical estimation in high-dimensional problems such as compressed sensing and low-rank matrix estimation. This paper analyzes the performance of AMP in the regime where the problem dimension is large but finite. For concreteness, we consider the setting of high-dimensional regression, where the goal is to estimate a high-dimensional vector $\beta_0$ from a noisy measurement $y=A \beta_0 + w$. AMP is a low-complexity, scalable algorithm for this problem. Under suitable assumptions on the measurement matrix $A$, AMP has the attractive feature that its performance can be accurately characterized in the large system limit by a simple scalar iteration called state evolution. Previous proofs of the validity of state evolution have all been asymptotic convergence results. In this paper, we derive a concentration inequality for AMP with i.i.d. Gaussian measurement matrices with finite size $n \times N$. The result shows that the probability of deviation from the state evolution prediction falls exponentially in $n$. This provides theoretical support for empirical findings that have demonstrated excellent agreement of AMP performance with state evolution predictions for moderately large dimensions. The concentration inequality also indicates that the number of AMP iterations $t$ can grow no faster than order $\frac{\log n}{\log \log n}$ for the performance to be close to the state evolution predictions with high probability. The analysis can be extended to obtain similar non-asymptotic results for AMP in other settings such as low-rank matrix estimation., Comment: To appear in IEEE Transactions on Information Theory

This paper considers the problem of estimating a high-dimensional vector of parameters $\boldsymbol{\theta} \in \mathbb{R}^n$ from a noisy observation. The noise vector is i.i.d. Gaussian with known variance. For a squared-error loss function, the James-Stein (JS) estimator is known to dominate the simple maximum-likelihood (ML) estimator when the dimension $n$ exceeds two. The JS-estimator shrinks the observed vector towards the origin, and the risk reduction over the ML-estimator is greatest for $\boldsymbol{\theta}$ that lie close to the origin. JS-estimators can be generalized to shrink the data towards any target subspace. Such estimators also dominate the ML-estimator, but the risk reduction is significant only when $\boldsymbol{\theta}$ lies close to the subspace. This leads to the question: in the absence of prior information about $\boldsymbol{\theta}$, how do we design estimators that give significant risk reduction over the ML-estimator for a wide range of $\boldsymbol{\theta}$? In this paper, we propose shrinkage estimators that attempt to infer the structure of $\boldsymbol{\theta}$ from the observed data in order to construct a good attracting subspace. In particular, the components of the observed vector are separated into clusters, and the elements in each cluster shrunk towards a common attractor. The number of clusters and the attractor for each cluster are determined from the observed vector. We provide concentration results for the squared-error loss and convergence results for the risk of the proposed estimators. The results show that the estimators give significant risk reduction over the ML-estimator for a wide range of $\boldsymbol{\theta}$, particularly for large $n$. Simulation results are provided to support the theoretical claims., Comment: Appeared in IEEE Transactions on Information Theory

Sparse superposition codes were recently introduced by Barron and Joseph for reliable communication over the AWGN channel at rates approaching the channel capacity. The codebook is defined in terms of a Gaussian design matrix, and codewords are sparse linear combinations of columns of the matrix. In this paper, we propose an approximate message passing decoder for sparse superposition codes, whose decoding complexity scales linearly with the size of the design matrix. The performance of the decoder is rigorously analyzed and it is shown to asymptotically achieve the AWGN capacity with an appropriate power allocation. Simulation results are provided to demonstrate the performance of the decoder at finite blocklengths. We introduce a power allocation scheme to improve the empirical performance, and demonstrate how the decoding complexity can be significantly reduced by using Hadamard design matrices., Comment: 25 pages, 4 figures. IEEE Transactions on Information Theory

This paper studies the performance of sparse regression codes for lossy compression with the squared-error distortion criterion. In a sparse regression code, codewords are linear combinations of subsets of columns of a design matrix. It is shown that with minimum-distance encoding, sparse regression codes achieve the Shannon rate-distortion function for i.i.d. Gaussian sources $R^*(D)$ as well as the optimal excess-distortion exponent. This completes a previous result which showed that $R^*(D)$ and the optimal exponent were achievable for distortions below a certain threshold. The proof of the rate-distortion result is based on the second moment method, a popular technique to show that a non-negative random variable $X$ is strictly positive with high probability. In our context, $X$ is the number of codewords within target distortion $D$ of the source sequence. We first identify the reason behind the failure of the standard second moment method for certain distortions, and illustrate the different failure modes via a stylized example. We then use a refinement of the second moment method to show that $R^*(D)$ is achievable for all distortion values. Finally, the refinement technique is applied to Suen's correlation inequality to prove the achievability of the optimal Gaussian excess-distortion exponent., Comment: 16 pages. IEEE Transactions on Information Theory

Consider two remote nodes having binary sequences $X$ and $Y$, respectively. $Y$ is an edited version of ${X}$, where the editing involves random deletions, insertions, and substitutions, possibly in bursts. The goal is for the node with $Y$ to reconstruct $X$ with minimal exchange of information over a noiseless link. The communication is measured in terms of both the total number of bits exchanged and the number of interactive rounds of communication. This paper focuses on the setting where the number of edits is $o(\tfrac{n}{\log n})$, where $n$ is the length of $X$. We first consider the case where the edits are a mixture of insertions and deletions (indels), and propose an interactive synchronization algorithm with near-optimal communication rate and average computational complexity of $O(n)$ arithmetic operations. The algorithm uses interaction to efficiently split the source sequence into substrings containing exactly one deletion or insertion. Each of these substrings is then synchronized using an optimal one-way synchronization code based on the single-deletion correcting channel codes of Varshamov and Tenengolts (VT codes). We then build on this synchronization algorithm in three different ways. First, it is modified to work with a single round of interaction. The reduction in the number of rounds comes at the expense of higher communication, which is quantified. Next, we present an extension to the practically important case where the insertions and deletions may occur in (potentially large) bursts. Finally, we show how to synchronize the sources to within a target Hamming distance. This feature can be used to differentiate between substitution and indel edits. In addition to theoretical performance bounds, we provide several validating simulation results for the proposed algorithms.

We study a new class of codes for Gaussian multi-terminal source and channel coding. These codes are designed using the statistical framework of high-dimensional linear regression and are called Sparse Superposition or Sparse Regression codes. Codewords are linear combinations of subsets of columns of a design matrix. These codes were recently introduced by Barron and Joseph and shown to achieve the channel capacity of AWGN channels with computationally feasible decoding. They have also recently been shown to achieve the optimal rate-distortion function for Gaussian sources. In this paper, we demonstrate how to implement random binning and superposition coding using sparse regression codes. In particular, with minimum-distance encoding/decoding it is shown that sparse regression codes attain the optimal information-theoretic limits for a variety of multi-terminal source and channel coding problems., Comment: 9 pages, appeared in the Proceedings of the 50th Annual Allerton Conference on Communication, Control, and Computing - 2012

We propose computationally efficient encoders and decoders for lossy compression using a Sparse Regression Code. The codebook is defined by a design matrix and codewords are structured linear combinations of columns of this matrix. The proposed encoding algorithm sequentially chooses columns of the design matrix to successively approximate the source sequence. It is shown to achieve the optimal distortion-rate function for i.i.d Gaussian sources under the squared-error distortion criterion. For a given rate, the parameters of the design matrix can be varied to trade off distortion performance with encoding complexity. An example of such a trade-off as a function of the block length n is the following. With computational resource (space or time) per source sample of O((n/\log n)^2), for a fixed distortion-level above the Gaussian distortion-rate function, the probability of excess distortion decays exponentially in n. The Sparse Regression Code is robust in the following sense: for any ergodic source, the proposed encoder achieves the optimal distortion-rate function of an i.i.d Gaussian source with the same variance. Simulations show that the encoder has good empirical performance, especially at low and moderate rates., Comment: 14 pages, to appear in IEEE Transactions on Information Theory

Many storage channels admit reading and rewriting of the content at a given cost. We consider rewritable channels with a hidden state which models the unknown characteristics of the memory cell. In addition to mitigating the effect of the write noise, rewrites can help the write controller obtain a better estimate of the hidden state. The paper has two contributions. The first is a lower bound on the capacity of a general rewritable channel with hidden state. The lower bound is obtained using a coding scheme that combines Gelfand-Pinsker coding with superposition coding. The rewritable AWGN channel is discussed as an example. The second contribution is a simple coding scheme for a rewritable channel where the write noise and hidden state are both uniformly distributed. It is shown that this scheme is asymptotically optimal as the number of rewrites gets large., Comment: 10 pages. Part of the paper appeared in the proceedings of the 2012 IEEE International Symposium on Information Theory

We study a new class of codes for lossy compression with the squared-error distortion criterion, designed using the statistical framework of high-dimensional linear regression. Codewords are linear combinations of subsets of columns of a design matrix. Called a Sparse Superposition or Sparse Regression codebook, this structure is motivated by an analogous construction proposed recently by Barron and Joseph for communication over an AWGN channel. For i.i.d Gaussian sources and minimum-distance encoding, we show that such a code can attain the Shannon rate-distortion function with the optimal error exponent, for all distortions below a specified value. It is also shown that sparse regression codes are robust in the following sense: a codebook designed to compress an i.i.d Gaussian source of variance $\sigma^2$ with (squared-error) distortion $D$ can compress any ergodic source of variance less than $\sigma^2$ to within distortion $D$. Thus the sparse regression ensemble retains many of the good covering properties of the i.i.d random Gaussian ensemble, while having having a compact representation in terms of a matrix whose size is a low-order polynomial in the block-length., Comment: This version corrects a typo in the statement of Theorem 2 of the published paper

A single-letter achievable rate region is proposed for the two-receiver discrete memoryless broadcast channel with generalized feedback. The coding strategy involves block-Markov superposition coding, using Marton's coding scheme for the broadcast channel without feedback as the starting point. If the message rates in the Marton scheme are too high to be decoded at the end of a block, each receiver is left with a list of messages compatible with its output. Resolution information is sent in the following block to enable each receiver to resolve its list. The key observation is that the resolution information of the first receiver is correlated with that of the second. This correlated information is efficiently transmitted via joint source-channel coding, using ideas similar to the Han-Costa coding scheme. Using the result, we obtain an achievable rate region for the stochastically degraded AWGN broadcast channel with noisy feedback from only one receiver. It is shown that this region is strictly larger than the no-feedback capacity region., Comment: To appear in IEEE Transactions on Information Theory. Contains example of AWGN Broadcast Channel with noisy feedback

This paper considers a binary channel with deletions and insertions, where each input bit is transformed in one of the following ways: it is deleted with probability d, or an extra bit is added after it with probability i, or it is transmitted unmodified with probability 1-d-i. A computable lower bound on the capacity of this channel is derived. The transformation of the input sequence by the channel may be viewed in terms of runs as follows: some runs of the input sequence get shorter/longer, some runs get deleted, and some new runs are added. It is difficult for the decoder to synchronize the channel output sequence to the transmitted codeword mainly due to deleted runs and new inserted runs. The main idea is a mutual information decomposition in terms of the rate achieved by a sub-optimal decoder that determines the positions of the deleted and inserted runs in addition to decoding the transmitted codeword. The mutual information between the channel input and output sequences is expressed as the sum of the rate achieved by this decoder and the rate loss due to its sub-optimality. Obtaining computable lower bounds on each of these quantities yields a lower bound on the capacity. The bounds proposed in this paper provide the first characterization of achievable rates for channels with general insertions, and for channels with both deletions and insertions. For the special case of the deletion channel, the proposed bound improves on the previous best lower bound for deletion probabilities up to 0.3., Comment: To appear in IEEE Transactions on Information Theory. For the deletion channel, the new capacity lower bound improves on the previous best bound for deletion probabilities up to 0.3

Let $\mathcal{X}$ and $\mathcal{Y}$ be finite alphabets and $P_{XY}$ a joint distribution over them, with $P_X$ and $P_Y$ representing the marginals. For any $\epsilon > 0$, the set of $n$-length sequences $x^n$ and $y^n$ that are jointly typical \cite{ckbook} according to $P_{XY}$ can be represented on a bipartite graph. We present a formal definition of such a graph, known as a \emph{typicality} graph, and study some of its properties.

A new single-letter achievable rate region is proposed for the two-user discrete memoryless multiple-access channel(MAC) with noiseless feedback. The proposed region includes the Cover-Leung rate region [1], and it is shown that the inclusion is strict. The proof uses a block-Markov superposition strategy based on the observation that the messages of the two users are correlated given the feedback. The rates of transmission are too high for each encoder to decode the other's message directly using the feedback, so they transmit correlated information in the next block to learn the message of one another. They then cooperate in the following block to resolve the residual uncertainty of the decoder. The coding scheme may be viewed as a natural generalization of the Cover-Leung scheme with a delay of one extra block and a pair of additional auxiliary random variables. We compute the proposed rate region for two different MACs and compare the results with other known rate regions for the MAC with feedback. Finally, we show how the coding scheme can be extended to obtain larger rate regions with more auxiliary random variables., Comment: appeared in IEEE Transactions on Information Theory

We study the problem of computing the rate-distortion function for sources with feed-forward and the capacity for channels with feedback. The formulas (involving directed information) for the optimal rate-distortion function with feed-forward and channel capacity with feedback are multi-letter expressions and cannot be computed easily in general. In this work, we derive conditions under which these can be computed for a large class of sources/channels with memory and distortion/cost measures. Illustrative examples are also provided., Comment: 5 pages, Submitted to ISIT 2007. Corrected typos, latest version that appeared in ISIT 2007