arXiv subjects

Ramji Venkataramanan

Publications and source records attributed to Ramji Venkataramanan.

At least 37 records · Page 2Linked to original sources

arXiv2022

Estimation in Rotationally Invariant Generalized Linear Models via Approximate Message Passing

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.

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arXiv2022

Near-Optimal Coding for Many-user Multiple Access Channels

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.

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arXiv2021

PCA Initialization for Approximate Message Passing in Rotationally Invariant Models

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.

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arXiv2021

Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models

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 $θ\hat{\boldsymbol x}^{\rm L}+\hat{\boldsymbol x}^{\rm s}$ 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.

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arXiv2021

Modulated Sparse Superposition Codes for the Complex AWGN Channel

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.

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arXiv2021

Capacity-achieving Spatially Coupled Sparse Superposition Codes with AMP Decoding

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.

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arXiv2021

A unifying tutorial on Approximate Message Passing

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.

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arXiv2021

Approximate Message Passing with Spectral Initialization for Generalized Linear Models

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.

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arXiv2020

Multilayer Codes for Synchronization from Deletions and Insertions

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.

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arXiv2019

Sparse Regression Codes

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.

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arXiv2019

Estimation of Low-Rank Matrices via Approximate Message Passing

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').

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arXiv2019

Boolean Functions with Biased Inputs: Approximation and Noise Sensitivity

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.

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arXiv2019

Coding for Deletion Channels with Multiple Traces

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.

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arXiv2019

The Error Probability of Sparse Superposition Codes with Approximate Message Passing Decoding

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.

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arXiv2018

Empirical Bayes Estimators for High-Dimensional Sparse Vectors

The problem of estimating a high-dimensional sparse vector $\boldsymbolθ \in \mathbb{R}^n$ 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 $\boldsymbolθ$, 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.

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arXiv2018

Efficient Systematic Encoding of Non-binary VT Codes

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 $n$.

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arXiv2018

Spatially Coupled Sparse Regression Codes: Design and State Evolution Analysis

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 $(ω, Λ)$, 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.

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arXiv2018

A strong converse bound for multiple hypothesis testing, with applications to high-dimensional estimation

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.

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