arXiv subjects

Arian Maleki

Publications and source records attributed to Arian Maleki.

At least 55 records · Page 3Linked to original sources

arXiv2018

Optimal Data Detection in Large MIMO

Large multiple-input multiple-output (MIMO) appears in massive multi-user MIMO and randomly-spread code-division multiple access (CDMA)-based wireless systems. In order to cope with the excessively high complexity of optimal data detection in such systems, a variety of efficient yet sub-optimal algorithms have been proposed in the past. In this paper, we propose a data detection algorithm that is computationally efficient and optimal in a sense that it is able to achieve the same error-rate performance as the individually optimal (IO) data detector under certain assumptions on the MIMO system matrix and constellation alphabet. Our algorithm, which we refer to as LAMA (short for large MIMO AMP), builds on complex-valued Bayesian approximate message passing (AMP), which enables an exact analytical characterization of the performance and complexity in the large-system limit via the state-evolution framework. We derive optimality conditions for LAMA and investigate performance/complexity trade-offs. As a byproduct of our analysis, we recover classical results of IO data detection for randomly-spread CDMA. We furthermore provide practical ways for LAMA to approach the theoretical performance limits in realistic, finite-dimensional systems at low computational complexity.

cs.IT↗

arXiv2018

Benefits of over-parameterization with EM

Expectation Maximization (EM) is among the most popular algorithms for maximum likelihood estimation, but it is generally only guaranteed to find its stationary points of the log-likelihood objective. The goal of this article is to present theoretical and empirical evidence that over-parameterization can help EM avoid spurious local optima in the log-likelihood. We consider the problem of estimating the mean vectors of a Gaussian mixture model in a scenario where the mixing weights are known. Our study shows that the global behavior of EM, when one uses an over-parameterized model in which the mixing weights are treated as unknown, is better than that when one uses the (correct) model with the mixing weights fixed to the known values. For symmetric Gaussians mixtures with two components, we prove that introducing the (statistically redundant) weight parameters enables EM to find the global maximizer of the log-likelihood starting from almost any initial mean parameters, whereas EM without this over-parameterization may very often fail. For other Gaussian mixtures, we provide empirical evidence that shows similar behavior. Our results corroborate the value of over-parameterization in solving non-convex optimization problems, previously observed in other domains.

cs.LG↗

arXiv2018

Approximate Leave-One-Out for High-Dimensional Non-Differentiable Learning Problems

Consider the following class of learning schemes: \begin{equation} \label{eq:main-problem1} \hat{\boldsymbolβ} := \underset{\boldsymbolβ \in \mathcal{C}}{\arg\min} \;\sum_{j=1}^n \ell(\boldsymbol{x}_j^\top\boldsymbolβ; y_j) + λR(\boldsymbolβ), \qquad \qquad \qquad (1) \end{equation} where $\boldsymbol{x}_i \in \mathbb{R}^p$ and $y_i \in \mathbb{R}$ denote the $i^{\rm th}$ feature and response variable respectively. Let $\ell$ and $R$ be the convex loss function and regularizer, $\boldsymbolβ$ denote the unknown weights, and $λ$ be a regularization parameter. $\mathcal{C} \subset \mathbb{R}^{p}$ is a closed convex set. Finding the optimal choice of $λ$ is a challenging problem in high-dimensional regimes where both $n$ and $p$ are large. We propose three frameworks to obtain a computationally efficient approximation of the leave-one-out cross validation (LOOCV) risk for nonsmooth losses and regularizers. Our three frameworks are based on the primal, dual, and proximal formulations of (1). Each framework shows its strength in certain types of problems. We prove the equivalence of the three approaches under smoothness conditions. This equivalence enables us to justify the accuracy of the three methods under such conditions. We use our approaches to obtain a risk estimate for several standard problems, including generalized LASSO, nuclear norm regularization, and support vector machines. We empirically demonstrate the effectiveness of our results for non-differentiable cases.

cs.LG↗

arXiv2018

Approximate Leave-One-Out for Fast Parameter Tuning in High Dimensions

Consider the following class of learning schemes: $$\hat{\boldsymbolβ} := \arg\min_{\boldsymbolβ}\;\sum_{j=1}^n \ell(\boldsymbol{x}_j^\top\boldsymbolβ; y_j) + λR(\boldsymbolβ),\qquad\qquad (1) $$ where $\boldsymbol{x}_i \in \mathbb{R}^p$ and $y_i \in \mathbb{R}$ denote the $i^{\text{th}}$ feature and response variable respectively. Let $\ell$ and $R$ be the loss function and regularizer, $\boldsymbolβ$ denote the unknown weights, and $λ$ be a regularization parameter. Finding the optimal choice of $λ$ is a challenging problem in high-dimensional regimes where both $n$ and $p$ are large. We propose two frameworks to obtain a computationally efficient approximation ALO of the leave-one-out cross validation (LOOCV) risk for nonsmooth losses and regularizers. Our two frameworks are based on the primal and dual formulations of (1). We prove the equivalence of the two approaches under smoothness conditions. This equivalence enables us to justify the accuracy of both methods under such conditions. We use our approaches to obtain a risk estimate for several standard problems, including generalized LASSO, nuclear norm regularization, and support vector machines. We empirically demonstrate the effectiveness of our results for non-differentiable cases.

stat.ML↗

arXiv2018

Approximate Message Passing for Amplitude Based Optimization

We consider an $\ell_2$-regularized non-convex optimization problem for recovering signals from their noisy phaseless observations. We design and study the performance of a message passing algorithm that aims to solve this optimization problem. We consider the asymptotic setting $m,n \rightarrow \infty$, $m/n \rightarrow δ$ and obtain sharp performance bounds, where $m$ is the number of measurements and $n$ is the signal dimension. We show that for complex signals the algorithm can perform accurate recovery with only $m=\left ( \frac{64}{π^2}-4\right)n\approx 2.5n$ measurements. Also, we provide sharp analysis on the sensitivity of the algorithm to noise. We highlight the following facts about our message passing algorithm: (i) Adding $\ell_2$ regularization to the non-convex loss function can be beneficial even in the noiseless setting; (ii) spectral initialization has marginal impact on the performance of the algorithm.

cs.IT↗

arXiv2018

Optimization-based AMP for Phase Retrieval: The Impact of Initialization and $\ell_2$-regularization

We consider an $\ell_2$-regularized non-convex optimization problem for recovering signals from their noisy phaseless observations. We design and study the performance of a message passing algorithm that aims to solve this optimization problem. We consider the asymptotic setting $m,n \rightarrow \infty$, $m/n \rightarrow δ$ and obtain sharp performance bounds, where $m$ is the number of measurements and $n$ is the signal dimension. We show that for complex signals the algorithm can perform accurate recovery with only $m= \left(\frac{64}{π^2}-4\right)n \approx 2.5n$ measurements. Also, we provide sharp analysis on the sensitivity of the algorithm to noise. We highlight the following facts about our message passing algorithm: (i) Adding $\ell_2$ regularization to the non-convex loss function can be beneficial. (ii) Spectral initialization has marginal impact on the performance of the algorithm. The sharp analyses in this paper, not only enable us to compare the performance of our method with other phase recovery schemes, but also shed light on designing better iterative algorithms for other non-convex optimization problems.

cs.IT↗

arXiv2018

Low noise sensitivity analysis of Lq-minimization in oversampled systems

The class of Lq-regularized least squares (LQLS) are considered for estimating a p-dimensional vector \b{eta} from its n noisy linear observations y = X\b{eta}+w. The performance of these schemes are studied under the high-dimensional asymptotic setting in which p grows linearly with n. In this asymptotic setting, phase transition diagrams (PT) are often used for comparing the performance of different estimators. Although phase transition analysis is shown to provide useful information for compressed sensing, the fact that it ignores the measurement noise not only limits its applicability in many application areas, but also may lead to misunderstandings. For instance, consider a linear regression problem in which n > p and the signal is not exactly sparse. If the measurement noise is ignored in such systems, regularization techniques, such as LQLS, seem to be irrelevant since even the ordinary least squares (OLS) returns the exact solution. However, it is well-known that if n is not much larger than p then the regularization techniques improve the performance of OLS. In response to this limitation of PT analysis, we consider the low-noise sensitivity analysis. We show that this analysis framework (i) reveals the advantage of LQLS over OLS, (ii) captures the difference between different LQLS estimators even when n > p, and (iii) provides a fair comparison among different estimators in high signal-to-noise ratios. As an application of this framework, we will show that under mild conditions LASSO outperforms other LQLS even when the signal is dense. Finally, by a simple transformation we connect our low-noise sensitivity framework to the classical asymptotic regime in which n/p goes to infinity and characterize how and when regularization techniques offer improvements over ordinary least squares, and which regularizer gives the most improvement when the sample size is large.

math.ST↗

arXiv2017

VLSI Design of a Nonparametric Equalizer for Massive MU-MIMO

Linear minimum mean-square error (L-MMSE) equalization is among the most popular methods for data detection in massive multi-user multiple-input multiple-output (MU-MIMO) wireless systems. While L-MMSE equalization enables near-optimal spectral efficiency, accurate knowledge of the signal and noise powers is necessary. Furthermore, corresponding VLSI designs must solve linear systems of equations, which requires high arithmetic precision, exhibits stringent data dependencies, and results in high circuit complexity. This paper proposes the first VLSI design of the NOnParametric Equalizer (NOPE), which avoids knowledge of the transmit signal and noise powers, provably delivers the performance of L-MMSE equalization for massive MU-MIMO systems, and is resilient to numerous system and hardware impairments due to its parameter-free nature. Moreover, NOPE avoids computation of a matrix inverse and only requires hardware-friendly matrix-vector multiplications. To showcase the practical advantages of NOPE, we propose a parallel VLSI architecture and provide synthesis results in 28nm CMOS. We demonstrate that NOPE performs on par with existing data detectors for massive MU-MIMO that require accurate knowledge of the signal and noise powers.

cs.IT↗

arXiv2017

Overcoming The Limitations of Phase Transition by Higher Order Analysis of Regularization Techniques

We study the problem of estimating $β\in \mathbb{R}^p$ from its noisy linear observations $y= Xβ+ w$, where $w \sim N(0, σ_w^2 I_{n\times n})$, under the following high-dimensional asymptotic regime: given a fixed number $δ$, $p \rightarrow \infty$, while $n/p \rightarrow δ$. We consider the popular class of $\ell_q$-regularized least squares (LQLS) estimators, a.k.a. bridge, given by the optimization problem: \begin{equation*} \hatβ (λ, q ) \in \arg\min_β\frac{1}{2} \|y-Xβ\|_2^2+ λ\|β\|_q^q, \end{equation*} and characterize the almost sure limit of $\frac{1}{p} \|\hatβ (λ, q )- β\|_2^2$. The expression we derive for this limit does not have explicit forms and hence are not useful in comparing different algorithms, or providing information in evaluating the effect of $δ$ or sparsity level of $β$. To simplify the expressions, researchers have considered the ideal "no-noise" regime and have characterized the values of $δ$ for which the almost sure limit is zero. This is known as the phase transition analysis. In this paper, we first perform the phase transition analysis of LQLS. Our results reveal some of the limitations and misleading features of the phase transition analysis. To overcome these limitations, we propose the study of these algorithms under the low noise regime. Our new analysis framework not only sheds light on the results of the phase transition analysis, but also makes an accurate comparison of different regularizers possible.

math.ST↗

arXiv2017

Optimally-Tuned Nonparametric Linear Equalization for Massive MU-MIMO Systems

This paper deals with linear equalization in massive multi-user multiple-input multiple-output (MU-MIMO) wireless systems. We first provide simple conditions on the antenna configuration for which the well-known linear minimum mean-square error (L-MMSE) equalizer provides near-optimal spectral efficiency, and we analyze its performance in the presence of parameter mismatches in the signal and/or noise powers. We then propose a novel, optimally-tuned NOnParametric Equalizer (NOPE) for massive MU-MIMO systems, which avoids knowledge of the transmit signal and noise powers altogether. We show that NOPE achieves the same performance as that of the L-MMSE equalizer in the large-antenna limit, and we demonstrate its efficacy in realistic, finite-dimensional systems. From a practical perspective, NOPE is computationally efficient and avoids dedicated training that is typically required for parameter estimation

cs.IT↗

arXiv2017

New approach to Bayesian high-dimensional linear regression

Consider the problem of estimating parameters $X^n \in \mathbb{R}^n $, generated by a stationary process, from $m$ response variables $Y^m = AX^n+Z^m$, under the assumption that the distribution of $X^n$ is known. This is the most general version of the Bayesian linear regression problem. The lack of computationally feasible algorithms that can employ generic prior distributions and provide a good estimate of $X^n$ has limited the set of distributions researchers use to model the data. In this paper, a new scheme called Q-MAP is proposed. The new method has the following properties: (i) It has similarities to the popular MAP estimation under the noiseless setting. (ii) In the noiseless setting, it achieves the "asymptotically optimal performance" when $X^n$ has independent and identically distributed components. (iii) It scales favorably with the dimensions of the problem and therefore is applicable to high-dimensional setups. (iv) The solution of the Q-MAP optimization can be found via a proposed iterative algorithm which is provably robust to the error (noise) in the response variables.

cs.IT↗

arXiv2017

An efficient algorithm for compression-based compressed sensing

Modern image and video compression codes employ elaborate structures existing in such signals to encode them into few number of bits. Compressed sensing recovery algorithms on the other hand use such signals' structures to recover them from few linear observations. Despite the steady progress in the field of compressed sensing, structures that are often used for signal recovery are still much simpler than those employed by state-of-the-art compression codes. The main goal of this paper is to bridge this gap through answering the following question: Can one employ a given compression code to build an efficient (polynomial time) compressed sensing recovery algorithm? In response to this question, the compression-based gradient descent (C-GD) algorithm is proposed. C-GD, which is a low-complexity iterative algorithm, is able to employ a generic compression code for compressed sensing and therefore elevates the scope of structures used in compressed sensing to those used by compression codes. The convergence performance of C-GD and its required number of measurements in terms of the rate-distortion performance of the compression code are theoretically analyzed. It is also shown that C-GD is robust to additive white Gaussian noise. Finally, the presented simulation results show that combining C-GD with commercial image compression codes such as JPEG2000 yields state-of-the-art performance in imaging applications.

cs.IT↗

arXiv2017

On the Gaussianity of Kolmogorov Complexity of Mixing Sequences

Let $ K(X_1, \ldots, X_n)$ and $H(X_n | X_{n-1}, \ldots, X_1)$ denote the Kolmogorov complexity and Shannon's entropy rate of a stationary and ergodic process $\{X_i\}_{i=-\infty}^\infty$. It has been proved that \[ \frac{K(X_1, \ldots, X_n)}{n} - H(X_n | X_{n-1}, \ldots, X_1) \rightarrow 0, \] almost surely. This paper studies the convergence rate of this asymptotic result. In particular, we show that if the process satisfies certain mixing conditions, then there exists $σ<\infty$ such that $$\sqrt{n}\left(\frac{K(X_{1:n})}{n}- H(X_0|X_1,\dots,X_{-\infty})\right) \rightarrow_d N(0,σ^2).$$ Furthermore, we show that under slightly stronger mixing conditions one may obtain non-asymptotic concentration bounds for the Kolmogorov complexity.

cs.IT↗

arXiv2016

Global analysis of Expectation Maximization for mixtures of two Gaussians

Expectation Maximization (EM) is among the most popular algorithms for estimating parameters of statistical models. However, EM, which is an iterative algorithm based on the maximum likelihood principle, is generally only guaranteed to find stationary points of the likelihood objective, and these points may be far from any maximizer. This article addresses this disconnect between the statistical principles behind EM and its algorithmic properties. Specifically, it provides a global analysis of EM for specific models in which the observations comprise an i.i.d. sample from a mixture of two Gaussians. This is achieved by (i) studying the sequence of parameters from idealized execution of EM in the infinite sample limit, and fully characterizing the limit points of the sequence in terms of the initial parameters; and then (ii) based on this convergence analysis, establishing statistical consistency (or lack thereof) for the actual sequence of parameters produced by EM.

math.ST↗

arXiv2016

On the Performance of Mismatched Data Detection in Large MIMO Systems

We investigate the performance of mismatched data detection in large multiple-input multiple-output (MIMO) systems, where the prior distribution of the transmit signal used in the data detector differs from the true prior. To minimize the performance loss caused by this prior mismatch, we include a tuning stage into our recently-proposed large MIMO approximate message passing (LAMA) algorithm, which allows us to develop mismatched LAMA algorithms with optimal as well as sub-optimal tuning. We show that carefully-selected priors often enable simpler and computationally more efficient algorithms compared to LAMA with the true prior while achieving near-optimal performance. A performance analysis of our algorithms for a Gaussian prior and a uniform prior within a hypercube covering the QAM constellation recovers classical and recent results on linear and non-linear MIMO data detection, respectively.

cs.IT↗

arXiv2016

Does $\ell_p$-minimization outperform $\ell_1$-minimization?

In many application areas we are faced with the following question: Can we recover a sparse vector $x_o \in \mathbb{R}^N$ from its undersampled set of noisy observations $y \in \mathbb{R}^n$, $y=A x_o+w$. The last decade has witnessed a surge of algorithms and theoretical results addressing this question. One of the most popular algorithms is the $\ell_p$-regularized least squares (LPLS) given by the following formulation: \[ \hat{x}(γ,p )\in \arg\min_x \frac{1}{2}\|y - Ax\|_2^2+γ\|x\|_p^p, \] where $p \in [0,1]$. Despite the non-convexity of these problems for $p<1$, they are still appealing because of the following folklores in compressed sensing: (i) $\hat{x}(γ,p )$ is closer to $x_o$ than $\hat{x}(γ,1)$. (ii) If we employ iterative methods that aim to converge to a local minima of LPLS, then under good initialization these algorithms converge to a solution that is closer to $x_o$ than $\hat{x}(γ,1)$. In spite of the existence of plenty of empirical results that support these folklore theorems, the theoretical progress to establish them has been very limited. This paper aims to study the above folklore theorems and establish their scope of validity. Starting with approximate message passing algorithm as a heuristic method for solving LPLS, we study the impact of initialization on the performance of AMP. Then, we employ the replica analysis to show the connection between the solution of AMP and $\hat{x}(γ, p)$ in the asymptotic settings. This enables us to compare the accuracy of $\hat{x}(γ,p)$ for $p \in [0,1]$. In particular, we will characterize the phase transition and noise sensitivity of LPLS for every $0\leq p\leq 1$ accurately. Our results in the noiseless setting confirm that LPLS exhibits the same phase transition for every $0\leq p <1$ and this phase transition is much higher than that of LASSO.

cs.IT↗

arXiv2016

From Denoising to Compressed Sensing

A denoising algorithm seeks to remove noise, errors, or perturbations from a signal. Extensive research has been devoted to this arena over the last several decades, and as a result, today's denoisers can effectively remove large amounts of additive white Gaussian noise. A compressed sensing (CS) reconstruction algorithm seeks to recover a structured signal acquired using a small number of randomized measurements. Typical CS reconstruction algorithms can be cast as iteratively estimating a signal from a perturbed observation. This paper answers a natural question: How can one effectively employ a generic denoiser in a CS reconstruction algorithm? In response, we develop an extension of the approximate message passing (AMP) framework, called Denoising-based AMP (D-AMP), that can integrate a wide class of denoisers within its iterations. We demonstrate that, when used with a high performance denoiser for natural images, D-AMP offers state-of-the-art CS recovery performance while operating tens of times faster than competing methods. We explain the exceptional performance of D-AMP by analyzing some of its theoretical features. A key element in D-AMP is the use of an appropriate Onsager correction term in its iterations, which coerces the signal perturbation at each iteration to be very close to the white Gaussian noise that denoisers are typically designed to remove.

cs.IT↗

arXiv2015

Consistent Parameter Estimation for LASSO and Approximate Message Passing

We consider the problem of recovering a vector $β_o \in \mathbb{R}^p$ from $n$ random and noisy linear observations $y= Xβ_o + w$, where $X$ is the measurement matrix and $w$ is noise. The LASSO estimate is given by the solution to the optimization problem $\hatβ_λ = \arg \min_β \frac{1}{2} \|y-Xβ\|_2^2 + λ\| β\|_1$. Among the iterative algorithms that have been proposed for solving this optimization problem, approximate message passing (AMP) has attracted attention for its fast convergence. Despite significant progress in the theoretical analysis of the estimates of LASSO and AMP, little is known about their behavior as a function of the regularization parameter $λ$, or the thereshold parameters $τ^t$. For instance the following basic questions have not yet been studied in the literature: (i) How does the size of the active set $\|\hatβ^λ\|_0/p$ behave as a function of $λ$? (ii) How does the mean square error $\|\hatβ_λ - β_o\|_2^2/p$ behave as a function of $λ$? (iii) How does $\|β^t - β_o \|_2^2/p$ behave as a function of $τ^1, \ldots, τ^{t-1}$? Answering these questions will help in addressing practical challenges regarding the optimal tuning of $λ$ or $τ^1, τ^2, \ldots$. This paper answers these questions in the asymptotic setting and shows how these results can be employed in deriving simple and theoretically optimal approaches for tuning the parameters $τ^1, \ldots, τ^t$ for AMP or $λ$ for LASSO. It also explores the connection between the optimal tuning of the parameters of AMP and the optimal tuning of LASSO.

math.ST↗