Adel Javanmard

Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing

We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by Krzakala [30], message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of nonzero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate δ exceeds the (upper) Rényi information dimension of the signal, d̅(pX). More precisely, for a sequence of signals of diverging dimension n whose empirical distribution converges to pX, reconstruction is with high probability successful from d̅(pX) n+o(n) measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension n and k(n) nonzero entries, this implies reconstruction from k(n)+o(n) measurements. For “discrete” signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from o(n) measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal pX.

Mobility Modeling, Spatial Traffic Distribution, and Probability of Connectivity for Sparse and Dense Vehicular Ad Hoc Networks

The mobility pattern of users is one of the distinct features of vehicular ad hoc networks (VANETs) compared with other types of mobile ad hoc networks (MANETs). This is due to the higher speed and the roadmap-restricted movement of vehicles. In this paper, we propose a new analytical mobility model for VANETs based on product-form queueing networks. In this model, we map the topology of the streets and the behavior of vehicles at both intersections and different parts of the streets onto different parameters of a BCMP open queueing network comprising M/G/infin nodes. This model represents a sparse situation for VANETs. To include the effect of dense situation on the mobility model, we modify the proposed queueing network as a new one comprising nodes with state-dependent service rates, i.e., M/G(n)/infin nodes. With respect to the product-form solution property of the proposed queueing networks, we are able to find the spatial traffic distribution for vehicles at both sparse and dense situations. Furthermore, we are able to modify the proposed queueing network to find the lower and upper bounds for the probability of connectivity. In the last part of this paper, we show the flexibility of the proposed model by several numerical examples and confirm our modeling approach by simulation.

Hypothesis Testing in High-Dimensional Regression Under the Gaussian Random Design Model: Asymptotic Theory

We consider linear regression in the high-dimensional regime where the number of observations n is smaller than the number of parameters p. A very successful approach in this setting uses 11-penalized least squares (also known as the Lasso) to search for a subset of s0 <; n parameters that best explain the data, while setting the other parameters to zero. Considerable amount of work has been devoted to characterizing the estimation and model selection problems within this approach. In this paper, we consider instead the fundamental, but far less understood, question of statistical significance. More precisely, we address the problem of computing p-values for single regression coefficients. On one hand, we develop a general upper bound on the minimax power of tests with a given significance level. We show that rigorous guarantees for earlier methods do not allow to achieve this bound, except in special cases. On the other, we prove that this upper bound is (nearly) achievable through a practical procedure in the case of random design matrices with independent entries. Our approach is based on a debiasing of the Lasso estimator. The analysis builds on a rigorous characterization of the asymptotic distribution of the Lasso estimator and its debiased version. Our result holds for optimal sample size, i.e., when n is at least on the order of s0 log(p/s0). We generalize our approach to random design matrices with independent identically distributed Gaussian rows xi ~ N(0, Σ). In this case, we prove that a similar distributional characterization (termed standard distributional limit) holds for n much larger than s0(log p)2.Our analysis assumes Σ is known. To cope with unknown Σ, we suggest a plug-in estimator for sparse covariances Σ and validate the method through numerical simulations. Finally, we show that for optimal sample size, n being at least of order s0 log(p/s0), the standard distributional limit for general Gaussian designs can be derived from the replica heuristics in statistical physics. This derivation suggests a stronger conjecture than the result we prove, and near-optimality of the statistical power for a large class of Gaussian designs.

Localization from Incomplete Noisy Distance Measurements

We consider the problem of positioning a cloud of points in the Euclidean space ℝd, using noisy measurements of a subset of pairwise distances. This task has applications in various areas, such as sensor network localization and reconstruction of protein conformations from NMR measurements. It is also closely related to dimensionality reduction problems and manifold learning, where the goal is to learn the underlying global geometry of a data set using local (or partial) metric information. Here we propose a reconstruction algorithm based on semidefinite programming. For a random geometric graph model and uniformly bounded noise, we provide a precise characterization of the algorithm’s performance: in the noiseless case, we find a radius r0 beyond which the algorithm reconstructs the exact positions (up to rigid transformations). In the presence of noise, we obtain upper and lower bounds on the reconstruction error that match up to a factor that depends only on the dimension d, and the average degree of the nodes in the graph.

Phase Transitions in Semidefinite Relaxations

Significance Modern data analysis requires solving hard optimization problems with a large number of parameters and a large number of constraints. A successful approach is to replace these hard problems by surrogate problems that are convex and hence tractable. Semidefinite programming relaxations offer a powerful method to construct such relaxations. In many instances it was observed that a semidefinite relaxation becomes very accurate when the noise level in the data decreases below a certain threshold. We develop a new method to compute these noise thresholds (or phase transitions) using ideas from statistical physics. Statistical inference problems arising within signal processing, data mining, and machine learning naturally give rise to hard combinatorial optimization problems. These problems become intractable when the dimensionality of the data is large, as is often the case for modern datasets. A popular idea is to construct convex relaxations of these combinatorial problems, which can be solved efficiently for large-scale datasets. Semidefinite programming (SDP) relaxations are among the most powerful methods in this family and are surprisingly well suited for a broad range of problems where data take the form of matrices or graphs. It has been observed several times that when the statistical noise is small enough, SDP relaxations correctly detect the underlying combinatorial structures. In this paper we develop asymptotic predictions for several detection thresholds, as well as for the estimation error above these thresholds. We study some classical SDP relaxations for statistical problems motivated by graph synchronization and community detection in networks. We map these optimization problems to statistical mechanics models with vector spins and use nonrigorous techniques from statistical mechanics to characterize the corresponding phase transitions. Our results clarify the effectiveness of SDP relaxations in solving high-dimensional statistical problems.

Debiasing the lasso: Optimal sample size for Gaussian designs

Performing statistical inference in high-dimension is an outstanding challenge. A major source of difficulty is the absence of precise information on the distribution of high-dimensional estimators. Here, we consider linear regression in the high-dimensional regime

Subsampling at information theoretically optimal rates

p\gg n

Nearly optimal sample size in hypothesis testing for high-dimensional regression

. In this context, we would like to perform inference on a high-dimensional parameters vector

1-bit matrix completion under exact low-rank constraint

\theta^*\in{\mathbb R}^p

Localization from incomplete noisy distance measurements

. Important progress has been achieved in computing confidence intervals for single coordinates