arXiv subjects

Sundeep Rangan

Publications and source records attributed to Sundeep Rangan.

151 records · Page 9Linked to original sources

arXiv2011

Orthogonal Matching Pursuit: A Brownian Motion Analysis

A well-known analysis of Tropp and Gilbert shows that orthogonal matching pursuit (OMP) can recover a k-sparse n-dimensional real vector from 4 k log(n) noise-free linear measurements obtained through a random Gaussian measurement matrix with a probability that approaches one as n approaches infinity. This work strengthens this result by showing that a lower number of measurements, 2 k log(n - k), is in fact sufficient for asymptotic recovery. More generally, when the sparsity level satisfies kmin <= k <= kmax but is unknown, 2 kmax log(n - kmin) measurements is sufficient. Furthermore, this number of measurements is also sufficient for detection of the sparsity pattern (support) of the vector with measurement errors provided the signal-to-noise ratio (SNR) scales to infinity. The scaling 2 k log(n - k) exactly matches the number of measurements required by the more complex lasso method for signal recovery with a similar SNR scaling.

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arXiv2011

Optimal Quantization for Compressive Sensing under Message Passing Reconstruction

We consider the optimal quantization of compressive sensing measurements following the work on generalization of relaxed belief propagation (BP) for arbitrary measurement channels. Relaxed BP is an iterative reconstruction scheme inspired by message passing algorithms on bipartite graphs. Its asymptotic error performance can be accurately predicted and tracked through the state evolution formalism. We utilize these results to design mean-square optimal scalar quantizers for relaxed BP signal reconstruction and empirically demonstrate the superior error performance of the resulting quantizers.

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arXiv2010

Belief Propagation Methods for Intercell Interference Coordination

We consider a broad class of interference coordination and resource allocation problems for wireless links where the goal is to maximize the sum of functions of individual link rates. Such problems arise in the context of, for example, fractional frequency reuse (FFR) for macro-cellular networks and dynamic interference management in femtocells. The resulting optimization problems are typically hard to solve optimally even using centralized algorithms but are an essential computational step in implementing rate-fair and queue stabilizing scheduling policies in wireless networks. We consider a belief propagation framework to solve such problems approximately. In particular, we construct approximations to the belief propagation iterations to obtain computationally simple and distributed algorithms with low communication overhead. Notably, our methods are very general and apply to, for example, the optimization of transmit powers, transmit beamforming vectors, and sub-band allocation to maximize the above objective. Numerical results for femtocell deployments demonstrate that such algorithms compute a very good operating point in typically just a couple of iterations.

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arXiv2010

Femto-Macro Cellular Interference Control with Subband Scheduling and Interference Cancelation

A significant technical challenge in deploying femtocells is controlling the interference from the underlay of femtos onto the overlay of macros. This paper presents a novel interference control method where the macrocell bandwidth is partitioned into subbands, and the short-range femtocell links adaptively allocate their power across the subbands based on a load-spillage power control method. The scheme can improve rate distribution in the macro network while also providing opportunities for short-range communication as well. Moreover, the proposed scheme requires minimal interference coordination communication between the femtos and macros, which is one of the main challenges in femtocell systems. Also, simulations show certain advantages over simpler orthogonalization schemes or power control schemes without subband partitioning. Further modest gains may also be possible with interference cancelation.

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arXiv2010

Estimation with Random Linear Mixing, Belief Propagation and Compressed Sensing

We apply Guo and Wang's relaxed belief propagation (BP) method to the estimation of a random vector from linear measurements followed by a componentwise probabilistic measurement channel. Relaxed BP uses a Gaussian approximation in standard BP to obtain significant computational savings for dense measurement matrices. The main contribution of this paper is to extend the relaxed BP method and analysis to general (non-AWGN) output channels. Specifically, we present detailed equations for implementing relaxed BP for general channels and show that relaxed BP has an identical asymptotic large sparse limit behavior as standard BP, as predicted by the Guo and Wang's state evolution (SE) equations. Applications are presented to compressed sensing and estimation with bounded noise.

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arXiv2009

On-Off Random Access Channels: A Compressed Sensing Framework

This paper considers a simple on-off random multiple access channel, where n users communicate simultaneously to a single receiver over m degrees of freedom. Each user transmits with probability lambda, where typically lambda n < m << n, and the receiver must detect which users transmitted. We show that when the codebook has i.i.d. Gaussian entries, detecting which users transmitted is mathematically equivalent to a certain sparsity detection problem considered in compressed sensing. Using recent sparsity results, we derive upper and lower bounds on the capacities of these channels. We show that common sparsity detection algorithms, such as lasso and orthogonal matching pursuit (OMP), can be used as tractable multiuser detection schemes and have significantly better performance than single-user detection. These methods do achieve some near-far resistance but--at high signal-to-noise ratios (SNRs)--may achieve capacities far below optimal maximum likelihood detection. We then present a new algorithm, called sequential OMP, that illustrates that iterative detection combined with power ordering or power shaping can significantly improve the high SNR performance. Sequential OMP is analogous to successive interference cancellation in the classic multiple access channel. Our results thereby provide insight into the roles of power control and multiuser detection on random-access signalling.

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arXiv2008

Necessary and Sufficient Conditions on Sparsity Pattern Recovery

The problem of detecting the sparsity pattern of a k-sparse vector in R^n from m random noisy measurements is of interest in many areas such as system identification, denoising, pattern recognition, and compressed sensing. This paper addresses the scaling of the number of measurements m, with signal dimension n and sparsity-level nonzeros k, for asymptotically-reliable detection. We show a necessary condition for perfect recovery at any given SNR for all algorithms, regardless of complexity, is m = Omega(k log(n-k)) measurements. Conversely, it is shown that this scaling of Omega(k log(n-k)) measurements is sufficient for a remarkably simple ``maximum correlation'' estimator. Hence this scaling is optimal and does not require more sophisticated techniques such as lasso or matching pursuit. The constants for both the necessary and sufficient conditions are precisely defined in terms of the minimum-to-average ratio of the nonzero components and the SNR. The necessary condition improves upon previous results for maximum likelihood estimation. For lasso, it also provides a necessary condition at any SNR and for low SNR improves upon previous work. The sufficient condition provides the first asymptotically-reliable detection guarantee at finite SNR.

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