A. Salman Avestimehr

Weighted ℓ 1 minimization for sparse recovery with prior information

In this paper we study the compressed sensing problem of recovering a sparse signal from a system of underdetermined linear equations when we have prior information about the probability of each entry of the unknown signal being nonzero. In particular, we focus on a model where the entries of the unknown vector fall into two sets, each with a different probability of being nonzero. We propose a weighted ℓ1 minimization recovery algorithm and analyze its performance using a Grassman angle approach. We compute explicitly the relationship between the system parameters (the weights, the number of measurements, the size of the two sets, the probabilities of being non-zero) so that an iid random Gaussian measurement matrix along with weighted ℓ1 minimization recovers almost all such sparse signals with overwhelming probability as the problem dimension increases. This allows us to compute the optimal weights. We also provide simulations to demonstrate the advantages of the method over conventional ℓ1 optimization.

Approximate capacity region of the two-pair bidirectional Gaussian relay network

We study the capacity of the Gaussian two-pair fullduplex directional (or two-way) relay network with a single-relay supporting the communication of the pairs. This network is a generalization of the well known bidirectional relay channel, where we have only one pair of users. We propose a novel transmission technique which is based on a specific superposition of lattice codes and random Gaussian codes at the source nodes. The relay attempts to decode the Gaussian codewords and the superposition of the lattice codewords of each pair. Then it forwards this information to all users. We analyze the achievable rate of this scheme and show that for all channel gains it achieves to within 2 bits/sec/Hz per user of the cut-set upper bound on the capacity region of the two-pair bidirectional relay network.

Capacity region of the deterministic multi-pair bi-directional relay network

In this paper we study the capacity region of the multi-pair bidirectional (or two-way) wireless relay network, in which a relay node facilitates the communication between multiple pairs of users. This network is a generalization of the well known bidirectional relay channel, where we have only one pair of users. We examine this problem in the context of the deterministic channel interaction model, which eliminates the channel noise and allows us to focus on the interaction between signals. We characterize the capacity region of this network when the relay is operating at either full-duplex mode or half-duplex mode (with non adaptive listen-transmit scheduling). In both cases we show that the cut-set upper bound is tight and, quite interestingly, the capacity region is achieved by a simple equation-forwarding strategy.

Improved sparse recovery thresholds with two-step reweighted ℓ 1 minimization

It is well known that ℓ<inf>1</inf> minimization can be used to recover sufficiently sparse unknown signals from compressed linear measurements. In fact, exact thresholds on the sparsity, as a function of the ratio between the system dimensions, so that with high probability almost all sparse signals can be recovered from iid Gaussian measurements, have been computed and are referred to as “weak thresholds” [4]. In this paper, we introduce a reweighted ℓ<inf>1</inf> recovery algorithm composed of two steps: a standard ℓ<inf>1</inf> minimization step to identify a set of entries where the signal is likely to reside, and a weighted ℓ<inf>1</inf> minimization step where entries outside this set are penalized. For signals where the non-sparse component has iid Gaussian entries, we prove a “strict” improvement in the weak recovery threshold. Simulations suggest that the improvement can be quite impressive—over 20% in the example we consider.

Breaking through the thresholds: an analysis for iterative reweighted ℓ 1 minimization via the Grassmann angle framework

It is now well understood that the ℓ<inf>1</inf> minimization algorithm is able to recover sparse signals from incomplete measurements [2], [1], [3] and sharp recoverable sparsity thresholds have also been obtained for the ℓ<inf>1</inf> minimization algorithm. However, even though iterative reweighted ℓ<inf>1</inf> minimization algorithms or related algorithms have been empirically observed to boost the recoverable sparsity thresholds for certain types of signals, no rigorous theoretical results have been established to prove this fact. In this paper, we try to provide a theoretical foundation for analyzing the iterative reweighted ℓ<inf>1</inf> algorithms. In particular, we show that for a nontrivial class of signals, the iterative reweighted ℓ<inf>1</inf> minimization can indeed deliver recoverable sparsity thresholds larger than that given in [1], [3]. Our results are based on a high-dimensional geometrical analysis (Grassmann angle analysis) of the null-space characterization for ℓ<inf>1</inf> minimization and weighted ℓ<inf>1</inf> minimization algorithms.

Fundamental tradeoff between computation and communication in distributed computing

We introduce a general distributed computing framework, motivated by commonly used structures like MapReduce, and formulate an information-theoretic tradeoff between computation and communication in such a framework. We characterize the optimal tradeoff to within a constant factor, for all system parameters. In particular, we propose a coded scheme, namely “Coded MapReduce” (CMR), which creates and exploits coding opportunities in data shuffling for distributed computing, reducing the communication load by a factor that is linearly proportional to the computation load. We then prove a lower bound on the minimum communication load, and demonstrate that CMR achieves this lower bound to within a constant factor. This result reveals a fundamental connection between computation and communication in distributed computing - the two are inverse-linearly proportional to each other.

The exact rate-memory tradeoff for caching with uncoded prefetching

We consider a cache network, where a single server is connected to multiple users via a shared bottleneck link. The server has a set of files, which can be cached by each user in a prefetching phase. In a following delivery phase, each user requests a file and the server delivers user demands as efficiently as possible by taking into account their cache contents. We focus on an important and commonly used class of prefetching schemes, where the caches are filled with uncoded data. We provide the exact characterization of rate-memory tradeoff for this problem, by deriving both the minimum average rate (for a uniform file popularity) and the minimum peak rate required on the bottleneck link for a given cache size available at each user. We propose a novel caching scheme, which strictly improves the state of the art by exploiting commonality among user demands. We then demonstrate the exact optimality of our proposed scheme through a matching converse, by dividing the set of all demands into types, and showing that the placement phase in the proposed caching scheme is universally optimal for all types. Using these techniques, we can also fully characterize the rate-memory tradeoff for a decentralized setting, in which users fill out their cache content without coordination.

Degrees of freedom of two-hop wireless networks: “Everyone gets the entire cake”

We show that fully connected two-hop wireless networks with K sources, K relays and K destinations have K degrees of freedom for almost all values of constant channel coefficients. Our main contribution is a new interference-alignment-based achievability scheme which we call aligned network diagonalization. This scheme allows the data streams transmitted by the sources to undergo a diagonal linear transformation from the sources to the destinations, thus being received free of interference by their intended destination.

Approximate Sum-Capacity of the Y-Channel

A network where three users want to establish multiple unicasts between each other via a relay is considered. This network is called the Y-channel and resembles an elemental ingredient of future wireless networks. The sum-capacity of this network is studied. A characterization of the sum-capacity within an additive gap of 2 bits, and a multiplicative gap of 4, for all values of channel gains and transmit powers is obtained. Contrary to similar setups where the cut-set bounds can be achieved within a constant gap, they cannot be achieved in our case, where they are dominated by our new genie-aided bounds. Furthermore, it is shown that a time-sharing strategy, in which at each time two users exchange information using coding strategies of the bidirectional relay channel, achieves the upper bounds to within a constant gap. This result is further extended to the K-user case, where it is shown that the same scheme achieves the sum-capacity within 2log(K-1) bits.

Coding for Distributed Fog Computing

Redundancy is abundant in fog networks (i.e., many computing and storage points) and grows linearly with network size. We demonstrate the transformational role of coding in fog computing for leveraging such redundancy to substantially reduce the bandwidth consumption and latency of computing. In particular, we discuss two recently proposed coding concepts, minimum bandwidth codes and minimum latency codes, and illustrate their impacts on fog computing. We also review a unified coding framework that includes the above two coding techniques as special cases, and enables a trade-off between computation latency and communication load to optimize system performance. At the end, we will discuss several open problems and future research directions.