Ilan Shomorony

Two-Unicast Wireless Networks: Characterizing the Degrees of Freedom

We consider two-source two-destination (i.e., two-unicast) multihop wireless networks that have a layered structure with arbitrary connectivity. We show that, if the channel gains are chosen independently according to continuous distributions, then, with probability 1, two-unicast layered Gaussian networks can only have 1, 3/2, or 2 sum degrees of freedom (unless both source-destination pairs are disconnected, in which case no degrees of freedom can be achieved). We provide sufficient and necessary conditions for each case based on network connectivity and a new notion of source-destination paths with manageable interference. Our achievability scheme is based on forwarding the received signals at all nodes, except for a small fraction of them in at most two key layers. Hence, we effectively create a “condensed network” that has at most four layers (including the sources layer and the destinations layer). We design the transmission strategies based on the structure of this condensed network. The converse results are obtained by developing information-theoretic inequalities that capture the structures of the network connectivity. Finally, we extend this result and characterize the full degrees of freedom region of two-unicast layered wireless networks.

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 both in the case of time-varying channel coefficients and constant channel coefficients (in which case the result holds for almost all values of constant channel coefficients). Our main contribution is a new 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. In addition, we extend our scheme to multihop networks with fully connected hops, and multihop networks with MIMO nodes, for which the degrees of freedom are also fully characterized.

Worst-Case Additive Noise in Wireless Networks

A classical result in information theory states that the Gaussian noise is the worst-case additive noise in point-to-point channels, meaning that, for a fixed noise variance, the Gaussian noise minimizes the capacity of an additive noise channel. In this paper, we significantly generalize this result and show that the Gaussian noise is also the worst-case additive noise in wireless networks with additive noises that are independent from the transmit signals. More specifically, we show that if we fix the noise variance at each node, then the capacity region with Gaussian noises is a subset of the capacity region with any other set of noise distributions. We prove this result by showing that a coding scheme that achieves a given set of rates on a network with Gaussian additive noises can be used to construct a coding scheme that achieves the same set of rates on a network that has the same topology and traffic demands, but with non-Gaussian additive noises.

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.

Is Gaussian noise the worst-case additive noise in wireless networks?

An important classical result in Information Theory states that the Gaussian noise is the worst-case additive noise in point-to-point channels. In this paper, we significantly generalize this result and show that, under very mild assumptions, Gaussian noise is also the worst-case additive noise in general wireless networks with additive noises that are independent from the transmit signals. More specifically, we prove that, given a coding scheme with finite reading precision for an AWGN network, one can build a coding scheme that achieves the same rates on an additive noise wireless network with the same topology, where the noise terms may have any distribution with same mean and variance as in the AWGN network.

A generalized cut-set bound for deterministic multi-flow networks and its applications

We present a new outer bound for the sum capacity of general multi-unicast deterministic networks. Intuitively, this bound can be understood as applying the cut-set bound to concatenated copies of the original network with a special restriction on the allowed transmit signal distributions. We first study applications to finite-field networks, where we obtain a general outer-bound expression in terms of ranks of the transfer matrices. We then show that, even though our outer bound is for deterministic networks, a result from [1] relating the capacity of AWGN K×K×K networks and the capacity of a deterministic counterpart allows us to establish an outer bound to the DoF of K×K×K wireless networks with general connectivity. This bound is tight in the case of the “adjacent-cell interference” topology, and yields graph-theoretic necessary and sufficient conditions for K DoF to be achievable in general topologies.

On the role of deterministic models in K × K × K wireless networks

This paper establishes a connection between the capacity region of the K × K × K wireless network under the AWGN channel model and under a truncated deterministic channel model, which allows any outer bound on the capacity region of the truncated network to be translated into an outer bound on the capacity region of the AWGN network. The result is obtained through the utilization of a recent worst-case noise theorem [1], which shows that perturbing the noise distribution in AWGN networks only increases the capacity region.

Sum degrees-of-freedom of two-unicast wireless networks

We consider two-source two-destination (i.e., two-unicast) multi-hop wireless networks that have a layered structure with arbitrary connectivity. We show that, if the channel gains are independently drawn from continuous distributions, then, with probability 1, two-unicast layered Gaussian networks can only have 1, 3/2 or 2 sum degrees-of-freedom1. We provide necessary and sufficient conditions for each case based on the network topology and a new notion of source-destination paths with manageable interference.

Do read errors matter for genome assembly

While most current high-throughput DNA sequencing technologies generate short reads with low error rates, emerging sequencing technologies generate long reads with high error rates. A basic question of interest is the tradeoff between read length and error rate in terms of the information needed for the perfect assembly of the genome. Using an adversarial erasure error model, we make progress on this problem by establishing a critical read length, as a function of the genome and the error rate, above which perfect assembly is guaranteed. For several real genomes, including those from the GAGE dataset, we verify that this critical read length is not significantly greater than the read length required for perfect assembly from reads without errors.

Network compression: Worst-case analysis

We consider the problem of communicating a distributed correlated memoryless source over a memoryless network, from source nodes to destination nodes, under quadratic distortion constraints. We show the following two complementary results: (a) for an arbitrary memoryless network, among all distributed memoryless sources with a particular correlation, Gaussian sources are the worst compressible, that is, they admit the smallest set of achievable distortion tuples, and (b) for any arbitrarily distributed memoryless source to be communicated over a memoryless additive noise network, among all noise processes with a fixed correlation, Gaussian noise admits the smallest achievable set of distortion tuples. In each case, given a coding scheme for the corresponding Gaussian problem, we provide a technique for the construction of a new coding scheme that achieves the same distortion at the destination nodes in a non-Gaussian scenario with the same correlation structure.