Guy Bresler

The Approximate Capacity of the Many-to-One and One-to-Many Gaussian Interference Channels

Recently, Etkin, Tse, and Wang found the capacity region of the two-user Gaussian interference channel to within 1 bit/s/Hz. A natural goal is to apply this approach to the Gaussian interference channel with an arbitrary number of users. We make progress towards this goal by finding the capacity region of the many-to-one and one-to-many Gaussian interference channels to within a constant number of bits. The result makes use of a deterministic model to provide insight into the Gaussian channel. The deterministic model makes explicit the dimension of signal level. A central theme emerges: the use of lattice codes for alignment of interfering signals on the signal level.

The two-user Gaussian interference channel: a deterministic view †

SUMMARY This paper explores the two-user Gaussian interference channel through the lens of a natural deterministic channel model. The main result is that the deterministic channel uniformly approximates the Gaussian channel, the capacity regions differing by a universal constant. The problem of finding the capacity of the Gaussian channel to within a constant error is therefore reduced to that of finding the capacity of the far simpler deterministic channel. Thus, the paper provides an alternative derivation of the recent constant gap capacity characterisation of Etkin, Tse and Wang. Additionally, the deterministic model gives significant insight towards the Gaussian channel. Copyright

Feasibility of Interference Alignment for the MIMO Interference Channel

We study vector space interference alignment for the multiple-input multiple-output interference channel with no time or frequency diversity, and no symbol extensions. We prove both necessary and sufficient conditions for alignment. In particular, we characterize the feasibility of alignment for the symmetric three-user channel where all users transmit along d dimensions, all transmitters have M antennas and all receivers have N antennas, as well as feasibility of alignment for the fully symmetric (M = N) channel with an arbitrary number of users. An implication of our results is that the total degrees of freedom available in a K-user interference channel, using only spatial diversity from the multiple antennas, is at most 2. This is in sharp contrast to the K/2 degrees of freedom shown to be possible by Cadambe and Jafar with arbitrarily large time or frequency diversity. Moving beyond the question of feasibility, we additionally discuss computation of the number of solutions using Schubert calculus in cases where there are a finite number of solutions.

Reconstruction of Markov Random Fields from Samples: Some Observations and Algorithms

Markov random fields are used to model high dimensional distributions in a number of applied areas. Much recent interest has been devoted to the reconstruction of the dependency structure from independent samples from the Markov random fields. We analyze a simple algorithm for reconstructing the underlying graph defining a Markov random field on nnodes and maximum degree dgiven observations. We show that under mild non-degeneracy conditions it reconstructs the generating graph with high probability using i¾?(dlogn) samples which is optimal up to a multiplicative constant. Our results seem to be the first results for general models that guarantee that thegenerating model is reconstructed. Furthermore, we provide an explicit O(dnd+ 2logn) running time bound. In cases where the measure on the graph has correlation decay, the running time is O(n2logn) for all fixed d. In the full-length version we also discuss the effect of observing noisy samples. There we show that as long as the noise level is low, our algorithm is effective. On the other hand, we construct an example where large noise implies non-identifiability even for generic noise and interactions. Finally, we briefly show that in some cases, models with hidden nodes can also be recovered.

Geometry of the 3-user MIMO interference channel

This paper studies vector space interference alignment for the three-user MIMO interference channel with no time or frequency diversity. The main result is a characterization of the feasibility of interference alignment in the symmetric case where all transmitters have M antennas and all receivers have N antennas. If N ≥ M and all users desire d transmit dimensions, then alignment is feasible if and only if (2r + 1)d ≤ max(rN, (r + 1)M) for all nonnegative integers r. The analogous result holds with M and N switched if M ≥ N. It turns out that, just as for the 3-user parallel interference channel [1], the length of alignment paths captures the essence of the problem. In fact, for each feasible value of M and N the maximum alignment path length dictates both the converse and achievability arguments. One of the implications of our feasibility criterion is that simply counting equations and comparing to the number of variables does not predict feasibility. Instead, a more careful investigation of the geometry of the alignment problem is required. The necessary condition obtained by counting equations is implied by our new feasibility criterion.

Optimal assembly for high throughput shotgun sequencing

We present a framework for the design of optimal assembly algorithms for shotgun sequencing under the criterion of complete reconstruction. We derive a lower bound on the read length and the coverage depth required for reconstruction in terms of the repeat statistics of the genome. Building on earlier works, we design a de Brujin graph based assembly algorithm which can achieve very close to the lower bound for repeat statistics of a wide range of sequenced genomes, including the GAGE datasets. The results are based on a set of necessary and sufficient conditions on the DNA sequence and the reads for reconstruction. The conditions can be viewed as the shotgun sequencing analogue of Ukkonen-Pevzners necessary and sufficient conditions for Sequencing by Hybridization.

Feasibility of interference alignment for the MIMO interference channel: The symmetric square case

Determining the feasibility conditions for vector space interference alignment in the K-user MIMO interference channel with constant channel coefficients has attracted much recent attention yet remains unsolved. The main result of this paper is restricted to the symmetric square case where all transmitters and receivers have N antennas, and each user desires d transmit dimensions. We prove that alignment is possible if and only if the number of antennas satisfies N ≥ d(K + 1)/2. We also show a necessary condition for feasibility of alignment with arbitrary system parameters. An algebraic geometry approach is central to the results.

Information Theory of DNA Shotgun Sequencing

DNA sequencing is the basic workhorse of modern day biology and medicine. Shotgun sequencing is the dominant technique used: many randomly located short fragments called reads are extracted from the DNA sequence, and these reads are assembled to reconstruct the original sequence. A basic question is: given a sequencing technology and the statistics of the DNA sequence, what is the minimum number of reads required for reliable reconstruction? This number provides a fundamental limit to the performance of any assembly algorithm. For a simple statistical model of the DNA sequence and the read process, we show that the answer admits a critical phenomenon in the asymptotic limit of long DNA sequences: if the read length is below a threshold, reconstruction is impossible no matter how many reads are observed, and if the read length is above the threshold, having enough reads to cover the DNA sequence is sufficient to reconstruct. The threshold is computed in terms of the Renyi entropy rate of the DNA sequence. We also study the impact of noise in the read process on the performance.

3 User interference channel: Degrees of freedom as a function of channel diversity

In this paper we characterize, in the context of vector space precoding strategies, the degrees of freedom of the parallel three-user interference channel as a function of the channel diversity L. A channel diversity of L is modeled by L independently fading real-valued parallel channels. Our results also apply to the case of parallel complex-valued channels, where the channel matrices Hij ∊ Cl×l are still diagonal but have complex entries. Here L = 2l is twice the number of parallel channels, and the resulting formulas (as a function of L) for the degrees of freedom are the same as in the real-valued case.

Mixing Time of Exponential Random Graphs

A variety of random graph models have been developed in recent years to study a range of problems on networks, driven by the wide availability of data from many social, telecommunication, biochemical and other networks. A key model, extensively used in the sociology literature, is the exponential random graph model. This model seeks to incorporate in random graphs the notion of reciprocity, that is, the larger than expected number of triangles and other small subgraphs. Sampling from these distributions is crucial for parameter estimation hypothesis testing, and more generally for understanding basic features of the network model itself. In practice sampling is typically carried out using Markov chain Monte Carlo, in particular either the Glauber dynamics or the Metropolis-Hasting procedure.In this paper we characterize the high and low temperature regimes of the exponential random graph model. We establish that in the high temperature regime the mixing time of the Glauber dynamics is Theta(n2 log n), where n is the number of vertices in the graph; in contrast, we show that in the low temperature regime the mixing is exponentially slow for any local Markov chain. Our results, moreover, give a rigorous basis for criticisms made of such models. In the high temperature regime, where sampling with MCMC is possible, we show that any finite collection of edges are asymptotically independent; thus, the model does not possess the desired reciprocity property, and is not appreciably different from the Erdos-Renyi random graph.