Capacity Upper Bounds for the Relay Channel via Reverse Hypercontractivity

Consider a primitive diamond relay channel, where a source X wants to send information to a destination with the help of two relays Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , and the two relays can communicate to the destination via error-free digital links of capacities C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> respectively, while Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> are conditionally independent given X. In this paper, we develop new upper bounds on the capacity of such primitive diamond relay channels that are tighter than the cut-set bound. Our results include both the Gaussian and the discrete memoryless case and build on the information inequalities recently developed in [6]-[8] that characterize the tension between information measures in a certain Markov chain.

A Gaussian channel with additive interference that is causally known to the transmitter is called a Dirty-Tape Channel (DTC). In this paper, we consider a state-dependent dirty-tape Gaussian relay channel with orthogonal channels from the source to the relay and from the source and relay to the destination. The orthogonal channels are corrupted by two independent additive interferences causally known to both the source and relay. The lower and upper bounds of the channel capacity are established. The lower bound is obtained by employing superposition coding at the source, Partial Decode-and-Forward (PDF) relaying at the relay, and a strategy similar to that used by Shannon at the source and relay. The explicit capacity is characterised when the power of the relay is sufficiently large. Finally, several numerical examples are provided to illustrate the impact of additive interferences and the role of the relay in information transmission and in removing the interference.

A multiple-input multiple-output (MIMO) relay network is composed of source, relay, and destination nodes, each of which is equipped with multiple antennas. Network information theory has shown that the use of multiple relay nodes in source and destination (S-D) communication makes the capacity of the S-D system logarithmically increase with the number of relay nodes. In [13], we proposed a MIMO relay scheme for a relay network with single source and destination pair; each of the multiple relay nodes performs QR decomposition of backward and forward channels in conjunction with phase control (QR-P-QR). In this paper, we extend this scheme to the MIMO relay network with multiple source and destination pairs. For this aim, we use the group nulling approach to decompose the multiple S-D MIMO relay channel to parallel independent S-D MIMO relay channels, and apply the QR-P-QR scheme in each of the decomposed MIMO relay links. Numerical examples show that the proposed relay scheme offers higher capacity than the existing zero-forcing (ZF) relaying scheme

: Calculating the amount of information about a random function contained in another random function has important uses in communication theory. An expression for the mutual information for continuous time random processes has been given by Gelfand and Yaglom, Chiang, and Perez by generalizing Shannon's result in a natural way. Under a condition of absolute continuity of measures the continuous time expression has the same form as Shannon's result. For two Gaussian processes Gelfand and Yaglom express the mutual information in terms of a mean square estimation error. We generalize this result to diffusion processes and express the solution in a different form which is more naturally related to a corresponding filtering problem. We also use these results to calculate some information rates.

We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.

We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a Boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010a). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.

A fundamental tool to prove inner bounds in classical network information theory is the so-called ‘conditional joint typicality lemma’. In addition to the lemma, one often uses unions and intersections of typical sets in the inner bound arguments without so much as giving them a second thought. These arguments do not work in the quantum setting. This bottleneck shows up in the fact that so-called ‘simultaneous decoders’, as opposed to ‘successive cancellation decoders’, are known for very few channels in quantum network information theory. Another manifestation of this bottleneck is the lack of so-called ‘simultaneous smoothing’ theorems for quantum states. In this paper, we overcome the bottleneck by proving for the first time a one-shot quantum joint typicality lemma with robust union and intersection properties. To do so we develop two novel tools in quantum information theory, which may be of independent interest. The first tool is a simple geometric idea called tilting, which increases the angles between a family of subspaces in orthogonal directions. The second tool, called smoothing and augmentation, is a way of perturbing a multipartite quantum state such that the partial trace over any subset of registers does not increase the operator norm much. Our joint typicality lemma allows us to construct simultaneous quantum decoders for many multiterminal quantum channels. It provides a powerful tool to extend many results in classical network information theory to the one-shot quantum setting.

Cloud radio access networks (C-RANs) provide a novel architecture for next-generation wireless cellular systems whereby the baseband processing is migrated from the base stations (BSs) to a control unit (CU) in the ?cloud.? The BSs, which operate as radio units (RUs), are connected via fronthaul links to the managing CU. The fronthaul links carry information about the baseband signals?in the uplink from the RUs to the CU and vice versa in the downlink?in the form of quantized in-phase and quadrature (IQ) samples. Due to the large bit rate produced by the quantized IQ signals, compression prior to transmission on the fronthaul links is deemed to be of critical importance and is receiving considerable attention. This article provides a survey of the work in this area with emphasis on advanced signal processing solutions based on network information theoretic concepts. Analysis and numerical results illustrate the considerable performance gains to be expected for standard cellular models.

The Cloud-Radio Access Network (C-RAN) cellular architecture relies on the transfer of complex baseband signals to and from a central unit (CU) over digital fronthaul links to enable the virtualization of the baseband processing functionalities of distributed radio units (RUs). The standard design of digital fronthauling is based on either scalar quantization or on more sophisticated point to-point compression techniques operating on baseband signals. Motivated by network-information theoretic results, techniques for fronthaul quantization and compression that improve over point-to-point solutions by allowing for joint processing across multiple fronthaul links at the CU have been recently proposed for both the uplink and the downlink. For the downlink, a form of joint compression, known in network information theory as multivariate compression, was shown to be advantageous under a non-constructive asymptotic information-theoretic framework. In this paper, instead, the design of a practical symbol-by-symbol fronthaul quantization algorithm that implements the idea of multivariate compression is investigated for the C-RAN downlink. As compared to current standards, the proposed multivariate quantization (MQ) only requires changes in the CU processing while no modification is needed at the RUs. The algorithm is extended to enable the joint optimization of downlink precoding and quantization, reduced-complexity MQ via successive block quantization, and variable-length compression. Numerical results, which include performance evaluations over standard cellular models, demonstrate the advantages of MQ and the merits of a joint optimization with precoding.

Cores of cooperative games are ubiquitous in information theory and arise most frequently in the characterization of fundamental limits in various scenarios involving multiple users. Examples include classical settings in network information theory such as Slepian-Wolf source coding and multiple access channels, classical settings in statistics such as robust hypothesis testing, and new settings at the intersection of networking and statistics such as distributed estimation problems for sensor networks. Cooperative game theory allows one to understand aspects of all these problems from a fresh and unifying perspective that treats users as players in a game, sometimes leading to new insights. At the heart of these analyses are fundamental dualities that have been long studied in the context of cooperative games; for information theoretic purposes, these are dualities between information inequalities on the one hand and properties of rate, capacity, or other resource allocation regions on the other.

This paper shows how network information theory can lead to practical algorithms for optimizing multipoint wireless networks. An information theoretic network objective function is formulated, which takes full advantage of channel reciprocity and multiple input multiple output (MIMO) channels. A numerical example is provided showing more than an order of magnitude performance improvement over a conventional network.

A strong converse shows that no procedure can beat the asymptotic (as blocklength n\to\infty ) fundamental limit of a given information-theoretic problem for any fixed error probability. A second-order converse strengthens this conclusion by showing that the asymptotic fundamental limit cannot be exceeded by more than O(\tfrac{1}{\sqrt{n}}) . While strong converses are achieved in a broad range of information-theoretic problems by virtue of the “blowing-up method” – a powerful methodology due to Ahlswede, Gács and Körner (1976) based on concentration of measure – this method is fundamentally unable to attain second-order converses and is restricted to finite-alphabet settings. Capitalizing on reverse hypercontractivity of Markov semigroups and functional inequalities, this paper develops the “smoothing-out” method, an alternative to the blowing-up approach that does not rely on finite alphabets and that leads to second-order converses in a variety of information-theoretic problems that were out of reach of previous methods.

This paper studies the achievable degrees of freedom (DoF) for multi-user multiple-input multiple-output (MIMO) two-way relay channels, where there are K source nodes, each equipped with M antennas, one relay node, equipped with N antennas, and each source node exchanges independent messages with an arbitrary set of other source nodes via the relay. By allowing an arbitrary information exchange pattern, the considered channel model is a unified one. It includes several existing channel models as special cases: 1) K-user MIMO Y channel; 2) multi-pair MIMO two-way relay channel; 3) generalized MIMO two-way X relay channel; and 4) L-cluster MIMO multiway relay channel. Previous studies mainly considered the achievability of the DoF cut-set bound 2N at the antenna configuration N <; 2M by applying signal alignment for network coding. This paper aims to investigate the achievability of the DoF cut-set bound K M for the case N 2M. To this end, we first derive tighter DoF upper bounds for three special cases of the considered channel model. Then, we propose a new transmission framework, generalized signal alignment (GSA), to approach these bounds. The notion of GSA is to form network-coded symbols by aligning every pair of signals to be exchanged in a compressed subspace at the relay. A necessary and sufficient condition to construct the relay compression matrix is given. We show that using GSA, the new DoF upper bound is achievable when: 1) N/m ∈ (0, 2+ 4/{K(K-1)}]∪[K-2, +∞) for the K-user MIMO Y channel; 2) N/M ∈ (0,2 +4/K]∪[K - 2, +∞) for the multi-pair MIMO two-way relay channel; and 3) N/M ∈ (0, 2+ 8/K <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ]∪[K-2, +∞) for the generalized MIMO two-way X relay channel. We also provide the antenna configuration regions for the general multi-user MIMO two-way relay channel to achieve the total DoF KM.

For a class C of graphs, #Sub(C) is the counting problem that, given a graph H from C and an arbitrary graph G, asks for the number of subgraphs of G isomorphic to H. It is known that if C has bounded vertex-cover number (equivalently, the size of the maximum matching in C is bounded), then #Sub(C) is polynomial-time solvable. We complement this result with a corresponding lower bound: if C is any recursively enumerable class of graphs with unbounded vertex-cover number, then #Sub(C) is #W[1]-hard parameterized by the size of H and hence not polynomial-time solvable and not even fixed-parameter tractable, unless FPT is equal to #W[1]. As a first step of the proof, we show that counting k-matchings in bipartite graphs is #W[1]-hard. Recently, Curticapean [ICALP 2013] proved the #W[1]-hardness of counting k-matchings in general graphs, our result strengthens this statement to bipartite graphs with a considerably simpler proof and even shows that, assuming the Exponential Time Hypothesis (ETH), there is no f(k)*no(k/log(k)) time algorithm for counting k-matchings in bipartite graphs for any computable function f. As a consequence, we obtain an independent and somewhat simpler proof of the classical result of Flum and Grohe [SICOMP 2004] stating that counting paths of length k is #W[1]-hard, as well as a similar almost-tight ETH-based lower bound on the exponent.

The $\gamma_2$ norm of a real $m\times n$ matrix $A$ is the minimum number $t$ such that the column vectors of $A$ are contained in a $0$-centered ellipsoid $E\subseteq\mathbb{R}^m$ which in turn is contained in the hypercube $[-t, t]^m$. We prove that this classical quantity approximates the \emph{hereditary discrepancy} $\mathrm{herdisc} A$ as follows: $\gamma_2(A) = {O(\log m)}\cdot \mathrm{herdisc} A$ and $\mathrm{herdisc} A = O(\sqrt{\log m}\,)\cdot\gamma_2(A) $. Since $\gamma_2$ is polynomial-time computable, this gives a polynomial-time approximation algorithm for hereditary discrepancy. Both inequalities are shown to be asymptotically tight. We then demonstrate on several examples the power of the $\gamma_2$ norm as a tool for proving lower and upper bounds in discrepancy theory. Most notably, we prove a new lower bound of $\Omega(\log^{d-1} n)$ for the \emph{$d$-dimensional Tusnady problem}, asking for the combinatorial discrepancy of an $n$-point set in $\mathbb{R}^d$ with respect to axis-parallel boxes. For $d>2$, this improves the previous best lower bound, which was of order approximately $\log^{(d-1)/2}n$, and it comes close to the best known upper bound of $O(\log^{d+1/2}n)$, for which we also obtain a new, very simple proof.