OpenRaiser/Intern-Atlas huggingface dataset:
Complexity of Networks (reprise)
Venue: arXiv cs.IT 2011
Year: 2011

Network or graph structures are ubiquitous in the study of complex systems. Often, we are interested in complexity trends of these system as it evolves under some dynamic. An example might be looking at the complexity of a food web as species enter an ecosystem via migration or speciation, and leave via extinction. In a previous paper, a complexity measure of networks was proposed based on the {\em complexity is information content} paradigm. To apply this paradigm to any object, one must fix two things: a representation language, in which strings of symbols from some alphabet describe, or stand for the objects being considered; and a means of determining when two such descriptions refer to the same object. With these two things set, the information content of an object can be computed in principle from the number of equivalent descriptions describing a particular object. The previously proposed representation language had the deficiency that the fully connected and empty networks were the most complex for a given number of nodes. A variation of this measure, called zcomplexity, applied a compression algorithm to the resulting bitstring representation, to solve this problem. Unfortunately, zcomplexity proved too computationally expensive to be practical. In this paper, I propose a new representation language that encodes the number of links along with the number of nodes and a representation of the linklist. This, like zcomplexity, exhibits minimal complexity for fully connected and empty networks, but is as tractable as the original measure. ...
On the Number of Errors Correctable with Codes on Graphs
Venue: arXiv cs.IT 2011
Year: 2011

We study ensembles of codes on graphs (generalized low-density parity-check, or LDPC codes) constructed from random graphs and fixed local constrained codes, and their extension to codes on hypergraphs. It is known that the average minimum distance of codes in these ensembles grows linearly with the code length. We show that these codes can correct a linearly growing number of errors under simple iterative decoding algorithms. In particular, we show that this property extends to codes constructed by parallel concatenation of Hamming codes and other codes with small minimum distance. Previously known results that proved this property for graph codes relied on graph expansion and required the choice of local codes with large distance relative to their length.
Analyzing Midpoint Subdivision
Venue: arXiv cs.GR 2011
Year: 2011

Midpoint subdivision generalizes the Lane-Riesenfeld algorithm for uniform tensor product splines and can also be applied to non regular meshes. For example, midpoint subdivision of degree 2 is a specific Doo-Sabin algorithm and midpoint subdivision of degree 3 is a specific Catmull-Clark algorithm. In 2001, Zorin and Schroeder were able to prove C1-continuity for midpoint subdivision surfaces analytically up to degree 9. Here, we develop general analysis tools to show that the limiting surfaces under midpoint subdivision of any degree >= 2 are C1-continuous at their extraordinary points.
A Local Spectral Method for Graphs: with Applications to Improving Graph Partitions and Exploring Data Graphs Locally
Venue: arXiv cs.DS 2011
Year: 2011

The second eigenvalue of the Laplacian matrix and its associated eigenvector are fundamental features of an undirected graph, and as such they have found widespread use in scientific computing, machine learning, and data analysis. In many applications, however, graphs that arise have several \emph{local} regions of interest, and the second eigenvector will typically fail to provide information fine-tuned to each local region. In this paper, we introduce a locally-biased analogue of the second eigenvector, and we demonstrate its usefulness at highlighting local properties of data graphs in a semi-supervised manner. To do so, we first view the second eigenvector as the solution to a constrained optimization problem, and we incorporate the local information as an additional constraint; we then characterize the optimal solution to this new problem and show that it can be interpreted as a generalization of a Personalized PageRank vector; and finally, as a consequence, we show that the solution can be computed in nearly-linear time. In addition, we show that this locally-biased vector can be used to compute an approximation to the best partition \emph{near} an input seed set in a manner analogous to the way in which the second eigenvector of the Laplacian can be used to obtain an approximation to the best partition in the entire input graph. Such a primitive is useful for identifying and refining clusters locally, as it allows us to focus on a local region of interest in a semi-supervised manner. Finally, we provide a detailed empirical evaluation of our method by showing how it can applied to finding locally-biased sparse cuts around an input vertex seed set in social and information networks.
Interference Alignment in Dense Wireless Networks
Venue: arXiv cs.IT 2011
Year: 2011

We consider arbitrary dense wireless networks, in which $n$ nodes are placed in an arbitrary (deterministic) manner on a square region of unit area and communicate with each other over Gaussian fading channels. We provide inner and outer bounds for the $n\times n$-dimensional unicast and the $n\times 2^n$-dimensional multicast capacity regions of such a wireless network. These inner and outer bounds differ only by a factor $O(\log(n))$, yielding a fairly tight scaling characterization of the entire regions. The communication schemes achieving the inner bounds use interference alignment as a central technique and are, at least conceptually, surprisingly simple.
Sparse Recovery from Combined Fusion Frame Measurements
Venue: arXiv cs.IT 2011
Year: 2011

Sparse representations have emerged as a powerful tool in signal and information processing, culminated by the success of new acquisition and processing techniques such as Compressed Sensing (CS). Fusion frames are very rich new signal representation methods that use collections of subspaces instead of vectors to represent signals. This work combines these exciting fields to introduce a new sparsity model for fusion frames. Signals that are sparse under the new model can be compressively sampled and uniquely reconstructed in ways similar to sparse signals using standard CS. The combination provides a promising new set of mathematical tools and signal models useful in a variety of applications. With the new model, a sparse signal has energy in very few of the subspaces of the fusion frame, although it does not need to be sparse within each of the subspaces it occupies. This sparsity model is captured using a mixed l1/l2 norm for fusion frames. A signal sparse in a fusion frame can be sampled using very few random projections and exactly reconstructed using a convex optimization that minimizes this mixed l1/l2 norm. The provided sampling conditions generalize coherence and RIP conditions used in standard CS theory. It is demonstrated that they are sufficient to guarantee sparse recovery of any signal sparse in our model. Moreover, a probabilistic analysis is provided using a stochastic model on the sparse signal that shows that under very mild conditions the probability of recovery failure decays exponentially with increasing dimension of the subspaces.
Ultimate Traces of Cellular Automata
Venue: arXiv cs.FL 2011
Year: 2011

A cellular automaton (CA) is a parallel synchronous computing model, which consists in a juxtaposition of finite automata (cells) whose state evolves according to that of their neighbors. Its trace is the set of infinite words representing the sequence of states taken by some particular cell. In this paper we study the ultimate trace of CA and partial CA (a CA restricted to a particular subshift). The ultimate trace is the trace observed after a long time run of the CA. We give sufficient conditions for a set of infinite words to be the trace of some CA and prove the undecidability of all properties over traces that are stable by ultimate coincidence.
Holant Problems for Regular Graphs with Complex Edge Functions
Venue: arXiv cs.CC 2011
Year: 2011

We prove a complexity dichotomy theorem for Holant Problems on 3-regular graphs with an arbitrary complex-valued edge function. Three new techniques are introduced: (1) higher dimensional iterations in interpolation; (2) Eigenvalue Shifted Pairs, which allow us to prove that a pair of combinatorial gadgets in combination succeed in proving #P-hardness; and (3) algebraic symmetrization, which significantly lowers the symbolic complexity of the proof for computational complexity. With holographic reductions the classification theorem also applies to problems beyond the basic model.
Multilevel Decoders Surpassing Belief Propagation on the Binary Symmetric Channel
Venue: arXiv cs.IT 2011
Year: 2011

In this paper, we propose a new class of quantized message-passing decoders for LDPC codes over the BSC. The messages take values (or levels) from a finite set. The update rules do not mimic belief propagation but instead are derived using the knowledge of trapping sets. We show that the update rules can be derived to correct certain error patterns that are uncorrectable by algorithms such as BP and min-sum. In some cases even with a small message set, these decoders can guarantee correction of a higher number of errors than BP and min-sum. We provide particularly good 3-bit decoders for 3-left-regular LDPC codes. They significantly outperform the BP and min-sum decoders, but more importantly, they achieve this at only a fraction of the complexity of the BP and min-sum decoders.
Vector Precoding for Gaussian MIMO Broadcast Channels: Impact of Replica Symmetry Breaking
Venue: arXiv cs.IT 2011
Year: 2011

The so-called "replica method" of statistical physics is employed for the large system analysis of vector precoding for the Gaussian multiple-input multiple-output (MIMO) broadcast channel. The transmitter is assumed to comprise a linear front-end combined with nonlinear precoding, that minimizes the front-end imposed transmit energy penalty. Focusing on discrete complex input alphabets, the energy penalty is minimized by relaxing the input alphabet to a larger alphabet set prior to precoding. For the common discrete lattice-based relaxation, the problem is found to violate the assumption of replica symmetry and a replica symmetry breaking ansatz is taken. The limiting empirical distribution of the precoder's output, as well as the limiting energy penalty, are derived for one-step replica symmetry breaking. For convex relaxations, replica symmetry is found to hold and corresponding results are obtained for comparison. Particularizing to a "zero-forcing" (ZF) linear front-end, and non-cooperative users, a decoupling result is derived according to which the channel observed by each of the individual receivers can be effectively characterized by the Markov chain u-x-y, where u, x, and y are the channel input, the equivalent precoder output, and the channel output, respectively. For discrete lattice-based alphabet relaxation, the impact of replica symmetry breaking is demonstrated for the energy penalty at the transmitter. An analysis of spectral efficiency is provided to compare discrete lattice-based relaxations against convex relaxations, as well as linear ZF and Tomlinson-Harashima precoding (THP). Focusing on quaternary phase shift-keying (QPSK), significant performance gains of both lattice and convex relaxations are revealed compared to linear ZF precoding, for medium to high signal-to-noise ratios (SNRs). THP is shown to be outperformed as well.
Improved Upper Bounds to the Causal Quadratic Rate-Distortion Function for Gaussian Stationary Sources
Venue: arXiv cs.IT 2011
Year: 2011

We improve the existing achievable rate regions for causal and for zero-delay source coding of stationary Gaussian sources under an average mean squared error (MSE) distortion measure. To begin with, we find a closed-form expression for the information-theoretic causal rate-distortion function (RDF) under such distortion measure, denoted by $R_{c}^{it}(D)$, for first-order Gauss-Markov processes. Rc^{it}(D) is a lower bound to the optimal performance theoretically attainable (OPTA) by any causal source code, namely Rc^{op}(D). We show that, for Gaussian sources, the latter can also be upper bounded as Rc^{op}(D)\leq Rc^{it}(D) + 0.5 log_{2}(2πe) bits/sample. In order to analyze $R_{c}^{it}(D)$ for arbitrary zero-mean Gaussian stationary sources, we introduce \bar{Rc^{it}}(D), the information-theoretic causal RDF when the reconstruction error is jointly stationary with the source. Based upon \bar{Rc^{it}}(D), we derive three closed-form upper bounds to the additive rate loss defined as \bar{Rc^{it}}(D) - R(D), where R(D) denotes Shannon's RDF. Two of these bounds are strictly smaller than 0.5 bits/sample at all rates. These bounds differ from one another in their tightness and ease of evaluation; the tighter the bound, the more involved its evaluation. We then show that, for any source spectral density and any positive distortion D\leq σ_{x}^{2}, \bar{Rc^{it}}(D) can be realized by an AWGN channel surrounded by a unique set of causal pre-, post-, and feedback filters. We show that finding such filters constitutes a convex optimization problem. In order to solve the latter, we propose an iterative optimization procedure that yields the optimal filters and is guaranteed to converge to \bar{Rc^{it}}(D). Finally, by establishing a connection to feedback quantization we design a causal and a zero-delay coding scheme which, for Gaussian sources, achieves...
Structure of Extreme Correlated Equilibria: a Zero-Sum Example and its Implications
Venue: arXiv cs.GT 2011
Year: 2011

We exhibit the rich structure of the set of correlated equilibria by analyzing the simplest of polynomial games: the mixed extension of matching pennies. We show that while the correlated equilibrium set is convex and compact, the structure of its extreme points can be quite complicated. In finite games the ratio of extreme correlated to extreme Nash equilibria can be greater than exponential in the size of the strategy spaces. In polynomial games there can exist extreme correlated equilibria which are not finitely supported; we construct a large family of examples using techniques from ergodic theory. We show that in general the set of correlated equilibrium distributions of a polynomial game cannot be described by conditions on finitely many moments (means, covariances, etc.), in marked contrast to the set of Nash equilibria which is always expressible in terms of finitely many moments.
Maximum $Δ$-edge-colorable subgraphs of class II graphs
Venue: arXiv cs.DM 2011
Year: 2011

A graph $G$ is class II, if its chromatic index is at least $Δ+1$. Let $H$ be a maximum $Δ$-edge-colorable subgraph of $G$. The paper proves best possible lower bounds for $\frac{|E(H)|}{|E(G)|}$, and structural properties of maximum $Δ$-edge-colorable subgraphs. It is shown that every set of vertex-disjoint cycles of a class II graph with $Δ\geq3$ can be extended to a maximum $Δ$-edge-colorable subgraph. Simple graphs have a maximum $Δ$-edge-colorable subgraph such that the complement is a matching. Furthermore, a maximum $Δ$-edge-colorable subgraph of a simple graph is always class I.
Removing Local Extrema from Imprecise Terrains
Venue: arXiv cs.CG 2011
Year: 2011

In this paper we consider imprecise terrains, that is, triangulated terrains with a vertical error interval in the vertices. In particular, we study the problem of removing as many local extrema (minima and maxima) as possible from the terrain. We show that removing only minima or only maxima can be done optimally in O(n log n) time, for a terrain with n vertices. Interestingly, however, removing both the minima and maxima simultaneously is NP-hard, and is even hard to approximate within a factor of O(log log n) unless P=NP. Moreover, we show that even a simplified version of the problem where vertices can have only two different heights is already NP-hard, a result we obtain by proving hardness of a special case of 2-Disjoint Connected Subgraphs, a problem that has lately received considerable attention from the graph-algorithms community.
When Should I Use Network Emulation?
Venue: arXiv cs.NI 2011
Year: 2011

The design and development of a complex system requires an adequate methodology and efficient instrumental support in order to early detect and correct anomalies in the functional and non-functional properties of the tested protocols. Among the various tools used to provide experimental support for such developments, network emulation relies on real-time production of impairments on real traffic according to a communication model, either realistically or not. This paper aims at simply presenting to newcomers in network emulation (students, engineers, ...) basic principles and practices illustrated with a few commonly used tools. The motivation behind is to fill a gap in terms of introductory and pragmatic papers in this domain. The study particularly considers centralized approaches, allowing cheap and easy implementation in the context of research labs or industrial developments. In addition, an architectural model for emulation systems is proposed, defining three complementary levels, namely hardware, impairment and model levels. With the help of this architectural framework, various existing tools are situated and described. Various approaches for modeling the emulation actions are studied, such as impairment-based scenarios and virtual architectures, real-time discrete simulation and trace-based systems. Those modeling approaches are described and compared in terms of services and we study their ability to respond to various designer needs to assess when emulation is needed.
Improved subspace estimation for multivariate observations of high dimension: the deterministic signals case
Venue: arXiv cs.IT 2011
Year: 2011

We consider the problem of subspace estimation in situations where the number of available snapshots and the observation dimension are comparable in magnitude. In this context, traditional subspace methods tend to fail because the eigenvectors of the sample correlation matrix are heavily biased with respect to the true ones. It has recently been suggested that this situation (where the sample size is small compared to the observation dimension) can be very accurately modeled by considering the asymptotic regime where the observation dimension $M$ and the number of snapshots $N$ converge to $+\infty$ at the same rate. Using large random matrix theory results, it can be shown that traditional subspace estimates are not consistent in this asymptotic regime. Furthermore, new consistent subspace estimate can be proposed, which outperform the standard subspace methods for realistic values of $M$ and $N$. The work carried out so far in this area has always been based on the assumption that the observations are random, independent and identically distributed in the time domain. The goal of this paper is to propose new consistent subspace estimators for the case where the source signals are modelled as unknown deterministic signals. In practice, this allows to use the proposed approach regardless of the statistical properties of the source signals. In order to construct the proposed estimators, new technical results concerning the almost sure location of the eigenvalues of sample covariance matrices of Information plus Noise complex Gaussian models are established. These results are believed to be of independent interest.
A fuzzified BRAIN algorithm for learning DNF from incomplete data
Venue: arXiv cs.IT 2011
Year: 2011

Aim of this paper is to address the problem of learning Boolean functions from training data with missing values. We present an extension of the BRAIN algorithm, called U-BRAIN (Uncertainty-managing Batch Relevance-based Artificial INtelligence), conceived for learning DNF Boolean formulas from partial truth tables, possibly with uncertain values or missing bits. Such an algorithm is obtained from BRAIN by introducing fuzzy sets in order to manage uncertainty. In the case where no missing bits are present, the algorithm reduces to the original BRAIN.
Multiuser MIMO Downlink Beamforming Design Based on Group Maximum SINR Filtering
Venue: arXiv cs.IT 2011
Year: 2011

In this paper we aim to solve the multiuser multi-input multi-output (MIMO) downlink beamforming problem where one multi-antenna base station broadcasts data to many users. Each user is assigned multiple data streams and has multiple antennas at its receiver. Efficient solutions to the joint transmit-receive beamforming and power allocation problem based on iterative methods are proposed. We adopt the group maximum signal-to-interference-plus-noise-ratio (SINR) filter bank (GSINR-FB) as our beamformer which exploits receiver diversity through cooperation between the data streams of a user. The data streams for each user are subject to an average SINR constraint, which has many important applications in wireless communication systems and serves as a good metric to measure the quality of service (QoS). The GSINR-FB also optimizes the average SINR of its output. Based on the GSINR-FB beamformer, we find an SINR balancing structure for optimal power allocation which simplifies the complicated power allocation problem to a linear one. Simulation results verify the superiority of the proposed algorithms over previous works with approximately the same complexity.
Bit Allocation Law for Multi-Antenna Channel Feedback Quantization: Single-User Case
Venue: arXiv cs.IT 2011
Year: 2011

This paper studies the design and optimization of a limited feedback single-user system with multiple-antenna transmitter and single-antenna receiver. The design problem is cast in form of the minimizing the average transmission power at the base station subject to the user's outage probability constraint. The optimization is over the user's channel quantization codebook and the transmission power control function at the base station. Our approach is based on fixing the outage scenarios in advance and transforming the design problem into a robust system design problem. We start by showing that uniformly quantizing the channel magnitude in dB scale is asymptotically optimal, regardless of the magnitude distribution function. We derive the optimal uniform (in dB) channel magnitude codebook and combine it with a spatially uniform channel direction codebook to arrive at a product channel quantization codebook. We then optimize such a product structure in the asymptotic regime of $B\rightarrow \infty$, where $B$ is the total number of quantization feedback bits. The paper shows that for channels in the real space, the asymptotically optimal number of direction quantization bits should be ${(M{-}1)}/{2}$ times the number of magnitude quantization bits, where $M$ is the number of base station antennas. We also show that the performance of the designed system approaches the performance of the perfect channel state information system as $2^{-\frac{2B}{M+1}}$. For complex channels, the number of magnitude and direction quantization bits are related by a factor of $(M{-}1)$ and the system performance scales as $2^{-\frac{B}{M}}$ as $B\rightarrow\infty$.
Computational Complexity of Iterated Maps on the Interval
Venue: arXiv cs.NA 2011
Year: 2011

The correct computation of orbits of discrete dynamical systems on the interval is considered. Therefore, an arbitrary-precision floating-point approach based on automatic error analysis is chosen and a general algorithm is presented. The correctness of the algorithm is shown and the computational complexity is analyzed. There are two main results. First, the computational complexity measure considered here is related to the Lyapunov exponent of the dynamical system under consideration. Second, the presented algorithm is optimal with regard to that complexity measure.