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As an algorithmic framework, message passing is extremely powerful and has wide applications in the context of different disciplines including communications, coding theory, statistics, signal processing, artificial intelligence and combinatorial optimization. In this paper, we investigate the performance of a message-passing algorithm called min-sum belief propagation (BP) for the vertex-disjoint shortest $k$ -path problem ($k$ -VDSP) on weighted directed graphs, and derive the iterative message-passing update rules. As the main result of this paper, we prove that for a weighted directed graph $G$ of order $n$ , BP algorithm converges to the unique optimal solution of $k$ -VDSP on $G$ within $O(n^{2}w_{max})$ iterations, provided that the weight $w_{e}$ is nonnegative integral for each arc $e\in E(G)$ , where $w_{max}=\max \{w_{e}: e\in E(G)\}$. To the best of our knowledge, this is the first instance where BP algorithm is proved correct for NP-hard problems. Additionally, we establish the extensions of $k$ -VDSP to the case of multiple sources or sinks. [ABSTRACT FROM AUTHOR]

Gossip protocols are popular methods for average consensus problems in distributed computing. We prove new convergence guarantees for a variety of such protocols, including path, clique, and synchronous pairwise gossip. These arise by exploiting the connection between these protocols and the block randomized Kaczmarz method for solving linear systems. Moreover, we extend existing convergence results for block randomized Kaczmarz to allow for a more general choice of blocks, rank-deficient systems, and provide a tighter convergence rate guarantee. Previous results for block randomized Kaczmarz assumed blocks formed a paving of the systems; our analysis, contrary to the title of our paper, generalizes to a broader class of blocks that we call coverings. We furthermore apply this analysis to inconsistent consensus models and obtain similar guarantees. An extensive empirical analysis of these methods is provided for a variety of synthetic and real networks. [ABSTRACT FROM AUTHOR]

This paper studies distributed algorithms for (strongly convex) composite optimization problems over mesh networks, subject to quantized communications. Instead of focusing on a specific algorithmic design, a black-box model is proposed, casting linearly convergent distributed algorithms in the form of fixed-point iterates. The algorithmic model is equipped with a novel random or deterministic Biased Compression (BC) rule on the quantizer design, and a new Adaptive encoding Non-uniform Quantizer (ANQ) coupled with a communication-efficient encoding scheme, which implements the BC-rule using a finite number of bits (below machine precision). This fills a gap existing in most state-of-the-art quantization schemes, such as those based on the popular compression rule, which rely on communication of some scalar signals with negligible quantization error (in practice quantized at the machine precision). A unified communication complexity analysis is developed for the black-box model, determining the average number of bits required to reach a solution of the optimization problem within a target accuracy. It is shown that the proposed BC-rule preserves linear convergence of the unquantized algorithms, and a trade-off between convergence rate and communication cost under ANQ-based quantization is characterized. Numerical results validate our theoretical findings and show that distributed algorithms equipped with the proposed ANQ have more favorable communication cost than algorithms using state-of-the-art quantization rules. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the convergence speed of the Arimoto-Blahut algorithm. For many channel matrices, the convergence speed is exponential, but for some channel matrices it is slower than exponential. By analyzing the Taylor expansion of the defining function of the Arimoto-Blahut algorithm, we will make the conditions clear for the exponential or slower convergence. The analysis of the slow convergence in this paper is new. Based on this analysis, we will compare the convergence speeds of the Arimoto-Blahut algorithm numerically with the values obtained in our theorems for several channel matrices. The purpose of this paper is to obtain a complete understanding of the convergence speed of the Arimoto-Blahut algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we consider a general K-user Gaussian multiple-input multiple-output (MIMO) broadcast channel (BC). We assume that the channel state is deterministic and known to all the nodes. While the private-message capacity region is well known to be achievable with dirty paper coding (DPC), we are interested in the simpler linearly precoded transmission schemes. In particular, we focus on linear precoding schemes combined with rate-splitting (RS). First, we derive an achievable rate region with minimum mean square error (MMSE) precoding at the transmitter and joint decoding of the sub-messages at the receivers. Then, we study the achievable sum rate of this scheme and obtain two findings: 1) an analytically tractable upper bound on the sum rate that is shown numerically to be a close approximation, and 2) how to reduce the number of active streams – crucial to the overall complexity – while preserving the sum rate to within a constant loss. The latter results in two practical algorithms: a stream elimination algorithm and a stream ordering algorithm. Finally, we investigate the constant-gap optimality of linearly precoded RS with respect to the capacity. Our result reveals that, while the achievable rate of linear precoding alone can be arbitrarily far from the capacity, the introduction of RS can help achieve the capacity region to within a constant gap in the two-user case. Nevertheless, we prove that the RS scheme’s constant-gap optimality does not extend to the three-user case. Specifically, we show, through a pathological example, that the gap between the sum rate and the sum capacity can be unbounded. [ABSTRACT FROM AUTHOR]

Data deduplication saves storage space by identifying and removing repeats in the data stream. Compared with traditional compression methods, data deduplication schemes are more computationally efficient and are thus widely used in large scale storage systems. In this paper, we provide an information-theoretic analysis of the performance of deduplication algorithms on data streams in which repeats are not exact. We introduce a source model in which probabilistic substitutions are considered. More precisely, each symbol in a repeated string is substituted with a given edit probability. Deduplication algorithms in both the fixed-length scheme and the variable-length scheme are studied. The fixed-length deduplication algorithm is shown to be unsuitable for the proposed source model as it does not take into account the edit probability. Two modifications are proposed and shown to have performances within a constant factor of optimal for a specific class of source models with the knowledge of model parameters. We also study the conventional variable-length deduplication algorithm and show that as source entropy becomes smaller, the size of the compressed string vanishes relative to the length of the uncompressed string, leading to high compression ratios. [ABSTRACT FROM AUTHOR]

Recent advances in noiseless non-adaptive group testing have led to a precise asymptotic characterization of the number of tests required for high-probability recovery in the sublinear regime $k = n^{\theta }$ (with $\theta \in (0,1)$), with $n$ individuals among which $k$ are infected. However, the required number of tests may increase substantially under real-world practical constraints, notably including bounds on the maximum number $\Delta $ of tests an individual can be placed in, or the maximum number $\Gamma $ of individuals in a given test. While previous works have given recovery guarantees for these settings, significant gaps remain between the achievability and converse bounds. In this paper, we substantially or completely close several of the most prominent gaps. In the case of $\Delta $ -divisible items, we show that the definite defectives (DD) algorithm coupled with a random regular design is asymptotically optimal in dense scaling regimes, and optimal to within a factor of e more generally; we establish this by strengthening both the best known achievability and converse bounds. In the case of $\Gamma $ -sized tests, we provide a comprehensive analysis of the regime $\Gamma = \Theta (1)$ , and again establish a precise threshold proving the asymptotic optimality of SCOMP (a slight refinement of DD) equipped with a tailored pooling scheme. Finally, for each of these two settings, we provide near-optimal adaptive algorithms based on sequential splitting, and provably demonstrate gaps between the performance of optimal adaptive and non-adaptive algorithms. [ABSTRACT FROM AUTHOR]

Given a binary sequence with length $n$ , determining its minimal rational fraction representation (MRFR) has important applications in the design and cryptanalysis of stream ciphers. There are many studies of this problem since Klapper and Goresky first introduced an adaptive rational approximation algorithm with a time complexity of $O(n^{2}\log n\log \log n)$. In this paper, we revisit this problem by considering both adaptive and non-adaptive efficient algorithms. Compared with the state-of-art methods, we make several contributions to the problem of finding MRFR. Firstly, we find a general and precise recursive relationship between the minimal bases for two adjacent lattices generated by successive truncation sequences. This enables us to improve the currently fastest adaptive algorithm proposed by Li et al.. Secondly, by optimizing a time-consuming step of the well-known Lagrange reduction algorithm for 2-dimensional lattices, we obtain a non-adaptive, and yet practically faster MRFR-solving algorithm named global Euclidean algorithm. Thirdly, we identify theoretical flaws on some non-adaptive methods in the literature by counter-examples and correct the problems by designing modified Euclidean algorithm named partial Euclidean algorithm. Meanwhile, we further reduce the time complexity of existing algorithm from $O(n^{2})$ to $O(n\log ^{2}n\log \log n)$ by invoking the half-gcd algorithm. We also conduct a comprehensive experimental comparative analysis on the above algorithms to validate our theoretical analysis. [ABSTRACT FROM AUTHOR]

The Binary Iterative Hard Thresholding (BIHT) algorithm is a popular reconstruction method for one-bit compressed sensing due to its simplicity and fast empirical convergence. Despite considerable research on this algorithm, a theoretical understanding of the corresponding approximation error and convergence rate still remains an open problem. This paper shows that the normalized version of BIHT (NBIHT) achieves an approximation error rate optimal up to logarithmic factors. More precisely, using $m$ one-bit measurements of an $s$ -sparse vector $x$ , we prove that the approximation error of NBIHT is of order $O \left ({\frac{1 }{ m }}\right)$ up to logarithmic factors, which matches the information-theoretic lower bound $\Omega \left ({\frac{1 }{ m }}\right)$ proved by Jacques, Laska, Boufounos, and Baraniuk in 2013. To our knowledge, this is the first theoretical analysis of a BIHT-type algorithm that explains the optimal rate of error decay empirically observed in the literature. This also makes NBIHT the first provable computationally-efficient one-bit compressed sensing algorithm that breaks the inverse square-root error decay rate $O \left ({\frac{1 }{ m^{1/2} }}\right)\vphantom {{\left ({\frac{1 }{ m^{1/2} }}\right)}^{'}}$. [ABSTRACT FROM AUTHOR]

Consider a massive random access scenario in which a small set of $k$ active users out of a large number of $n$ potential users need to be scheduled in $b\ge k$ slots. What is the minimum common feedback to the users needed to ensure that scheduling is collision-free? Instead of a naive scheme of listing the indices of the $k$ active users in the order in which they should transmit, at a cost of $k\log (n)$ bits, this paper shows that for the case of $b=k$ , the rate of the minimum fixed-length common feedback code scales only as $k \log (e)$ bits, plus an additive term that scales in $n$ as $\Theta (\log \log (n))$ for fixed $k$. If a variable-length code can be used, assuming uniform activity among the users, the minimum average common feedback rate still requires $k \log (e)$ bits, but the dependence on $n$ can be reduced to $O(1)$. When $b>k$ , the number of feedback bits needed for collision-free scheduling can be significantly further reduced. Moreover, a similar scaling on the minimum feedback rate is derived for the case of scheduling $m$ users per slot, when $k \le mb$. The problem of constructing a minimum collision-free feedback scheduling code is connected to that of constructing a perfect hashing family, which allows practical feedback scheduling codes to be constructed from perfect hashing algorithms. [ABSTRACT FROM AUTHOR]

This paper proposes a new algorithm, named Householder Dice (HD), for simulating dynamics on dense random matrix ensembles with rotational invariance. Examples include the Gaussian ensemble, the Haar-distributed random orthogonal ensemble, and their complex-valued counterparts. A “direct” approach to the simulation, where one first generates a dense $n \times n$ matrix from the ensemble, requires at least $\mathcal {O}(n^{2})$ resource in space and time. The HD algorithm overcomes this $\mathcal {O}(n^{2})$ bottleneck by using the principle of deferred decisions: rather than fixing the entire random matrix in advance, it lets the randomness unfold with the dynamics. At the heart of this matrix-free algorithm is an adaptive and recursive construction of (random) Householder reflectors. These orthogonal transformations exploit the group symmetry of the matrix ensembles, while simultaneously maintaining the statistical correlations induced by the dynamics. The memory and computation costs of the HD algorithm are $\mathcal {O}(nT)$ and $\mathcal {O}(nT^{2})$ , respectively, with $T$ being the number of iterations. When $T \ll n$ , which is nearly always the case in practice, the new algorithm leads to significant reductions in runtime and memory footprint. Numerical results demonstrate the promise of the HD algorithm as a new computational tool in the study of high-dimensional random systems. [ABSTRACT FROM AUTHOR]

The basic goal of threshold group testing is to identify up to $d$ defective items among a population of $n$ items, where $d$ is usually much smaller than $n$. The outcome of a test on a subset of items is positive if the subset has at least $u$ defective items, negative if it has up to $\ell $ defective items, where $0 \leq \ell < u$ , and arbitrary otherwise. This is called threshold group testing. The parameter $g = u - \ell - 1$ is called the gap. In this paper, we focus on the case $g > 0$ , i.e., threshold group testing with a gap. Note that the results presented here are also applicable to the case $g = 0$ ; however, the results are not as efficient as those in related work. Currently, a few reported studies have investigated test designs and decoding algorithms for identifying defective items. Most of the previous studies have not been feasible because there are numerous constraints on their problem settings or the decoding complexities of their proposed schemes are relatively large. Therefore, it is compulsory to reduce the number of tests as well as the decoding complexity, i.e., the time for identifying the defective items, for achieving practical schemes. The work presented here makes five contributions. The first is a more accurate theorem for a non-adaptive algorithm for threshold group testing proposed by Chen and Fu. The second is an improvement in the construction of disjunct matrices, which are the main tools for tackling (threshold) group testing and other tasks such as constructing cover-free families or learning hidden graphs. Specifically, we present a better exact upper bound on the number of tests for disjunct matrices compared with that in related work. The third and fourth contributions are a reduced exact upper bound on the number of tests and a reduced asymptotic bound on the decoding time for identifying defective items in a noisy setting on test outcomes. The fifth contribution is a simulation on the number of tests of the resulting improvements for previous work and the proposed theorems. [ABSTRACT FROM AUTHOR]

In image detection, one problem is to test whether the set, though mainly consisting of uniformly scattered points, also contains a small fraction of points sampled from some (a priori unknown) curve, for example, a curve with Cα-norm bounded by β. One approach is to analyze the data by counting membership in multiscale multianisotropic strips, which involves an algorithm that delves into the length of the path connecting many consecutive “significant” nodes. In this paper, we develop the mathematical formalism of this algorithm and analyze the statistical property of the length of the longest significant run. The rate of convergence is derived. Using percolation theory and random graph theory, we present a novel probabilistic model named, pseudo-tree model. Based on the asymptotic results for the pseudo-tree model, we further study the length of the longest significant run in an “inflating” Bernoulli net. We find that the probability parameter p of significant node plays an important role: there is a threshold pc, such that in the cases of p < pc and p > pc, very different asymptotic behaviors of the length of the significant runs are observed. We apply our results to the detection of an underlying curvilinear feature and prove that the test based on our proposed longest run theory is asymptotically powerful. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of encoding data into repeat-free sequences in which sequences are imposed to contain any k-tuple at most once (for predefined k). First, the capacity of the repeat-free constraint are calculated. Then, an efficient algorithm, which uses two bits of redundancy, is presented to encode length-n sequences for k = 2 + 2 log (n). This algorithm is then improved to support any value of k of the form k = a log (n), for 1 < a, while its redundancy is o(n). We also calculate the capacity of repeat-free sequences when combined with local constraints which are given by a constrained system, and the capacity of multi-dimensional repeat-free codes. [ABSTRACT FROM AUTHOR]

Line spectral estimation is a classical signal processing problem that aims to estimate the line spectra from their signal which is contaminated by deterministic or random noise. Despite a large body of research on this subject, the theoretical understanding of this problem is still elusive. In this paper, we introduce and quantitatively characterize the two resolution limits for the line spectral estimation problem under deterministic noise: one is the minimum separation distance between the line spectra that is required for exact detection of their number, and the other is the minimum separation distance between the line spectra that is required for a stable recovery of their supports. The quantitative results imply a phase transition phenomenon in each of the two recovery problems, and also the subtle difference between the two. We further propose a sweeping singular-value-thresholding algorithm for the number detection problem and conduct numerical experiments. The numerical results confirm the phase transition phenomenon in the number detection problem. [ABSTRACT FROM AUTHOR]

This paper studies a large random matrix system (LRMS) model involving an arbitrary signal distribution and forward error control (FEC) coding. We establish an area property based on the approximate message passing (AMP) algorithm. Under the assumption that the state evolution for AMP is correct for the coded system, the achievable rate of AMP is analyzed. We prove that AMP achieves the constrained capacity of the LRMS with an arbitrary signal distribution provided that a matching condition is satisfied. As a byproduct, we provide an alternative derivation for the constraint capacity of an LRMS using a proved property of AMP. We discuss realization techniques for the matching principle of binary signaling using irregular low-density parity-check (LDPC) codes and provide related numerical results. We show that the optimized codes demonstrate significantly better performance over un-matched ones under AMP. For quadrature phase shift keying (QPSK) modulation, bit error rate (BER) performance within 1 dB from the constrained capacity limit is observed. [ABSTRACT FROM AUTHOR]

This paper revisits the ordered statistics decoding (OSD). It provides a comprehensive analysis of the OSD algorithm by characterizing the statistical properties, evolution and the distribution of the Hamming distance and weighted Hamming distance from codeword estimates to the received sequence in the reprocessing stages of the OSD algorithm. We prove that the Hamming distance and weighted Hamming distance distributions can be characterized as mixture models capturing the decoding error probability and code weight enumerator. Simulation and numerical results show that our proposed statistical approaches can accurately describe the distance distributions. Based on these distributions and with the aim to reduce the decoding complexity, several techniques, including stopping rules and discarding rules, are proposed, and their decoding error performance and complexity are accordingly analyzed. Simulation results for decoding various eBCH codes demonstrate that the proposed techniques can significantly reduce the decoding complexity with a negligible loss in the decoding error performance. [ABSTRACT FROM AUTHOR]

Communication for omniscience (CO) refers to the problem where the users in a finite set V observe a discrete multiple random source and want to exchange data over broadcast channels to reach omniscience, the state where everyone recovers the entire source. This paper studies how to improve the computational complexity for the problem of minimizing the sum-rate for attaining omniscience in V. While the existing algorithms rely on the submodular function minimization (SFM) techniques and complete in O(|V|2⋅SFM (|V|) time, we prove the strict strong map property of the nesting SFM problem. We propose a parametric (PAR) algorithm that utilizes the parametric SFM techniques and reduces the complexity to O(|V| ⋅SFM (|V|). We propose efficient solutions to the successive omniscience (SO): attaining omniscience successively in user subsets. We first focus on how to determine a complimentary subset X* ⊄ V in the existing two-stage SO such that if the local omniscience in X* is reached first, the global omniscience whereafter can still be attained with the minimum sum-rate. It is shown that such a subset can be extracted at one of the iterations of the PAR algorithm. We then propose a novel multi-stage SO strategy: a nesting sequence of complimentary user subsets X*(1) ⊄ . . . ⊆ X*(K) = V, the omniscience in which is attained progressively by the monotonic rate vectors rV(1) ≤ . . . rV(K). We propose algorithms to obtain this K-stage SO from the returned results by the PAR algorithm. The run time of these algorithms is the same as the PAR algorithm. [ABSTRACT FROM AUTHOR]

The Demirci-Selçuk’s meet-in-the-middle attack is one of the most important methods among all the cryptanalytic vectors, which gives the best result against the round-reduced AES with respect to the rounds, and tradeoffs between data, time and memory. While we have already built provable security models against the differential cryptanalysis, linear cryptanalysis cryptanalysis, impossible differential and zero-correlation linear cryptanalysis, the provable security against the meet-in-the-middle attack is missing. In this paper, we propose the subset representation of function based on which we could give an algorithm to compute the exact number of parameters of the Demirci-Selçuk’s distinguisher given the input and output, respectively. Experiments show that this algorithm can be more efficient than the automatical tool presented by Shi et al. at Asiacrypt 2018. We further extract a formula based on this algorithm and show an upper bound for the length of the Demirci-Selçuk’s distinguisher of an iterative SPN cipher. We prove that for an SPN block cipher whose block size equals the key size, an effective Demirci-Selçuk-type meet-in-the-middle distinguisher covers at most twice the maximum of the primitive indexes of the linear layer and its inverse. As a result, we show that the known length of the Demirci-Selçuk’s distinguisher of the AES-128 cannot be improved unless the details of the S-boxes are exploited, which demonstrates that the AES has a provable security against the Demirci-Selçuk’s meet-in-the-middle attack. [ABSTRACT FROM AUTHOR]