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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]

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]

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]

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]

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]

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]

We present an efficient list decoding algorithm in the style of Guruswami-Sudan for algebraic geometry codes. Our decoder can decode any such code using $\tilde{\mathcal {O}} (s\ell ^{\omega }\mu ^{\omega -1}(n+g))$ operations in the underlying finite field, where $n$ is the code length, $g$ is the genus of the function field used to construct the code, $s$ is the multiplicity parameter, $\ell $ is the designed list size and $\mu $ is the smallest positive element in the Weierstrass semigroup at some chosen place; the “soft-O” notation $\tilde{\mathcal {O}} (\cdot)$ is similar to the “big-O” notation $\mathcal {O}(\cdot)$ , but ignores logarithmic factors. For the interpolation step, which constitutes the computational bottleneck of our approach, we use known algorithms for univariate polynomial matrices, while the root-finding step is solved using existing algorithms for root-finding over univariate power series. [ABSTRACT FROM AUTHOR]

Consider the compressed sensing setup where the support $\mathbf {s}^{\ast}$ of an $m$ -sparse $d$ -dimensional signal $\mathbf {x}$ is to be recovered from $n$ linear measurements with a given algorithm. Suppose that the measurements are such that the algorithm does not guarantee perfect support recovery and that true features may be missed. Can they efficiently be retrieved? We address this question through a simple error-correction module referred to as LiRE. LiRE takes as input an estimate $\mathbf {s}_{\text {in}}$ of the true support $\mathbf {s}^{\ast}$ , and outputs a refined support estimate $\mathbf {s}_{\text {out}}$. We establish sufficient conditions under which LiRE is guaranteed to recover the entire support, that is $\mathbf {s}_{\text {out}}\supseteq \mathbf {s}^{\ast} $. These conditions imply, for instance, that in high dimension LiRE can correct a sublinear in $m$ number of errors made by Orthogonal Matching Pursuit (OMP). The computational complexity of LiRE is ${\mathcal{O}}(m n d)$. Experimental results with random Gaussian design matrices show that LiRE substantially reduces the number of measurements needed for perfect support recovery via Compressive Sampling Matching Pursuit, Basis Pursuit (BP), and OMP. Interestingly, adding LiRE to OMP yields a support recovery procedure that is more accurate and significantly faster than BP. This observation carries over in the noisy measurement setup where the combination of LiRE and OMP is faster and more accurate than LASSO. Finally, as a standalone support recovery algorithm with a random initialization, experiments show that LiRE’s support recovery performance lies between OMP and BP. These results show that LiRE can be used generically, on top of any suboptimal baseline support recovery algorithm, to improve support recovery or to operate with a smaller number of measurements, at the cost of a relatively small computational overhead. Alternatively, LiRE may be used as a standalone support recovery algorithm that is competitive with respect to OMP. [ABSTRACT FROM AUTHOR]

Trace reconstruction considers the task of recovering an unknown string $\mathbf {x}\in \{0,1\}^{n}$ given a number of independent “traces”, i.e., subsequences of $\mathbf {x}$ obtained by randomly and independently deleting every symbol of $\mathbf {x}$ with some probability $p$. The information-theoretic limit of the number of traces needed to recover a string of length $n$ is still unknown. This limit is essentially the same as the number of traces needed to determine, given strings $\mathbf {x}$ and $\mathbf {y}$ and traces of one of them, which string is the source. The most-studied class of algorithms for the worst-case version of the problem are “mean-based” algorithms. These are a restricted class of distinguishers that only use the mean value of each coordinate on the given samples. In this work we study limitations of mean-based algorithms on strings at small Hamming or edit distance. We show that, on the one hand, distinguishing strings that are nearby in Hamming distance is “easy” for such distinguishers. On the other hand, we show that distinguishing strings that are nearby in edit distance is “hard” for mean-based algorithms. Along the way, we also describe a connection to the famous Prouhet-Tarry-Escott (PTE) problem, which shows a barrier to finding explicit hard-to-distinguish strings: namely such strings would imply explicit short solutions to the PTE problem, a well-known difficult problem in number theory. Furthermore, we show that the converse is also true, thus, finding explicit solutions to the PTE problem is equivalent to the problem of finding explicit strings that are hard-to-distinguish by mean-based algorithms. Our techniques rely on complex analysis arguments that involve careful trigonometric estimates, and algebraic techniques that include applications of Descartes’ rule of signs for polynomials over the reals. [ABSTRACT FROM AUTHOR]

The Recently proposed Vector Approximate Message Passing (VAMP) algorithm demonstrates a great reconstruction potential at solving compressed sensing related linear inverse problems. VAMP provides high per-iteration improvement, can utilize powerful denoisers like BM3D, has rigorously defined dynamics and is able to recover signals measured by highly undersampled and ill-conditioned linear operators. Yet, its applicability is limited to relatively small problem sizes due to the necessity to compute the expensive LMMSE estimator at each iteration. In this work we consider the problem of upscaling VAMP by utilizing Conjugate Gradient (CG) to approximate the intractable LMMSE estimator. We propose a rigorous method for correcting and tuning CG withing CG-VAMP to achieve a stable and efficient reconstruction. To further improve the performance of CG-VAMP, we design a warm-starting scheme for CG and develop theoretical models for the Onsager correction and the State Evolution of Warm-Started CG-VAMP (WS-CG-VAMP). Additionally, we develop robust and accurate methods for implementing the WS-CG-VAMP algorithm. The numerical experiments on large-scale image reconstruction problems demonstrate that WS-CG-VAMP requires much fewer CG iterations compared to CG-VAMP to achieve the same or superior level of reconstruction. [ABSTRACT FROM AUTHOR]

Lattices are a popular field of study in mathematical research, but also in more practical areas like cryptology or multiple-input/multiple-output (MIMO) transmission. In mathematical theory, most often lattices over real numbers are considered. However, in communications, complex-valued processing is usually of interest. Besides, by the use of dual-polarized transmission as well as by the combination of two time slots or frequencies, four-dimensional (quaternion-valued) approaches become more and more important. Hence, to account for this fact, well-known lattice algorithms and related concepts are generalized in this work. To this end, a brief review of complex arithmetic, including the sets of Gaussian and Eisenstein integers, and an introduction to quaternion-valued numbers, including the sets of Lipschitz and Hurwitz integers, are given. On that basis, generalized variants of two important algorithms are derived: first, of the polynomial-time LLL algorithm, resulting in a reduced basis of a lattice by performing a special variant of the Euclidean algorithm defined for matrices, and second, of an algorithm to calculate the successive minima—the norms of the shortest independent vectors of a lattice—and its related lattice points. Generalized bounds for the quality of the particular results are established and the asymptotic complexities of the algorithms are assessed. These findings are extensively compared to conventional real-valued processing. It is shown that the generalized approaches outperform their real-valued counterparts in complexity and/or quality aspects. Moreover, the application of the generalized algorithms to MIMO communications is studied, particularly in the field of lattice-reduction-aided and integer-forcing equalization. [ABSTRACT FROM AUTHOR]

We establish sufficient conditions of exact and almost full recovery of the node partition in Bipartite Stochastic Block Model (BSBM) using polynomial time algorithms. First, we improve upon the known conditions of almost full recovery by spectral clustering algorithms in BSBM. Next, we propose a new computationally simple and fast procedure achieving exact recovery under milder conditions than the state of the art. Namely, if the vertex sets $V_{1}$ and $V_{2}$ in BSBM have sizes $n_{1}$ and $n_{2}$ , we show that the condition $ p = \Omega \left ({\max \left ({\sqrt {\frac {\log {n_{1}}}{n_{1}n_{2}}},\frac {\log {n_{1}}}{n_{2}}}\right)}\right)$ on the edge intensity $p$ is sufficient for exact recovery within $V_{1}$. This condition exhibits an elbow at $n_{2} \asymp n_{1}\log {n_{1}}$ between the low-dimensional and high-dimensional regimes. The suggested procedure is a variant of Lloyd’s iterations initialized with a well-chosen spectral estimator leading to what we expect to be the optimal condition for exact recovery in BSBM. The optimality conjecture is supported by showing that, for a supervised oracle procedure, such a condition is necessary to achieve exact recovery. The key elements of the proof techniques are different from classical community detection tools on random graphs. Numerical studies confirm our theory, and show that the suggested algorithm is both very fast and achieves almost the same performance as the supervised oracle. Finally, using the connection between planted satisfiability problems and the BSBM, we improve upon the sufficient number of clauses to completely recover the planted assignment. [ABSTRACT FROM AUTHOR]

Minkowski proved that any $n$ -dimensional lattice of unit determinant has a nonzero vector of Euclidean norm at most $\sqrt {n}$ ; in fact, there are $2^{\Omega (n)}$ such lattice vectors. Lattices whose minimum distances come close to Minkowski’s bound provide excellent sphere packings and error-correcting codes in ${\mathbb {R}}^{n}$. The focus of this work is a certain family of efficiently constructible $n$ -dimensional lattices due to Barnes and Sloane, whose minimum distances are within an $O(\sqrt {\log n})$ factor of Minkowski’s bound. Our primary contribution is a polynomial-time algorithm that list decodes this family to distances approaching $1/\sqrt {2}$ of the minimum distance. The main technique is to decode Reed-Solomon codes under error measured in the Euclidean norm, using the Koetter-Vardy “soft decision” variant of the Guruswami-Sudan list-decoding algorithm. [ABSTRACT FROM AUTHOR]

Privacy amplification (PA) is an indispensable component in classical and quantum cryptography. Error correction (EC) and data compression (DC) algorithms are also indispensable in classical and quantum information theory. We here study these three algorithms (PA, EC, and DC) in the presence of quantum side information, and show that they all become equivalent in the one-shot scenario. As an application of this equivalence, we take previously known security bounds of PA, and translate them into coding theorems for EC and DC which have not been obtained previously. Further, we apply these results to simplify and improve our previous result that the two prevalent approaches to the security proof of quantum key distribution (QKD) are equivalent. We also propose a new method to simplify the security proof of QKD. [ABSTRACT FROM AUTHOR]

Suppose that $\mathcal {P}$ is a property that may be satisfied by a random code $C \subset \Sigma ^{n}$. For example, for some $p \in (0,1)$ , $\mathcal {P}$ might be the property that there exist three elements of $C$ that lie in some Hamming ball of radius $pn$. We say that $R^{\ast}$ is the threshold rate for $\mathcal {P}$ if a random code of rate $R^{\ast} + \varepsilon $ is very likely to satisfy $\mathcal {P}$ , while a random code of rate $R^{\ast} - \varepsilon $ is very unlikely to satisfy $\mathcal {P}$. While random codes are well-studied in coding theory, even the threshold rates for relatively simple properties like the one above are not well understood. We characterize threshold rates for a rich class of properties. These properties, like the example above, are defined by the inclusion of specific sets of codewords which are also suitably “symmetric.” For properties in this class, we show that the threshold rate is in fact equal to the lower bound that a simple first-moment calculation obtains. Our techniques not only pin down the threshold rate for the property $\mathcal {P}$ above, they give sharp bounds on the threshold rate for list-recovery in several parameter regimes, as well as an efficient algorithm for estimating the threshold rates for list-recovery in general. [ABSTRACT FROM AUTHOR]

In this work we present a new framework for the analysis and design of randomized gossip algorithms for solving the average consensus problem. We show how classical randomized iterative methods for solving linear systems can be interpreted as gossip algorithms when applied to special systems encoding the underlying network and explain in detail their decentralized nature. Our general framework recovers a comprehensive array of well-known gossip algorithms as special cases, including the pairwise randomized gossip algorithm and path averaging gossip, and allows for the development of provably faster variants. The flexibility of the new approach enables the design of a number of new specific gossip methods. For instance, we propose and analyze novel block and the first provably accelerated randomized gossip protocols, and dual randomized gossip algorithms. From a numerical analysis viewpoint, our work is the first that explores in depth the decentralized nature of randomized iterative methods for linear systems and proposes them as methods for solving the average consensus problem. We evaluate the performance of the proposed gossip protocols by performing extensive experimental testing on typical wireless network topologies. [ABSTRACT FROM AUTHOR]

In this work, we study the generalization capability of algorithms from an information-theoretic perspective. It has been shown that the expected generalization error of an algorithm is bounded from above by a function of the relative entropy between the conditional probability distribution of the algorithm’s output hypothesis, given the dataset with which it was trained, and its marginal probability distribution. We build upon this fact and introduce a mathematical formulation to obtain upper bounds on this relative entropy. Assuming that the data is discrete, we then develop a strategy using this formulation, based on the method of types and typicality, to find explicit upper bounds on the generalization error of stable algorithms, i.e., algorithms that produce similar output hypotheses given similar input datasets. In particular, we show the bounds obtained with this strategy for the case of $\epsilon $ -DP and $\mu $ -GDP algorithms. [ABSTRACT FROM AUTHOR]

We consider a multi-armed bandit framework where the rewards obtained by pulling different arms are correlated. We develop a unified approach to leverage these reward correlations and present fundamental generalizations of classic bandit algorithms to the correlated setting. We present a unified proof technique to analyze the proposed algorithms. Rigorous analysis of C-UCB (the correlated bandit versions of Upper-confidence-bound) reveals that the algorithm end up pulling certain sub-optimal arms, termed as non-competitive, only $\mathrm {O}(1)$ times, as opposed to the $\mathrm {O}(\log T)$ pulls required by classic bandit algorithms such as UCB, TS etc. We present regret-lower bound and show that when arms are correlated through a latent random source, our algorithms obtain order-optimal regret. We validate the proposed algorithms via experiments on the MovieLens and Goodreads datasets, and show significant improvement over classical bandit algorithms. [ABSTRACT FROM AUTHOR]

Compute and Forward (CF) is a coding scheme which enables receivers to decode linear combinations of simultaneously transmitted messages while exploiting the linear properties of lattice codes and the additive nature of a shared medium. The scheme was originally designed for relay networks, yet, it was found useful in other communication problems, such as MIMO communication. Works in the current literature assume a fixed number of transmitters and receivers in the system. However, following the increase in communication networks density, it is interesting to investigate the performance of CF when the number of transmitters is large. In this work, we show that as the number of transmitters, L, grows, CF becomes degenerated, in the sense that a relay prefers to decode only one (strongest) user instead of any other linear combination of the transmitted codewords, treating the other users as noise. Moreover, the system’s sum-rate tends to zero as well. This makes scheduling necessary in order to maintain the superior abilities CF provides. We thus examine the problem of scheduling for CF. We start with insights on why good scheduling opportunities can be found. Then, we provide an asymptotically optimal, polynomial-time scheduling algorithm and analyze its performance. We conclude that with proper scheduling, CF is not merely non-degenerated, but, in fact, provides a gain for the system sum-rate, up to the optimal scaling law of O(log log L). [ABSTRACT FROM AUTHOR]

We give a linear-time erasure list-decoding algorithm for expander codes. More precisely, let r > 0 be any integer. Given an inner code C0 of length d, and a d-regular bipartite expander graph G with n vertices on each side, we give an algorithm to list-decode the code C = C(G, C0) of length nd from approximately δ δr nd erasures in time n ⋅ poly (d2r/δ), where δ and δr are the relative distance and the r’th generalized relative distance of C0, respectively. To the best of our knowledge, this is the first linear-time algorithm that can list-decode expander codes from erasures beyond their (designed) distance of approximately δ2nd. To obtain our results, we show that an approach similar to that of (Hemenway and Wootters, Information and Computation, 2018) can be used to obtain such an erasure-list-decoding algorithm with an exponentially worse dependence of the running time on r and δ; then we show how to improve the dependence of the running time on these parameters. [ABSTRACT FROM AUTHOR]

Motivated by applications such as discovering strong ties in social networks and assembling genome subsequences in biology, we study the problem of recovering a hidden 2k-nearest neighbor (NN) graph in an n-vertex complete graph, whose edge weights are independent and distributed according to Pn for edges in the hidden 2k-NN graph and Qn otherwise. The special case of Bernoulli distributions corresponds to a variant of the Watts-Strogatz small-world graph. We focus on two types of asymptotic recovery guarantees as n → ∞ : (1) exact recovery: all edges are classified correctly with probability tending to one; (2) almost exact recovery: the expected number of misclassified edges is o(nk). We show that the maximum likelihood estimator achieves (1) exact recovery for 2 ≤ k ≤ no(1) if lim inf 2αn/log n > 1; (2) almost exact recovery for 1 ≤ k ≤ o (log n/log log n) if lim inf kD(Pn||Qn)/log n > 1, where αn ≜ −2 log ∫ √d Pn d Qn is the Rényi divergence of order 1/2 and D(Pn||Qn) is the Kullback-Leibler divergence. Under mild distributional assumptions, these conditions are shown to be information-theoretically necessary for any algorithm to succeed. A key challenge in the analysis is the enumeration of 2k-NN graphs that differ from the hidden one by a given number of edges. We also analyze several computationally efficient algorithms and provide sufficient conditions under which they achieve exact/almost exact recovery. In particular, we develop a polynomial-time algorithm that attains the threshold for exact recovery under the small-world model. [ABSTRACT FROM AUTHOR]

We speed up existing decoding algorithms for three code classes in different metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved Gabidulin codes in the subspace metric, and linearized Reed–Solomon codes in the sum-rank metric. The speed-ups are achieved by new algorithms that reduce the cores of the underlying computational problems of the decoders to one common tool: computing left and right approximant bases of matrices over skew polynomial rings. To accomplish this, we describe a skew-analogue of the existing PM-Basis algorithm for matrices over ordinary polynomials. This captures the bulk of the work in multiplication of skew polynomials, and the complexity benefit comes from existing algorithms performing this faster than in classical quadratic complexity. The new algorithms for the various decoding-related computational problems are interesting in their own and have further applications, in particular parts of decoders of several other codes and foundational problems related to the remainder-evaluation of skew polynomials. [ABSTRACT FROM AUTHOR]