OpenRaiser/Intern-Atlas huggingface dataset:
Revisiting Asynchronous Linear Solvers: Provable Convergence Rate Through Randomization
Venue: arXiv cs.DC 2015
Year: 2015

Asynchronous methods for solving systems of linear equations have been researched since Chazan and Miranker's pioneering 1969 paper on chaotic relaxation. The underlying idea of asynchronous methods is to avoid processor idle time by allowing the processors to continue to make progress even if not all progress made by other processors has been communicated to them. Historically, the applicability of asynchronous methods for solving linear equations was limited to certain restricted classes of matrices, such as diagonally dominant matrices. Furthermore, analysis of these methods focused on proving convergence in the limit. Comparison of the asynchronous convergence rate with its synchronous counterpart and its scaling with the number of processors were seldom studied, and are still not well understood. In this paper, we propose a randomized shared-memory asynchronous method for general symmetric positive definite matrices. We rigorously analyze the convergence rate and prove that it is linear, and is close to that of the method's synchronous counterpart if the processor count is not excessive relative to the size and sparsity of the matrix. We also present an algorithm for unsymmetric systems and overdetermined least-squares. Our work presents a significant improvement in the applicability of asynchronous linear solvers as well as in their convergence analysis, and suggests randomization as a key paradigm to serve as a foundation for asynchronous methods.
Interval exchanges, admissibility and branching Rauzy induction
Venue: arXiv cs.DM 2015
Year: 2015

We introduce a definition of admissibility for subintervals in interval exchange transformations. Using this notion, we prove a property of the natural codings of interval exchange transformations, namely that any derived set of a regular interval exchange set is a regular interval exchange set with the same number of intervals. Derivation is taken here with respect to return words. We characterize the admissible intervals using a branching version of the Rauzy induction. We also study the case of regular interval exchange transformations defined over a quadratic field and show that the set of factors of such a transformation is primitive morphic. The proof uses an extension of a result of Boshernitzan and Carroll.
Multi-Sided Boundary Labeling
Venue: arXiv cs.CG 2015
Year: 2015

In the Boundary Labeling problem, we are given a set of $n$ points, referred to as sites, inside an axis-parallel rectangle $R$, and a set of $n$ pairwise disjoint rectangular labels that are attached to $R$ from the outside. The task is to connect the sites to the labels by non-intersecting rectilinear paths, so-called leaders, with at most one bend. In this paper, we study the Multi-Sided Boundary Labeling problem, with labels lying on at least two sides of the enclosing rectangle. We present a polynomial-time algorithm that computes a crossing-free leader layout if one exists. So far, such an algorithm has only been known for the cases in which labels lie on one side or on two opposite sides of $R$ (here a crossing-free solution always exists). The case where labels may lie on adjacent sides is more difficult. We present efficient algorithms for testing the existence of a crossing-free leader layout that labels all sites and also for maximizing the number of labeled sites in a crossing-free leader layout. For two-sided boundary labeling with adjacent sides, we further show how to minimize the total leader length in a crossing-free layout.
Communicating over Filter-and-Forward Relay Networks with Channel Output Feedback
Venue: arXiv cs.IT 2015
Year: 2015

Relay networks aid in increasing the rate of communication from source to destination. However, the capacity of even a three-terminal relay channel is an open problem. In this work, we propose a new lower bound for the capacity of the three-terminal relay channel with destination-to-source feedback in the presence of correlated noise. Our lower bound improves on the existing bounds in the literature. We then extend our lower bound to general relay network configurations using an arbitrary number of filter-and-forward relay nodes. Such network configurations are common in many multi-hop communication systems where the intermediate nodes can only perform minimal processing due to limited computational power. Simulation results show that significant improvements in the achievable rate can be obtained through our approach. We next derive a coding strategy (optimized using post processed signal-to-noise ratio as a criterion) for the three-terminal relay channel with noisy channel output feedback for two transmissions. This coding scheme can be used in conjunction with open-loop codes for applications like automatic repeat request (ARQ) or hybrid-ARQ.
Rules with parameters in modal logic I
Venue: arXiv cs.LO 2015
Year: 2015

We study admissibility of inference rules and unification with parameters in transitive modal logics (extensions of K4), in particular we generalize various results on parameter-free admissibility and unification to the setting with parameters. Specifically, we give a characterization of projective formulas generalizing Ghilardi's characterization in the parameter-free case, leading to new proofs of Rybakov's results that admissibility with parameters is decidable and unification is finitary for logics satisfying suitable frame extension properties (called cluster-extensible logics in this paper). We construct explicit bases of admissible rules with parameters for cluster-extensible logics, and give their semantic description. We show that in the case of finitely many parameters, these logics have independent bases of admissible rules, and determine which logics have finite bases. As a sideline, we show that cluster-extensible logics have various nice properties: in particular, they are finitely axiomatizable, and have an exponential-size model property. We also give a rather general characterization of logics with directed (filtering) unification. In the sequel, we will use the same machinery to investigate the computational complexity of admissibility and unification with parameters in cluster-extensible logics, and we will adapt the results to logics with unique top cluster (e.g., S4.2) and superintuitionistic logics.
Asymptotic Collusion-Proofness of Voting Rules: The Case of Large Number of Candidates
Venue: arXiv cs.GT 2015
Year: 2015

Classical results in voting theory show that strategic manipulation by voters is inevitable if a voting rule simultaneously satisfy certain desirable properties. Motivated by this, we study the relevant question of how often a voting rule is manipulable. It is well known that elections with a large number of voters are rarely manipulable under impartial culture (IC) assumption. However, the manipulability of voting rules when the number of candidates is large has hardly been addressed in the literature and our paper focuses on this problem. First, we propose two properties (1) asymptotic strategy-proofness and (2) asymptotic collusion-proofness, with respect to new voters, which makes the two notions more relevant from the perspective of computational problem of manipulation. In addition to IC, we explore a new culture of society where all score vectors of the candidates are equally likely. This new notion has its motivation in computational social choice and we call it impartial scores culture (ISC) assumption. We study asymptotic strategy-proofness and asymptotic collusion-proofness for plurality, veto, $k$-approval, and Borda voting rules under IC as well as ISC assumptions. Specifically, we prove bounds for the fraction of manipulable profiles when the number of candidates is large. Our results show that the size of the coalition and the tie-breaking rule play a crucial role in determining whether or not a voting rule satisfies the above two properties.
Higher-order Segmentation via Multicuts
Venue: arXiv cs.CV 2015
Year: 2015

Multicuts enable to conveniently represent discrete graphical models for unsupervised and supervised image segmentation, in the case of local energy functions that exhibit symmetries. The basic Potts model and natural extensions thereof to higher-order models provide a prominent class of such objectives, that cover a broad range of segmentation problems relevant to image analysis and computer vision. We exhibit a way to systematically take into account such higher-order terms for computational inference. Furthermore, we present results of a comprehensive and competitive numerical evaluation of a variety of dedicated cutting-plane algorithms. Our approach enables the globally optimal evaluation of a significant subset of these models, without compromising runtime. Polynomially solvable relaxations are studied as well, along with advanced rounding schemes for post-processing.
Sofic-Dyck shifts
Venue: arXiv cs.FL 2015
Year: 2015

We define the class of sofic-Dyck shifts which extends the class of Markov-Dyck shifts introduced by Inoue, Krieger and Matsumoto. Sofic-Dyck shifts are shifts of sequences whose finite factors form unambiguous context-free languages. We show that they correspond exactly to the class of shifts of sequences whose sets of factors are visibly pushdown languages. We give an expression of the zeta function of a sofic-Dyck shift.
A Generalized Channel Coding Theory for Distributed Communication
Venue: arXiv cs.IT 2015
Year: 2015

This paper presents generalized channel coding theorems for a time-slotted distributed communication system where a transmitter-receiver pair is communicating in parallel with other transmitters. Assume that the channel code of each transmitter is chosen arbitrarily in each time slot. The coding choice of a transmitter is denoted by a code index parameter, which is known neither to other transmitters nor to the receiver. Fundamental performance limitation of the system is characterized using an achievable region defined in the space of the code index vectors. As the codeword length is taken to infinity, for all code index vectors inside the region, the receiver will decode the message reliably, while for all code index vectors outside the region, the receiver will report a collision reliably. A generalized system error performance measure is defined as the weighted sum of probabilities of different types of communication error events. Assume that the receiver chooses an "operation region" and intends to decode the message if the code index vector is inside the operation region. Achievable bounds on the tradeoff between the operation region and the generalize error performance measure are obtained under the assumption of a finite codeword length.
Sparse Auto-Regressive: Robust Estimation of AR Parameters
Venue: arXiv cs.AI 2015
Year: 2015

In this paper I present a new approach for regression of time series using their own samples. This is a celebrated problem known as Auto-Regression. Dealing with outlier or missed samples in a time series makes the problem of estimation difficult, so it should be robust against them. Moreover for coding purposes I will show that it is desired the residual of auto-regression be sparse. To these aims, I first assume a multivariate Gaussian prior on the residual and then obtain the estimation. Two simple simulations have been done on spectrum estimation and speech coding.
Upper-bounding $\ell_1$-optimization sectional thresholds
Venue: arXiv cs.IT 2015
Year: 2015

In this paper we look at a particular problem related to under-determined linear systems of equations with sparse solutions. $\ell_1$-minimization is a fairly successful polynomial technique that can in certain statistical scenarios find sparse enough solutions of such systems. Barriers of $\ell_1$ performance are typically referred to as its thresholds. Depending if one is interested in a typical or worst case behavior one then distinguishes between the \emph{weak} thresholds that relate to a typical behavior on one side and the \emph{sectional} and \emph{strong} thresholds that relate to the worst case behavior on the other side. Starting with seminal works \cite{CRT,DonohoPol,DOnoho06CS} a substantial progress has been achieved in theoretical characterization of $\ell_1$-minimization statistical thresholds. More precisely, \cite{CRT,DOnoho06CS} presented for the first time linear lower bounds on all of these thresholds. Donoho's work \cite{DonohoPol} (and our own \cite{StojnicCSetam09,StojnicUpper10}) went a bit further and essentially settled the $\ell_1$'s \emph{weak} thresholds. At the same time they also provided fairly good lower bounds on the values on the \emph{sectional} and \emph{strong} thresholds. In this paper, we revisit the \emph{sectional} thresholds and present a simple mechanism that can be used to create solid upper bounds as well. The method we present relies on a seemingly simple but substantial progress we made in studying Hopfield models in \cite{StojnicHopBnds10}.
SL2 homomorphic hash functions: Worst case to average case reduction and short collision search
Venue: arXiv cs.CR 2015
Year: 2015

We study homomorphic hash functions into SL(2,q), the 2x2 matrices with determinant 1 over the field with $q$ elements. Modulo a well supported number theoretic hypothesis, which holds in particular for concrete homomorphisms proposed thus far, we provide a worst case to average case reduction for these hash functions: upto a logarithmic factor, a random homomorphism is as secure as _any_ concrete homomorphism. For a family of homomorphisms containing several concrete proposals in the literature, we prove that collisions of length O(log(q)) can be found in running time O(sqrt(q)). For general homomorphisms we offer an algorithm that, heuristically and according to experiments, in running time O(sqrt(q)) finds collisions of length O(log(q)) for q even, and length O(log^2(q)/loglog(q))$ for arbitrary q. While exponetial time, our algorithms are faster in practice than all earlier generic algorithms, and produce much shorter collisions.
Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas
Venue: arXiv cs.DS 2015
Year: 2015

We investigate the approximability of several classes of real-valued functions by functions of a small number of variables ({\em juntas}). Our main results are tight bounds on the number of variables required to approximate a function $f:\{0,1\}^n \rightarrow [0,1]$ within $\ell_2$-error $ε$ over the uniform distribution: 1. If $f$ is submodular, then it is $ε$-close to a function of $O(\frac{1}{ε^2} \log \frac{1}ε)$ variables. This is an exponential improvement over previously known results. We note that $Ω(\frac{1}{ε^2})$ variables are necessary even for linear functions. 2. If $f$ is fractionally subadditive (XOS) it is $ε$-close to a function of $2^{O(1/ε^2)}$ variables. This result holds for all functions with low total $\ell_1$-influence and is a real-valued analogue of Friedgut's theorem for boolean functions. We show that $2^{Ω(1/ε)}$ variables are necessary even for XOS functions. As applications of these results, we provide learning algorithms over the uniform distribution. For XOS functions, we give a PAC learning algorithm that runs in time $2^{poly(1/ε)} poly(n)$. For submodular functions we give an algorithm in the more demanding PMAC learning model (Balcan and Harvey, 2011) which requires a multiplicative $1+γ$ factor approximation with probability at least $1-ε$ over the target distribution. Our uniform distribution algorithm runs in time $2^{poly(1/(γε))} poly(n)$. This is the first algorithm in the PMAC model that over the uniform distribution can achieve a constant approximation factor arbitrarily close to 1 for all submodular functions. As follows from the lower bounds in (Feldman et al., 2013) both of these algorithms are close to optimal. We also give applications for proper learning, testing and agnostic learning with value queries of these classes.
Polynomial-Time Algorithms for Quadratic Isomorphism of Polynomials: The Regular Case
Venue: arXiv cs.SC 2015
Year: 2015

Let $\mathbf{f}=(f\_1,\ldots,f\_m)$ and $\mathbf{g}=(g\_1,\ldots,g\_m)$ be two sets of $m\geq 1$ nonlinear polynomials over $\mathbb{K}[x\_1,\ldots,x\_n]$ ($\mathbb{K}$ being a field). We consider the computational problem of finding -- if any -- an invertible transformation on the variables mapping $\mathbf{f}$ to $\mathbf{g}$. The corresponding equivalence problem is known as {\tt Isomorphism of Polynomials with one Secret} ({\tt IP1S}) and is a fundamental problem in multivariate cryptography. The main result is a randomized polynomial-time algorithm for solving {\tt IP1S} for quadratic instances, a particular case of importance in cryptography and somewhat justifying {\it a posteriori} the fact that {\it Graph Isomorphism} reduces to only cubic instances of {\tt IP1S} (Agrawal and Saxena). To this end, we show that {\tt IP1S} for quadratic polynomials can be reduced to a variant of the classical module isomorphism problem in representation theory, which involves to test the orthogonal simultaneous conjugacy of symmetric matrices. We show that we can essentially {\it linearize} the problem by reducing quadratic-{\tt IP1S} to test the orthogonal simultaneous similarity of symmetric matrices; this latter problem was shown by Chistov, Ivanyos and Karpinski to be equivalent to finding an invertible matrix in the linear space $\mathbb{K}^{n \times n}$ of $n \times n$ matrices over $\mathbb{K}$ and to compute the square root in a matrix algebra. While computing square roots of matrices can be done efficiently using numerical methods, it seems difficult to control the bit complexity of such methods. However, we present exact and polynomial-time algorithms for computing the square root in $\mathbb{K}^{n \times n}$ for various fields (including finite fields). We then consider \\#{\tt IP1S}, the counting version of {\tt IP1S} for quadratic instances. In particular, we provide a (complete) characterization of the automorphism group of homogeneous quadratic polynomials. Finally, we also consider the more general {\it Isomorphism of Polynomials} ({\tt IP}) problem where we allow an invertible linear transformation on the variables \emph{and} on the set of polynomials. A randomized polynomial-time algorithm for solving {\tt IP} when \(\mathbf{f}=(x\_1^d,\ldots,x\_n^d)\) is presented. From an algorithmic point of view, the problem boils down to factoring the determinant of a linear matrix (\emph{i.e.}\ a matrix whose components are linear polynomials). This extends to {\tt IP} a result of Kayal obtained for {\tt PolyProj}.
A Near-Optimal Dynamic Learning Algorithm for Online Matching Problems with Concave Returns
Venue: arXiv cs.DS 2015
Year: 2015

We consider an online matching problem with concave returns. This problem is a significant generalization of the Adwords allocation problem and has vast applications in online advertising. In this problem, a sequence of items arrive sequentially and each has to be allocated to one of the bidders, who bid a certain value for each item. At each time, the decision maker has to allocate the current item to one of the bidders without knowing the future bids and the objective is to maximize the sum of some concave functions of each bidder's aggregate value. In this work, we propose an algorithm that achieves near-optimal performance for this problem when the bids arrive in a random order and the input data satisfies certain conditions. The key idea of our algorithm is to learn the input data pattern dynamically: we solve a sequence of carefully chosen partial allocation problems and use their optimal solutions to assist with the future decision. Our analysis belongs to the primal-dual paradigm, however, the absence of linearity of the objective function and the dynamic feature of the algorithm makes our analysis quite unique.
On active and passive testing
Venue: arXiv cs.DS 2015
Year: 2015

Given a property of Boolean functions, what is the minimum number of queries required to determine with high probability if an input function satisfies this property or is "far" from satisfying it? This is a fundamental question in Property Testing, where traditionally the testing algorithm is allowed to pick its queries among the entire set of inputs. Balcan, Blais, Blum and Yang have recently suggested to restrict the tester to take its queries from a smaller random subset of polynomial size of the inputs. This model is called active testing, and in the extreme case when the size of the set we can query from is exactly the number of queries performed it is known as passive testing. We prove that passive or active testing of k-linear functions (that is, sums of k variables among n over Z_2) requires Theta(k*log n) queries, assuming k is not too large. This extends the case k=1, (that is, dictator functions), analyzed by Balcan et. al. We also consider other classes of functions including low degree polynomials, juntas, and partially symmetric functions. Our methods combine algebraic, combinatorial, and probabilistic techniques, including the Talagrand concentration inequality and the Erdos--Rado theorem on Delta-systems.
Multi-agent Systems with Compasses
Venue: arXiv cs.MA 2015
Year: 2015

This paper investigates agreement protocols over cooperative and cooperative--antagonistic multi-agent networks with coupled continuous-time nonlinear dynamics. To guarantee convergence for such systems, it is common in the literature to assume that the vector field of each agent is pointing inside the convex hull formed by the states of the agent and its neighbors, given that the relative states between each agent and its neighbors are available. This convexity condition is relaxed in this paper, as we show that it is enough that the vector field belongs to a strict tangent cone based on a local supporting hyperrectangle. The new condition has the natural physical interpretation of requiring shared reference directions in addition to the available local relative states. Such shared reference directions can be further interpreted as if each agent holds a magnetic compass indicating the orientations of a global frame. It is proven that the cooperative multi-agent system achieves exponential state agreement if and only if the time-varying interaction graph is uniformly jointly quasi-strongly connected. Cooperative--antagonistic multi-agent systems are also considered. For these systems, the relation has a negative sign for arcs corresponding to antagonistic interactions. State agreement may not be achieved, but instead it is shown that all the agents' states asymptotically converge, and their limits agree componentwise in absolute values if and in general only if the time-varying interaction graph is uniformly jointly strongly connected.
Bounded-Choice Statements for User Interaction in Imperative and Object-Oriented Programming
Venue: arXiv cs.PL 2015
Year: 2015

Adding versatile interactions to imperative programming -- C, Java and Android -- is an essential task. Unfortunately, existing languages provide only limited constructs for user interaction. These constructs are usually in the form of $unbounded$ quantification. For example, existing languages can take the keyboard input from the user only via the $read(x)/scan(x)$ construct. Note that the value of $x$ is unbounded in the sense that $x$ can have any value. This construct is thus not useful for applications with bounded inputs. To support bounded choices, we propose new bounded-choice statements for user interation. Each input device (the keyboard, the mouse, the touch, $...$) naturally requires a new bounded-choice statement. To make things simple, however, we focus on a bounded-choice statement for keyboard -- kchoose -- to allow for more controlled and more guided participation from the user. It is straightforward to adjust our idea to other input devices. We illustrate our idea via Java(BI), an extension of the core Java with a new bounded-choice statement for the keyboard.