Chains are vector-valued signals sampling a curve. They are important to motion signal processing and to many scientific applications including location sensors. We propose a novel measure of smoothness for chains curves by generalizing the scalar-valued concept of monotonicity. Monotonicity can be defined by the connectedness of the inverse image of balls. This definition is coordinate-invariant and can be computed efficiently over chains. Monotone curves can be discontinuous, but continuous monotone curves are differentiable a.e. Over chains, a simple sphere-preserving filter shown to never decrease the degree of monotonicity. It outperforms moving average filters over a synthetic data set. Applications include Time Series Segmentation, chain reconstruction from unordered data points, Optical Character Recognition, and Pattern Matching.
@cite_4 @cite_14 @cite_10 , the authors chose to define monotonicity for curves or chains with an arbitrary direction vector: a curve is monotone if its projection on a line is does not backtrack. While this is a sensible choice given the lack of definition elsewhere, we argue that not all applications support an arbitrary direction that can be used to define monotonicity.
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Monotonicity in concurrent systems stipulates that, in any global state, extant system actions remain executable when new processes are added to the state. This concept is not only natural and common in multi-threaded software, but also useful: if every thread’s memory is finite, monotonicity often guarantees the decidability of safety property verification even when the number of running threads is unknown. In this paper, we show that the act of obtaining finite-data thread abstractions for model checking can be at odds with monotonicity: Predicate-abstracting certain widely used monotone software results in non-monotone multi-threaded Boolean programs — the monotonicity is lost in the abstraction. As a result, well-established sound and complete safety checking algorithms become inapplicable; in fact, safety checking turns out to be undecidable for the obtained class of unbounded-thread Boolean programs. We demonstrate how the abstract programs can be modified into monotone ones, without affecting safety properties of the non-monotone abstraction. This significantly improves earlier approaches of enforcing monotonicity via overapproximations. Maximal monotone operators are set-valued mappings which extend (but are not limited to) the notion of subdifferential of a convex function. The proximal point algorithm is a method for finding a zero of a maximal monotone operator. The algorithm consists in fixed point iterations of a mapping called the resolvent which depends on the maximal monotone operator of interest. The paper investigates a stochastic version of the algorithm where the resolvent used at iteration k is associated to one realization of a random maximal monotone operator. We establish the almost sure ergodic convergence of the iterates to a zero of the expectation (in the Aumann sense) of the latter random operator. Application to constrained stochastic optimization is considered. In a companion paper we described what intuitively would seem to be the most general possible way to generate Coarsening at Random mechanisms, a sequential procedure called randomized monotone coarsening. Counter-examples showed that CAR mechanisms exist which cannot be represented in this way. Here, we further develop these results in two directions. Firstly, we consider what happens when data is coarsened at random in two or more phases. We show that the resulting coarsening mechanism is not CAR anymore, but under suitable assumptions is identified and can provide interesting alternative analysis of data under a non-CAR model. Secondly, we look at sequential mechanisms for generating MAR data, missing components of a multivariate random vector. Randomised monotone missingness schemes, in which one variable at a time is observed and depending on its value, another variable is chosen or the procedure is terminated, supply in our opinion the broadest class of physically interpretable MAR mechanisms. We show that every randomised monotone missingness scheme can be represented by a Markov monotone missingness scheme, in which the choice of which variable to observe next only depends on the set of previously observed variables and their values, not on the sequence in which they were measured. We also show that MAR mechanisms exist which cannot be represented sequentially. We show a directed and robust analogue of a boolean isoperimetric-type theorem of Talagrand [Geom. Funct. Anal., 3 (1993), pp. 295--314]. As an application, we give a monotonicity testing algorithm that makes @math nonadaptive queries to a function @math , always accepts a monotone function, and rejects a function that is @math -far from being monotone with constant probability. Monotonic surfaces spanning finite regions of Zd arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. We explore how we can sample these surfaces when the distribution is biased to favor higher surfaces. We show that a natural local chain is rapidly mixing with any bias for regions in Z2, and for bias λ > d2 in Zd, when d > 2. Moreover, our bounds on the mixing time are optimal on d-dimensional hyper-cubic regions. The proof uses a geometric distance function and introduces a variant of path coupling in order to handle distances that are exponentially large. This paper is motivated by two applications, namely i) generalizations of cuckoo hashing, a computationally simple approach to assigning keys to objects, and ii) load balancing in content distribution networks, where one is interested in determining the impact of content replication on performance. These two problems admit a common abstraction: in both scenarios, performance is characterized by the maximum weight of a generalization of a matching in a bipartite graph, featuring node and edge capacities. Our main result is a law of large numbers characterizing the asymptotic maximum weight matching in the limit of large bipartite random graphs, when the graphs admit a local weak limit that is a tree. This result specializes to the two application scenarios, yielding new results in both contexts. In contrast with previous results, the key novelty is the ability to handle edge capacities with arbitrary integer values. An analysis of belief propagation algorithms (BP) with multivariate belief vectors underlies the proof. In particular, we show convergence of the corresponding BP by exploiting monotonicity of the belief vectors with respect to the so-called upshifted likelihood ratio stochastic order. This auxiliary result can be of independent interest, providing a new set of structural conditions which ensure convergence of BP. In this paper we investigate the monotonicity of various optimality criteria during the exploration phase of an active SLAM algorithm. Optimality criteria such as A-opt, D-opt or E-opt are used in active SLAM to account for uncertainty in the map or the robot's pose, and these criteria are usually part of utility functions which help active SLAM algorithms decide where the robot should move next. The monotonicity of the optimality criteria is of utmost importance. During the exploration phase, i.e. when the robot is traversing new territory or cannot perform a loop closure, the most common way of estimating the pose of the robot is through dead-reckoning. Correctly accounting for the uncertainty is important for an active SLAM algorithm and in particular for a dead-reckoning scenario, where by definition the uncertainty in the robot's pose grows. If monotonicity does not hold in this scenario, active SLAM algorithms can execute actions under the false belief that the uncertainty has reduced. We show analytically and experimentally some conditions in which the A-opt and E-opt criteria lose monotonicity in a dead-reckoning scenario, where the propagation of the robot's pose is done using a linearized framework. We also show analytically and experimentally that under the same conditions the D-opt does not lose monotonicity and, in general for the linearized framework under consideration, D-opt does not break monotonicity. We show a directed and robust analogue of a boolean isoperimetric type theorem of Talagrand. As an application, we give a monotonicity testing algorithm that makes, up to polylog factors, (square root of n) a#x03B5;2 non-adaptive queries to a function f:0, 1n -> 0, 1, always accepts a monotone function and rejects a function that is a#x03B5;-far from being monotone with constant probability. In this paper we study several monotonicity axioms in approval-based multi-winner voting rules. We consider monotonicity with respect to the support received by the winners and also monotonicity in the size of the committee. Monotonicity with respect to the support is studied when the set of voters does not change and when new voters enter the election. For each of these two cases we consider a strong and a weak version of the axiom. We observe certain incompatibilities between the monotonicity axioms and well-known representation axioms (extended proportional justified representation) for the voting rules that we analyze and provide formal proofs of incompatibility between some of these axioms and perfect representation. We discuss a Continuous Curvelet Transform (CCT), a transform f → Γf (a, b, θ) of functions f(x1, x2) on R^2, into a transform domain with continuous scale a > 0, location b ∈ R^2, and orientation θ ∈ [0, 2π). The transform is defined by Γf (a, b, θ) = f, γabθ where the inner products project f onto analyzing elements called curvelets γ_(abθ) which are smooth and of rapid decay away from an a by √a rectangle with minor axis pointing in direction θ. We call them curvelets because this anisotropic behavior allows them to ‘track’ the behavior of singularities along curves. They are continuum scale space orientation analogs of the discrete frame of curvelets discussed in Candes and Donoho (2002). We use the CCT to analyze several objects having singularities at points, along lines, and along smooth curves. These examples show that for fixed (x0, θ0), Γf (a, x0, θ0) decays rapidly as a → 0 if f is smooth near x0, or if the singularity of f at x0 is oriented in a different direction than θ_0. Generalizing these examples, we state general theorems showing that decay properties of Γf (a, x0, θ0) for fixed (x0, θ0), as a → 0 can precisely identify the wavefront set and the H^m- wavefront set of a distribution. In effect, the wavefront set of a distribution is the closure of the set of (x0, θ0) near which Γf (a, x, θ) is not of rapid decay as a → 0; the H^m-wavefront set is the closure of those points (x0, θ0) where the ‘directional parabolic square function’ S^m(x, θ) = ( ʃ|Γf (a, x, θ)|^2 ^(da) _a^3+^(2m))^(1 2) is not locally integrable. The CCT is closely related to a continuous transform used by Hart Smith in his study of Fourier Integral Operators. Smith’s transform is based on strict affine parabolic scaling of a single mother wavelet, while for the transform we discuss, the generating wavelet changes (slightly) scale by scale. The CCT can also be compared to the FBI (Fourier-Bros-Iagolnitzer) and Wave Packets (Cordoba-Fefferman) transforms. We describe their similarities and differences in resolving the wavefront set. We consider the problem of monotonicity testing over graph products. Monotonicity testing is one of the central problems studied in the field of property testing. We present a testing approach that enables us to use known monotonicity testers for given graphs G1, G2, to test monotonicity over their product G1 × G2. Such an approach of reducing monotonicity testing over a graph product to monotonicity testing over the original graphs, has been previously used in the special case of monotonicity testing over [n]d for a limited type of testers; in this article, we show that this approach can be applied to allow modular design of testers in many interesting cases: this approach works whenever the functions are boolean, and also in certain cases for functions with a general range. We demonstrate the usefulness of our results by showing how a careful use of this approach improves the query complexity of known testers. Specifically, based on our results, we provide a new analysis for the known tester for [n]d which significantly improves its query complexity analysis in the low-dimensional case. For example, when d = O(1), we reduce the best known query complexity from O(log 2n-e) to O(log n-e). © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008 An extended abstract of this paper appeared in ICALP 2004 [23]. This paper provides sufficient conditions for the optimal value in a discrete-time, finite, partially observed Markov decision process to be monotone on the space of state probability vectors ordered by likelihood ratios. The paper also presents sufficient conditions for the optimal policy to be monotone in a simple machine replacement problem, and, in the general case, for the optimal policy to be bounded from below by an easily calculated monotone function. We consider monotonicity testing of functions f:[n]d→ 0,1 , in the property testing framework of Rubinfeld and Sudan [23] and Goldreich, Goldwasser and Ron [14]. Specifically, we consider the framework of distribution-free property testing, where the distance between functions is measured with respect to a fixed but unknown distribution D on the domain and the testing algorithms have an oracle access to random sampling from the domain according to this distribution D. We show that, though in the uniform distribution case, testing of boolean functions defined over the boolean hypercube can be done using query complexity that is polynomial in @math and in the dimension d, in the distribution-free setting such testing requires a number of queries that is exponential in d. Therefore, in the high-dimensional case (in oppose to the low-dimensional case), the gap between the query complexity for the uniform and the distribution-free settings is exponential. We prove existence, uniqueness, duality results and give a characterization of optimal measure-preserving maps for a class of optimal transportation problems with several marginals with compact support in R under the requirement that the cost function satisfles a so-called monotonicity of order 2 condition. Explicit formulas for the minimizers are given and links with some rearrangement inequalities are sketched. We consider Glauber dynamics (starting from an extremal configuration) in a monotone spin system, and show that interjecting extra updates cannot increase the expected Hamming distance or the total variation distance to the stationary distribution. We deduce that for monotone Markov random fields, when block dynamics contracts a Hamming metric, single-site dynamics mixes in O(n log n) steps on an n-vertex graph. In particular, our result completes work of Kenyon, Mossel and Peres concerning Glauber dynamics for the Ising model on trees. Our approach also shows that on bipartite graphs, alternating updates systematically between odd and even vertices cannot improve the mixing time by more than a factor of log n compared to updates at uniform random locations on an n-vertex graph. Our result is especially effective in comparing block and single-site dynamics; it has already been used in works of Martinelli, Toninelli, Sinclair, Mossel, Sly, Ding, Lubetzky, and Peres in various combinations. Abstract This work proves the existence and multiplicity results of monotonic traveling wave solutions for some lattice differential equations by using the monotone iteration method. Our results include the model of cellular neural networks (CNN). In addition to the monotonic traveling wave solutions, non-monotonic and oscillating traveling wave solutions in the delay type of CNN are also obtained. In this paper, we study the problem of maximizing continuous submodular functions that naturally arise in many learning applications such as those involving utility functions in active learning and sensing, matrix approximations and network inference. Despite the apparent lack of convexity in such functions, we prove that stochastic projected gradient methods can provide strong approximation guarantees for maximizing continuous submodular functions with convex constraints. More specifically, we prove that for monotone continuous DR-submodular functions, all fixed points of projected gradient ascent provide a factor @math approximation to the global maxima. We also study stochastic gradient methods and show that after @math iterations these methods reach solutions which achieve in expectation objective values exceeding @math . An immediate application of our results is to maximize submodular functions that are defined stochastically, i.e. the submodular function is defined as an expectation over a family of submodular functions with an unknown distribution. We will show how stochastic gradient methods are naturally well-suited for this setting, leading to a factor @math approximation when the function is monotone. In particular, it allows us to approximately maximize discrete, monotone submodular optimization problems via projected gradient ascent on a continuous relaxation, directly connecting the discrete and continuous domains. Finally, experiments on real data demonstrate that our projected gradient methods consistently achieve the best utility compared to other continuous baselines while remaining competitive in terms of computational effort. We present improved algorithms for testing monotonicity of functions. Namely, given the ability to query an unknown function f: Σ n ↦ Ξ, where Σ and Ξ are finite ordered sets, the test always accepts a monotone f, and rejects f with high probability if it is e-far from being monotone (i.e., every monotone function differs from f on more than an e fraction of the domain). For any e > 0, the query complexity of the test is O((n e) · log ∣Σ ∣ · log ∣Ξ∣). The previous best known bound was ( O ((n^2 ) ^2 ) ). We consider the maximal displacement of one dimensional branching Brownian motion with (macroscopically) time varying profiles. For monotone decreasing variances, we show that the correction from linear displacement is not logarithmic but rather proportional to T 1 3. We conjecture that this is the worse case correction possible. The primary problem in property testing is to decide whether a given function satisfies a certain property or is far from any function satisfying it. This crucially requires a notion of distance between functions. The most prevalent notion is the Hamming distance over the uniform distribution on the domain. This restriction to uniformity is rather limiting, and it is important to investigate distances induced by more general distributions. In this article, we provide simple and optimal testers for bounded derivative properties over arbitrary product distributions. Bounded derivative properties include fundamental properties, such as monotonicity and Lipschitz continuity. Our results subsume almost all known results (upper and lower bounds) on monotonicity and Lipschitz testing over arbitrary ranges. We prove an intimate connection between bounded derivative property testing and binary search trees (BSTs). We exhibit a tester whose query complexity is the sum of expected depths of optimal BSTs for each marginal. Furthermore, we show that this sum-of-depths is also a lower bound. A technical contribution of our work is an optimal dimension reduction theorem for all bounded derivative properties that relates the distance of a function from the property to the distance of restrictions of the function to random lines. Such a theorem has been elusive even for monotonicity, and our theorem is an exponential improvement to the previous best-known result.