Influence maximization, defined as finding a small subset of nodes that maximizes spread of influence in social networks, is NP-hard under both Independent Cascade (IC) and Linear Threshold (LT) models, where many gre...
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Influence maximization, defined as finding a small subset of nodes that maximizes spread of influence in social networks, is NP-hard under both Independent Cascade (IC) and Linear Threshold (LT) models, where many greedy-based algorithms have been proposed with the best approximation guarantee. However, existing greedy-based algorithms are inefficient on large networks, as it demands heavy Monte-Carlo simulations of the spread functions for each node at the initial step [ 7]. In this paper, we establish new upper bounds to significantly reduce the number of Monte-Carlo simulations in greedy-based algorithms, especially at the initial step. We theoretically prove that the bound is tight and convergent when the summation of weights towards (or from) each node is less than 1. Based on the bound, we propose a new Upper Bound based Lazy Forward algorithm (UBLF in short) for discovering the top-k influential nodes in social networks. We test and compare UBLF with prior greedy algorithms, especially CELF [ 30]. Experimental results show that UBLF reduces more than 95 percent Monte-Carlo simulations of CELF and achieves about 2-10 times speedup when the seed set is small.
A purely greedy algorithm and an orthogonal greedy algorithm are studied. It is established that the set of objective functions for which a greedy algorithm can be "realized correctly" has second category fo...
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A purely greedy algorithm and an orthogonal greedy algorithm are studied. It is established that the set of objective functions for which a greedy algorithm can be "realized correctly" has second category for discrete dictionaries.
This paper focuses on sensor scheduling for state estimation, which consists of a network of noisy sensors and a discrete-time linear system with process noise. As an energy constraint, only a subset of sensors can ta...
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This paper focuses on sensor scheduling for state estimation, which consists of a network of noisy sensors and a discrete-time linear system with process noise. As an energy constraint, only a subset of sensors can take a measurement at each time step. These measurements are fused into a common state estimate using a Kalman filter and the goal is to schedule the sensors to minimize the estimation error at a terminal time. A simple approach is to greedily choose sensors at each time step to minimize the estimation error at the next time step. Recent work has shown that this greedy algorithm outperforms other well known approaches. Results have been established to show that the estimation error is a submodular function of the sensor schedule;submodular functions have a diminishing returns property that ensures the greedy algorithm yields near optimal performance. As a negative result, we show that most commonly-used estimation error metrics are not, in general, submodular functions. This disproves an established result. We then provide sufficient conditions on the system for which the estimation error is a submodular function of the sensor schedule, and thus the greedy algorithm yields performance guarantees. (C) 2015 Elsevier Ltd. All rights reserved.
In a separable Hilbert space H, greedy algorithms iteratively define nz-term approximants to a given vector from a complete redundant dictionary D. With very large dictionaries, the pure greedy algorithm cannot be imp...
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In a separable Hilbert space H, greedy algorithms iteratively define nz-term approximants to a given vector from a complete redundant dictionary D. With very large dictionaries, the pure greedy algorithm cannot be implemented and must be replaced with a weak greedy algorithm. In numerical applications, partially greedy algorithms have been introduced to reduce the numerical complexity. A conjecture about their convergence arises naturally from the observation of numerical experiments. We introduce, study and disprove this conjecture. (C) 2001 Academic Press.
For appropriate matrix ensembles, greedy algorithms have proven to be an efficient means of solving the combinatorial optimization problem associated with compressed sensing. This paper describes an implementation for...
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For appropriate matrix ensembles, greedy algorithms have proven to be an efficient means of solving the combinatorial optimization problem associated with compressed sensing. This paper describes an implementation for graphics processing units (GPU) of hard thresholding, iterative hard thresholding, normalized iterative hard thresholding, hard thresholding pursuit, and a two-stage thresholding algorithm based on compressive sampling matching pursuit and subspace pursuit. The GPU acceleration of the former bottleneck, namely the matrix-vector multiplications, transfers a significant portion of the computational burden to the identification of the support set. The software solves high-dimensional problems in fractions of a second which permits large-scale testing at dimensions currently unavailable in the literature. The GPU implementations exhibit up to 70x acceleration over standard Matlab central processing unit implementations using automatic multi-threading.
The finite-impulse-response (FIR) decision feedback equalisers (DFEs) with a large number of taps are used to eliminate the intersymbol interference. In this Letter, a hybrid optimisation approach based on reweighted ...
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The finite-impulse-response (FIR) decision feedback equalisers (DFEs) with a large number of taps are used to eliminate the intersymbol interference. In this Letter, a hybrid optimisation approach based on reweighted l(1)-norm minimisation and the greedy algorithm is proposed to get a better estimation of the non-zero taps. First, the authors transform the problem of designing sparse FIR multiple-input multiple-output DFEs into an l(0)-norm minimisation problem, then use the proposed approach, which involves two stages as the preliminary selection of non-zero tap positions and re-optimisation with non-zero taps, to determine the positions and values of the non-zero taps for the FIR DFEs. The simulation results demonstrate the effectiveness of the proposed hybrid optimisation approach.
We consider some theoretical greedy algorithms for approximation in Banach spaces with respect to a general dictionary. We prove convergence of the algorithms for Banach spaces which satisfy certain smoothness assumpt...
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We consider some theoretical greedy algorithms for approximation in Banach spaces with respect to a general dictionary. We prove convergence of the algorithms for Banach spaces which satisfy certain smoothness assumptions. We compare the algorithms and their rates of convergence when the Banach space is L-p (T-d) (1 < p < infinity) and the dictionary is the trigonometric system.
For the minimization knapsack problem with Boolean variables, primal and dual greedy algorithms are formally described. Their relations to the corresponding algorithms for the maximization knapsack problem are shown. ...
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For the minimization knapsack problem with Boolean variables, primal and dual greedy algorithms are formally described. Their relations to the corresponding algorithms for the maximization knapsack problem are shown. The average behavior of primal and dual algorithms for the minimization problem is analyzed. It is assumed that the coefficients of the objective function and the constraint are independent identically distributed random variables on [ 0, 1] with an arbitrary distribution having a density and that the right-hand side d is deterministic and proportional to the number of variables (i.e., d = mu n). A condition on mu is found under which the primal and dual greedy algorithms have an asymptotic error of t
We study nonlinear m-term approximation with regard to a redundant dictionary D in a Hilbert space H. It is known that the Pure greedy Algorithm ( or, more generally, the Weak greedy Algorithm) provides for each f is ...
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We study nonlinear m-term approximation with regard to a redundant dictionary D in a Hilbert space H. It is known that the Pure greedy Algorithm ( or, more generally, the Weak greedy Algorithm) provides for each f is an element of H and any dictionary D an expansion into a series f = Sigma(infinity)(j=1) c(j)(f)phi(j)(f), phi(j)(f) is an element of D, j = 1,2,... with the Parseval property: parallel to f parallel to(2) = Sigma(j) | c(j) ( f)|(2). Following the paper of A. Lutoborski and the second author we study analogs of the above expansions for a given finite number of functions f(1),..., f(N) with a requirement that the dictionary elements phi(j) of these expansions are the same for all f(i), i = 1,..., N. We study convergence and rate of convergence of such expansions which we call simultaneous expansions.
The paper studies weak greedy algorithms for finding sparse solutions of convex optimization problems in Banach spaces. We consider the concept of duality gap, the values of which are implicitly calculated at the step...
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The paper studies weak greedy algorithms for finding sparse solutions of convex optimization problems in Banach spaces. We consider the concept of duality gap, the values of which are implicitly calculated at the step of choosing the direction of the fastest descent at each iteration of the greedy algorithm. We show that these values give upper bounds for the difference between the values of the objective function in the current state and the optimal point. Since the value of the objective function at the optimal point is not known in advance, the current values of the duality gap can be used, for example, in the stopping criteria for the greedy algorithm. In the paper, we find estimates of the duality gap values depending on the number of iterations for the weak greedy algorithms under consideration.
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