A randomized algorithm is proposed for solving the problem of finding hyper-rectangles, sufficiently approximating the true region in each class. This method yields a suboptimal solution, but is more efficient than pr...
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A randomized algorithm is proposed for solving the problem of finding hyper-rectangles, sufficiently approximating the true region in each class. This method yields a suboptimal solution, but is more efficient than previous methods. The performance is analysed based on a criterion of PAC (Probably Approximately Correct) learning. Experimental results show that the proposed method can solve large problems which were not able to be solved previously.
We analyze a sublinear RAlSFA (randomized algorithm for Sparse Fourier analysis) that finds a near-optimal B-term, Sparse representation R for a given discrete signal S of length N, in time and space poly(B, log(N)), ...
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We analyze a sublinear RAlSFA (randomized algorithm for Sparse Fourier analysis) that finds a near-optimal B-term, Sparse representation R for a given discrete signal S of length N, in time and space poly(B, log(N)), following the approach given in [A.C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, M. Strauss, Near-Optimal Sparse Fourier Representations via Sampling, STOC, 2002]. Its time cost poly(log(N)) should be compared with the superlinear Omega(N logN) time requirement of the Fast Fourier Transform (FFT). A straightforward implementation of the RAlSFA, as presented in the theoretical paper [A.C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, M. Strauss, Near-Optimal Sparse Fourier Representations via Sampling, STOC, 2002], turns out to be very slow in practice. Our main result is a greatly improved and practical RAlSFA. We introduce several new ideas and techniques that speed up the algorithm. Both rigorous and heuristic arguments for parameter choices are presented. Our RAlSFA constructs, with probability at least 1 - delta, a near-optimal B-term representation R in time poly(B) log(N) log(1/delta)/is an element of(2) log(M) such that vertical bar vertical bar S - R vertical bar vertical bar(2)(2) <= (1 + is an element of) vertical bar vertical bar S - R(opt)vertical bar vertical bar(2)(2). Furthermore, this RAlSFA implementation already beats the FFTW for not unreasonably large N. We extend the algorithm to higher dimensional cases both theoretically and numerically. The crossover point lies at N similar or equal to 70, 000 in one dimension, and at N similar or equal to 900 for data on a N x N grid in two dimensions for small B signals where there is noise. (c) 2005 Elsevier Inc. All rights reserved.
Tensors admitting an expression as the sum of at most s rank 1 tensors can be considered points of the sth order secant variety of a Segre variety, yet the most basic invariant of this variety-its dimension-is not yet...
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Tensors admitting an expression as the sum of at most s rank 1 tensors can be considered points of the sth order secant variety of a Segre variety, yet the most basic invariant of this variety-its dimension-is not yet fully understood. A conjecture was nevertheless proposed by Abo et al. (2009, Induction for secant varieties of Segre varieties. Trans. Amer. Math. Soc., 361, 767-792), which was proved to be correct for s6. We propose a numerical randomized algorithm for testing whether a mathematically exact and structured matrix is singular, requiring only the availability of an approximate matrix-vector product. Using this method, we test whether the aforementioned conjecture is true. The proposed method requires several orders of magnitude less memory than approaches based on symbolic arithmetic, thus greatly increasing the number of varieties that can be handled. Our experiments establish that the Segre varieties PC boolean AND{n1} x PC boolean AND{n2} x ... x PC boolean AND{nd} embedded in PC n_1 x ... x n_d with expected generic rank s55 obey the conjecture;the probability that a defective variety is incorrectly classified as nondefective is less than < 10(-55).
We introduce a novel algorithm for approximating the logarithm of the determinant of a symmetric positive definite (SPD) matrix. The algorithm is randomized and approximates the traces of a small number of matrix powe...
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We introduce a novel algorithm for approximating the logarithm of the determinant of a symmetric positive definite (SPD) matrix. The algorithm is randomized and approximates the traces of a small number of matrix powers of a specially constructed matrix, using the method of Avron and Toledo [1]. From a theoretical perspective, we present additive and relative error bounds for our algorithm. Our additive error bound works for any SPD matrix, whereas our relative error bound works for SPD matrices whose eigenvalues lie in the interval (theta(1),1), with 0 < theta(1) < 1;the latter setting was proposed in [16]. From an empirical perspective, we demonstrate that a C++ implementation of our algorithm can approximate the logarithm of the determinant of large matrices very accurately in a matter of seconds. (C) 2017 Elsevier Inc. All rights reserved.
The paper is devoted to demonstrating a randomized algorithm for determining a dominating set in a given graph having a maximum degree of five. The algorithm follows the Las Vegas technique. Furthermore, the concept o...
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The paper is devoted to demonstrating a randomized algorithm for determining a dominating set in a given graph having a maximum degree of five. The algorithm follows the Las Vegas technique. Furthermore, the concept of a 2-separated collection of subsets of vertices in graphs is used. The suggested algorithm is based on a condition of the upper bound of the cardinality of a local dominating set. If the condition is not satisfied, then the algorithm halts with an appropriate message. Otherwise, the algorithm determines the dominating set. The given algorithm is considered a polynomial-time approximation one. (C) 2009 Elsevier B.V. All rights reserved.
Discretization of the electric field integral equation (EFIE) generally leads to dense impedance matrix. The resulting matrix, however, can be compressed in sparsity based on hierarchical structure and low rank approx...
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Discretization of the electric field integral equation (EFIE) generally leads to dense impedance matrix. The resulting matrix, however, can be compressed in sparsity based on hierarchical structure and low rank approximation. In this paper, we propose an HSS-matrix-based fast direct solver for surface integral equation (SIE) that has a compression complexity of O(rN(2)) to analysis the large-scale electromagnetic problems, where r is a modest integer. The proposed solver efficiently compresses the dense matrices using a randomized algorithm and requires modest memory. Efficiency and accuracy is validated by numerical simulations. In addition, being an algebraic method, the HSS-matrix-based fast solver employs Green's kernels and hence is suitable for other integral equations in electromagnetism.
The minimum rainbow subgraph problem arises in bioinformatics. The graph is given as an edge-colored undirected graph. Our goal is to find a subgraph with minimum number of vertices such that there is exactly one edge...
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The minimum rainbow subgraph problem arises in bioinformatics. The graph is given as an edge-colored undirected graph. Our goal is to find a subgraph with minimum number of vertices such that there is exactly one edge from each color class. The currently best approximation ratio achieved by a deterministic approximation algorithm is O(Delta). (Here Delta is the max degree of a graph.) In [4] he proposes a randomized algorithm which achieves an approximation ratio of O (root Delta ln Delta). However, we find that there is a flaw in his probability analysis which renders this approximation ratio invalid. We present a simple example to show why his analysis does not work. Instead, we propose an alternative analysis for his randomized algorithm. Our estimate shows that this randomized algorithm may achieve approximation ratio of O(Delta) in general. However, if the number of colors is Theta(n Delta(r)) for some positive r <= 1, his randomized algorithm can beat the bound of O(Delta). Moreover, through our analysis, we also find that if we impose an extra constraint on the color function, the bound O(root Delta ln Delta) still holds. (C) 2015 Elsevier B.V. All rights reserved.
Detecting circles from a digital image is very important in shape recognition. In this paper, an efficient randomized algorithm (RCD) for detecting circles is presented, which is not based on the Hough transform (HT)....
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Detecting circles from a digital image is very important in shape recognition. In this paper, an efficient randomized algorithm (RCD) for detecting circles is presented, which is not based on the Hough transform (HT). Instead of using an accumulator for saving the information of the related parameters in the HT-based methods, the proposed RCD does not need an accumulator. The main concept used in the proposed RCD is that we first randomly select four edge pixels in the image and define a distance criterion to determine whether there is a possible circle in the image: after finding a possible circle, we apply an evidence-collecting process to further determine whether the possible circle is a true circle or not. Some synthetic images with different levels of noises and some realistic images containing circular objects with some occluded circles and missing edges have been taken to test the performance. Experimental results demonstrate that the proposed RCD is faster than other HT-based methods for the noise level between the light level and the modest level. For a heavy noise level, the randomized HT could be faster than the proposed RCD, but at the expense of massive memory requirements. (C) 2001 Academic Press.
Cluster analysis is one of the most important research issues in data mining and machine learning. To date, numerous clustering algorithms have been proposed to tackle the fixed -length vector data. In many real appli...
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Cluster analysis is one of the most important research issues in data mining and machine learning. To date, numerous clustering algorithms have been proposed to tackle the fixed -length vector data. In many real applications, we need to detect clusters from a set of discrete sequences in which each sequence is an ordered list of items. Due to the sequential and discrete nature, the discrete sequence clustering problem is more challenging and most of existing vector data clustering algorithms cannot be directly employed. In this paper, we present a stochastic algorithm for clustering discrete sequences. Our method first quickly generates a set of random partitions over the sequential data set and then merges these random clustering results via weighted graph construction and partition. We perform extensive empirical comparisons on real data sets to show that our method is comparable to those state-of-the-art clustering algorithms with respect to both accuracy and efficiency.
This paper studies the fault detection problem in ship propulsion systems based on randomized algorithms. The nominal propulsion system model, model with uncertainties and model with additive and multiplicative faults...
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This paper studies the fault detection problem in ship propulsion systems based on randomized algorithms. The nominal propulsion system model, model with uncertainties and model with additive and multiplicative faults are first addressed in the form of normalized left coprime factorization (LCF), respectively. The -gap metric is then introduced to measure how far the system deviates from the nominal operation. To reduce the conservatism in the norm-based threshold, a threshold setting law and the estimation of fault detection rate (FDR) are formulated on the probabilistic assumption of uncertain and faulty parameters. The simulation results on the ship propulsion system show that the randomized technique is an efficient solution to deal with the fault detection issues.
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