This paper explores and analyzes two randomized designs for robust principal component analysis employing lowdimensional data sketching. In one design, a data sketch is constructed using random column sampling followe...
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This paper explores and analyzes two randomized designs for robust principal component analysis employing lowdimensional data sketching. In one design, a data sketch is constructed using random column sampling followed by lowdimensional embedding, while in the other, sketching is based on random column and rowsampling. Both designs are shown to bring about substantial savings in complexity andmemory requirements for robust subspace learning over conventional approaches that use the full scale data. A characterization of the sample and computational complexity of both designs is derived in the context of two distinct outliermodels, namely, sparse and independent outlier models. The proposed randomized approach can provably recover the correct subspace with computational and sample complexity which depend only weakly on the size of the data (only through the coherence parameters). The results of the mathematical analysis are confirmed through numerical simulations using both synthetic and real data.
In this paper, we study the following randomized broadcasting protocol. At some time t an information r is placed at one of the nodes of a graph. In the succeeding steps, each informed node chooses one neighbor, indep...
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In this paper, we study the following randomized broadcasting protocol. At some time t an information r is placed at one of the nodes of a graph. In the succeeding steps, each informed node chooses one neighbor, independently and uniformly at random, and informs this neighbor by sending a copy of r to it. We begin by developing tight lower and upper bounds on the runtime of the algorithm described above. First, it is shown that on Delta-regular graphs this algorithm requires at least log(2-1/Delta) n + log((Delta/Delta-1)Delta) n - o(log n) approximate to 1.69 log(2) n rounds to inform all n nodes. Together with a result of Pittel [B. Pittel, On spreading a rumor, SIAM journal on Applied Mathematics, 47 (1) (1987) 213-223] this bound implies that the algorithm has the best performance on complete graphs among all regular graphs. For general graphs, we prove a slightly weaker lower bound of log(2-1/Delta) n+log4 n-o(log n) approximate to 1.5 log(2) n, where Delta denotes the maximum degree of G. We also prove two general upper bounds, (1 + o(1))n ln n and O(n Delta/delta), respectively, where delta denotes the minimum degree. The second part of this paper is devoted to the analysis of fault-tolerance. We show that if the informed nodes are allowed to fail in some step with probability 1 - p, then the broadcasting time increases by at most a factor 6/p. As a by-product, we determine the performance of agent based broadcasting in certain graphs and obtain bounds for the runtime of randomized broadcasting on Cartesian products of graphs. (C) 2008 Elsevier B.V. All rights reserved.
We propose and analyze two randomized local election algorithms in an asynchronous anonymous graph. (C) 2001 Elsevier Science B.V. All rights reserved.
We propose and analyze two randomized local election algorithms in an asynchronous anonymous graph. (C) 2001 Elsevier Science B.V. All rights reserved.
The low -rank approximation of big data matrices and tensors plays a pivotal role in many modern applications. Although, a truncated version of the singular value decomposition (SVD) furnishes the best approximation, ...
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The low -rank approximation of big data matrices and tensors plays a pivotal role in many modern applications. Although, a truncated version of the singular value decomposition (SVD) furnishes the best approximation, its computation is challenging on modern, multicore architectures. Recently, the randomized subspace iteration has shown to be a powerful tool in approximating large-scale matrices. In this paper we present a two-sided variant of the randomized subspace iteration. Novel in our work is the exploitation of the unpivoted QR factorization, rather than the SVD, for factorizing the compressed matrix. Hence our algorithm is a randomized rank -revealing URV decomposition. We prove the rank-revealingness of our algorithm by establishing bounds for the singular values as well as the other blocks of the compressed matrix. We further provide bounds on the error of the low -rank approximations of the proposed algorithm, in both 2- and Frobenius norm. In addition, we employ the proposed algorithm to efficiently compute low rank tensor decompositions: we present two randomized algorithms, one for the truncated higher -order SVD, and the other for the tensor SVD. We conduct numerical tests on (i) various classes of matrices, and (ii) synthetic tensors and real datasets to demonstrate the efficacy of the proposed algorithms.
Given an undirected network G(V, E) and a set of traffic requests R, the minimum power-cost routing problem requires that each R-k is an element of R, be routed along a single path to minimize Sigma(e is an element of...
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Given an undirected network G(V, E) and a set of traffic requests R, the minimum power-cost routing problem requires that each R-k is an element of R, be routed along a single path to minimize Sigma(e is an element of E)(l(e))(alpha), where l(e) is the traffic load on edge e and alpha is a constant greater than 1. Typically, alpha is an element of (1,3]. This problem is important in optimizing the energy consumption of networks. To address this problem, we propose a randomized oblivious routing algorithm. An oblivious routing algorithm makes decisions independently of the current traffic in the network. This feature enables the efficient implementation of our algorithm in a distributed manner, which is desirable for large-scale high-capacity networks. An important feature of our work is that our algorithm can satisfy the integral constraint, which requires that each traffic request Rk should follow a single path. We prove that, given this constraint, no randomized oblivious routing algorithm can guarantee a competitive ratio bounded by o(vertical bar E vertical bar(alpha-1/alpha+1)). By contrast, our approach provides a competitive ratio of O(vertical bar E vertical bar(alpha-1/alpha+1) log(2 alpha/alpha+1) vertical bar V vertical bar . log(alpha-1) D), where D is the maximum demand of traffic requests. Furthermore, our results also hold for a more general case where the objective is to minimize Sigma(e)(l(e))(p), where p >= I is an arbitrary unknown parameter with a given upper bound alpha >1. The theoretical results established in proving these bounds can be further generalized to a framework of designing and analyzing oblivious integral routing algorithms, which is significant for research on minimizing Sigma(e)(l(e))(alpha) in specific scenarios with simplified problem settings. For instance, we prove that this framework can generate an oblivious integral routing algorithm whose competitive ratio can be bounded by O(log(alpha) vertical bar V vertical bar . log(a
We present the randomized QLP (RQLP) algorithm and its enhanced version (ERQLP) for computing a low-rank approximation to a matrix A of size m x n such that A approximate to QLP(T) , where L is a rank-k lower-triangul...
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We present the randomized QLP (RQLP) algorithm and its enhanced version (ERQLP) for computing a low-rank approximation to a matrix A of size m x n such that A approximate to QLP(T) , where L is a rank-k lower-triangular matrix, Q and P are column orthogonal matrices. The implementation of RQLP and ERQLP needs O(mnk) flops, which are mainly applied in BLAS-3 operations. We derive the lower bounds on the L-values, which can track the singular values of A approximately. Our claims are supported by numerical experiments. (C) 2020 Elsevier Inc. All rights reserved.
This paper contains a simple, randomized algorithm for constructing the convex bull of a set of n points in the plane with expected running time O(n log h) where h is the number of points on the convex hull.
This paper contains a simple, randomized algorithm for constructing the convex bull of a set of n points in the plane with expected running time O(n log h) where h is the number of points on the convex hull.
This paper introduces the Nystro"\m preconditioned conjugate gradient (PCG) algo-rithm for solving a symmetric positive-definite linear system. The algorithm applies the randomized Nystro"\m method to form a...
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This paper introduces the Nystro"\m preconditioned conjugate gradient (PCG) algo-rithm for solving a symmetric positive-definite linear system. The algorithm applies the randomized Nystro"\m method to form a low-rank approximation of the matrix, which leads to an efficient pre -conditioner that can be deployed with the conjugate gradient algorithm. Theoretical analysis shows that the preconditioned system has constant condition number as soon as the rank of the approx-imation is comparable with the number of effective degrees of freedom in the matrix. The paper also develops adaptive methods that provably achieve similar performance without knowledge of the effective dimension. Numerical tests show that Nystro"\m PCG can rapidly solve large linear systems that arise in data analysis problems, and it surpasses several competing methods from the literature.
For large-scale data fitting, the least-squares progressive iterative approximation is a widely used method in many applied domains because of its intuitive geometric meaning and efficiency. In this work, we present a...
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For large-scale data fitting, the least-squares progressive iterative approximation is a widely used method in many applied domains because of its intuitive geometric meaning and efficiency. In this work, we present a randomized progressive iterative approximation (RPIA) for the B-spline curve and surface fittings. In each iteration, RPIA locally adjusts the control points according to a random criterion of index selections. The difference for each control point is computed concerning the randomized block coordinate descent method. From geometric and algebraic aspects, the illustrations of RPIA are provided. We prove that RPIA constructs a series of fitting curves (resp., surfaces), whose limit curve (resp., surface) can converge in expectation to the least-squares fitting result of the given data points. Numerical experiments are given to confirm our results and show the benefits of RPIA.
This paper describes a randomized algorithm for computing mixed Nash equilibrium (NE) in bimatrix games. The new algorithm transforms the NE problem of bimatrix games into a new form by using a fuzzification technique...
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This paper describes a randomized algorithm for computing mixed Nash equilibrium (NE) in bimatrix games. The new algorithm transforms the NE problem of bimatrix games into a new form by using a fuzzification technique, it calculates mixed NE points in the new bimatrix games. This paper proves that the solution of the new bimatrix games is also a solution of the original bimatrix games. It proves that the computational complexity of the new algorithm for computing mixed NE points in bimatrix games is in the class of randomized polynomial time (RP). Examples are given that show the new algorithm can find two times more mixed NE points than Lemke-Howson algorithm. Numerical experiments indicate that the average computational time of the new algorithm is four hundred time faster than Lemke-Howson algorithm for calculating the first mixed NE point.
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