This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fractio...
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This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fraction of the cost of deterministic algorithms, easing the computational challenges arising in the area of "big data." The idea is to derive a small matrix from the high-dimensional data, which is then used to efficiently compute the dynamic modes and eigenvalues. The algorithm is presented in a modular probabilistic framework, and the approximation quality can be controlled via oversampling and power iterations. The effectiveness of the resulting randomized DMD algorithm is demonstrated on several benchmark examples of increasing complexity, providing an accurate and efficient approach to extract spatiotemporal coherent structures from big data in a framework that scales with the intrinsic rank of the data, rather than the ambient measurement dimension. For this work we assume that the dynamics of the problem under consideration is evolving on a low-dimensional subspace that is well characterized by a quickly decaying singular value spectrum.
Jerrum, Sinclair, and Vigoda [J. ACM, 51 (2004), pp. 671-697] showed that the permanent of any square matrix can be estimated in polynomial time. This computation can be viewed as approximating the partition function ...
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Jerrum, Sinclair, and Vigoda [J. ACM, 51 (2004), pp. 671-697] showed that the permanent of any square matrix can be estimated in polynomial time. This computation can be viewed as approximating the partition function of edge-weighted matchings in a bipartite graph. Equivalently, this may be viewed as approximating the partition function of vertex-weighted independent sets in the line graph of a bipartite graph. Line graphs of bipartite graphs are perfect graphs and are known to be precisely the class of (claw, diamond, odd hole)-free graphs. So how far does the result of Jerrum, Sinclair, and Vigoda extend? We first show that it extends to (claw, odd hole)-free graphs, and then show that it extends to the even larger class of (fork, odd hole)-free graphs. Our techniques are based on graph decompositions, which have been the focus of much recent work in structural graph theory, and on structural results of Chvatal and Sbihi [J. Combin. Theory Ser. B, 44 (1988)], Maffray and Reed [J. Combin. Theory Ser. B, 75 (1999)], and Lozin and Milanic [J. Discrete algorithms, 6 (2008), pp. 595-604].
Model reduction of large Markov chains is an essential step in a wide array of techniques for understanding complex systems and for efficiently learning structures from high-dimensional data. We present a novel aggreg...
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Model reduction of large Markov chains is an essential step in a wide array of techniques for understanding complex systems and for efficiently learning structures from high-dimensional data. We present a novel aggregation algorithm for compressing such chains that exploits a specific low rank structure in the transition matrix which, e.g., is present in metastable systems, among others. It enables the recovery of the aggregates from a vastly undersampled transition matrix which in practical applications may gain a speedup of several orders of magnitude over methods that require the full transition matrix. Moreover, we show that the new technique is robust under perturbation of the transition matrix. The practical applicability of the new method is demonstrated by identifying a reduced model for the large-scale traffic flow patterns from real-world taxi trip data. (C) 2020 The Authors. Published by Elsevier B.V.
This paper describes a new algorithm for computing a low-Tucker-rank approximation of a tensor. The method applies a randomized linear map to the tensor to obtain a sketch that captures the important directions within...
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This paper describes a new algorithm for computing a low-Tucker-rank approximation of a tensor. The method applies a randomized linear map to the tensor to obtain a sketch that captures the important directions within each mode, as well as the interactions among the modes. The sketch can be extracted from streaming or distributed data or with a single pass over the tensor, and it uses storage proportional to the degrees of freedom in the output Tucker approximation. The algorithm does not require a second pass over the tensor, although it can exploit another view to compute a superior approximation. The paper provides a rigorous theoretical guarantee on the approximation error. Extensive numerical experiments show that the algorithm produces useful results that improve on the state-of-the-art for streaming Tucker decomposition.
This survey provides an introduction to the use of randomization in the design of fast algorithms for numerical linear algebra. These algorithms typically examine only a subset of the input to solve basic problems app...
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This survey provides an introduction to the use of randomization in the design of fast algorithms for numerical linear algebra. These algorithms typically examine only a subset of the input to solve basic problems approximately, including matrix multiplication, regression and low-rank approximation. The survey describes the key ideas and gives complete proofs of the main results in the field. A central unifying idea is sampling the columns (or rows) of a matrix according to their squared lengths.
randomized Monte Carlo algorithms are constructed by a combination of a basic probabilistic model and its random parameters to investigate parametric distributions of linear functionals. An optimization of the algorit...
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randomized Monte Carlo algorithms are constructed by a combination of a basic probabilistic model and its random parameters to investigate parametric distributions of linear functionals. An optimization of the algorithms with a statistical kernel estimator for the probability density is presented. A randomized projection algorithm for estimating a nonlinear functional distribution is formulated and applied to the investigation of the criticality fluctuations of a particle multiplication process in a random medium.
We describe an algorithm for the application of the forward and inverse spherical harmonic transforms. It is based on a new method for rapidly computing the forward and inverse associated Legendre transforms by hierar...
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We describe an algorithm for the application of the forward and inverse spherical harmonic transforms. It is based on a new method for rapidly computing the forward and inverse associated Legendre transforms by hierarchically applying the interpolative decomposition butterfly factorization. Experimental evidence suggests that the complexity of our method-including all necessary precomputations-is O(N-2 log(3) N) in terms of both flops and memory, where N is the order of the transform. This is nearly asymptotically optimal. Moreover, unlike existing algorithms which are asymptotically optimal or nearly so, the constants in the running time and memory costs of our algorithm are small enough to make it competitive with state-of-the-art O(N-3) methods at relatively small values of N (e.g., N = 1024). Numerical results are provided to demonstrate the effectiveness and numerical stability of the new framework.
For any T >= 1, there are constants R = R(T) > 1 and zeta = zeta((T) > 0 and a randomized algorithm that takes as input an integer n and two strings x, y of length at most n, and runs in time O(n(1+1/T)) and ...
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ISBN:
(纸本)9781450369794
For any T >= 1, there are constants R = R(T) > 1 and zeta = zeta((T) > 0 and a randomized algorithm that takes as input an integer n and two strings x, y of length at most n, and runs in time O(n(1+1/T)) and outputs an upper bound U on the edit distance of d(edit)(x, y) that with high probability, satisfies U <= R(d(edit)(x, y) + n(1-zeta)). particular, on any input with d(edit)(x, y) >= n(1-zeta) the algorithm outputs a constant factor approximation with high probability. A similar result has been proven independently by Brakensiek and Rubinstein (this proceedings).
randomized Monte Carlo algorithms are constructed by jointly realizing a baseline probabilistic model of the problem and its random parameters (random medium) in order to study a parametric distribution of linear func...
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randomized Monte Carlo algorithms are constructed by jointly realizing a baseline probabilistic model of the problem and its random parameters (random medium) in order to study a parametric distribution of linear functionals. This work relies on statistical kernel estimation of the multidimensional distribution density with a homogeneous kernel and on a splitting method, according to which a certain number of baseline trajectories are modeled for each medium realization. The optimal value of is estimated using a criterion for computational complexity formulated in this work. Analytical estimates of the corresponding computational efficiency are obtained with the help of rather complicated calculations.
In this paper,we propose a randomized primal–dual proximal block coordinate updating framework for a general multi-block convex optimization model with coupled objective function and linear *** mere convexity,we esta...
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In this paper,we propose a randomized primal–dual proximal block coordinate updating framework for a general multi-block convex optimization model with coupled objective function and linear *** mere convexity,we establish its O(1/t)convergence rate in terms of the objective value and feasibility *** framework includes several existing algorithms as special cases such as a primal–dual method for bilinear saddle-point problems(PD-S),the proximal Jacobian alternating direction method of multipliers(Prox-JADMM)and a randomized variant of the ADMM for multi-block convex *** analysis recovers and/or strengthens the convergence properties of several existing *** example,for PD-S our result leads to the same order of convergence rate without the previously assumed boundedness condition on the constraint sets,and for Prox-JADMM the new result provides convergence rate in terms of the objective value and the feasibility *** is well known that the original ADMM may fail to converge when the number of blocks exceeds *** result shows that if an appropriate randomization procedure is invoked to select the updating blocks,then a sublinear rate of convergence in expectation can be guaranteed for multi-block ADMM,without assuming any strong *** new approach is also extended to solve problems where only a stochastic approximation of the subgradient of the objective is available,and we establish an O(1/√t)convergence rate of the extended approach for solving stochastic programming.
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