1,063 research outputs found
Improved scaling of Time-Evolving Block-Decimation algorithm through Reduced-Rank Randomized Singular Value Decomposition
When the amount of entanglement in a quantum system is limited, the relevant
dynamics of the system is restricted to a very small part of the state space.
When restricted to this subspace the description of the system becomes
efficient in the system size. A class of algorithms, exemplified by the
Time-Evolving Block-Decimation (TEBD) algorithm, make use of this observation
by selecting the relevant subspace through a decimation technique relying on
the Singular Value Decomposition (SVD). In these algorithms, the complexity of
each time-evolution step is dominated by the SVD. Here we show that, by
applying a randomized version of the SVD routine (RRSVD), the power law
governing the computational complexity of TEBD is lowered by one degree,
resulting in a considerable speed-up. We exemplify the potential gains in
efficiency at the hand of some real world examples to which TEBD can be
successfully applied to and demonstrate that for those system RRSVD delivers
results as accurate as state-of-the-art deterministic SVD routines.Comment: 14 pages, 5 figure
Online Tensor Methods for Learning Latent Variable Models
We introduce an online tensor decomposition based approach for two latent
variable modeling problems namely, (1) community detection, in which we learn
the latent communities that the social actors in social networks belong to, and
(2) topic modeling, in which we infer hidden topics of text articles. We
consider decomposition of moment tensors using stochastic gradient descent. We
conduct optimization of multilinear operations in SGD and avoid directly
forming the tensors, to save computational and storage costs. We present
optimized algorithm in two platforms. Our GPU-based implementation exploits the
parallelism of SIMD architectures to allow for maximum speed-up by a careful
optimization of storage and data transfer, whereas our CPU-based implementation
uses efficient sparse matrix computations and is suitable for large sparse
datasets. For the community detection problem, we demonstrate accuracy and
computational efficiency on Facebook, Yelp and DBLP datasets, and for the topic
modeling problem, we also demonstrate good performance on the New York Times
dataset. We compare our results to the state-of-the-art algorithms such as the
variational method, and report a gain of accuracy and a gain of several orders
of magnitude in the execution time.Comment: JMLR 201
Novel Monte Carlo Methods for Large-Scale Linear Algebra Operations
Linear algebra operations play an important role in scientific computing and data analysis. With increasing data volume and complexity in the Big Data era, linear algebra operations are important tools to process massive datasets. On one hand, the advent of modern high-performance computing architectures with increasing computing power has greatly enhanced our capability to deal with a large volume of data. One the other hand, many classical, deterministic numerical linear algebra algorithms have difficulty to scale to handle large data sets.
Monte Carlo methods, which are based on statistical sampling, exhibit many attractive properties in dealing with large volume of datasets, including fast approximated results, memory efficiency, reduced data accesses, natural parallelism, and inherent fault tolerance. In this dissertation, we present new Monte Carlo methods to accommodate a set of fundamental and ubiquitous large-scale linear algebra operations, including solving large-scale linear systems, constructing low-rank matrix approximation, and approximating the extreme eigenvalues/ eigenvectors, across modern distributed and parallel computing architectures. First of all, we revisit the classical Ulam-von Neumann Monte Carlo algorithm and derive the necessary and sufficient condition for its convergence. To support a broad family of linear systems, we develop Krylov subspace Monte Carlo solvers that go beyond the use of Neumann series. New algorithms used in the Krylov subspace Monte Carlo solvers include (1) a Breakdown-Free Block Conjugate Gradient algorithm to address the potential rank deficiency problem occurred in block Krylov subspace methods; (2) a Block Conjugate Gradient for Least Squares algorithm to stably approximate the least squares solutions of general linear systems; (3) a BCGLS algorithm with deflation to gain convergence acceleration; and (4) a Monte Carlo Generalized Minimal Residual algorithm based on sampling matrix-vector products to provide fast approximation of solutions. Secondly, we design a rank-revealing randomized Singular Value Decomposition (R3SVD) algorithm for adaptively constructing low-rank matrix approximations to satisfy application-specific accuracy. Thirdly, we study the block power method on Markov Chain Monte Carlo transition matrices and find that the convergence is actually depending on the number of independent vectors in the block. Correspondingly, we develop a sliding window power method to find stationary distribution, which has demonstrated success in modeling stochastic luminal Calcium release site. Fourthly, we take advantage of hybrid CPU-GPU computing platforms to accelerate the performance of the Breakdown-Free Block Conjugate Gradient algorithm and the randomized Singular Value Decomposition algorithm. Finally, we design a Gaussian variant of Freivalds’ algorithm to efficiently verify the correctness of matrix-matrix multiplication while avoiding undetectable fault patterns encountered in deterministic algorithms
Randomized Dynamic Mode Decomposition
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 fast decaying singular
value spectrum
Efficient Randomized Algorithms for the Fixed-Precision Low-Rank Matrix Approximation
Randomized algorithms for low-rank matrix approximation are investigated,
with the emphasis on the fixed-precision problem and computational efficiency
for handling large matrices. The algorithms are based on the so-called QB
factorization, where Q is an orthonormal matrix. Firstly, a mechanism for
calculating the approximation error in Frobenius norm is proposed, which
enables efficient adaptive rank determination for large and/or sparse matrix.
It can be combined with any QB-form factorization algorithm in which B's rows
are incrementally generated. Based on the blocked randQB algorithm by P.-G.
Martinsson and S. Voronin, this results in an algorithm called randQB EI. Then,
we further revise the algorithm to obtain a pass-efficient algorithm, randQB
FP, which is mathematically equivalent to the existing randQB algorithms and
also suitable for the fixed-precision problem. Especially, randQB FP can serve
as a single-pass algorithm for calculating leading singular values, under
certain condition. With large and/or sparse test matrices, we have empirically
validated the merits of the proposed techniques, which exhibit remarkable
speedup and memory saving over the blocked randQB algorithm. We have also
demonstrated that the single-pass algorithm derived by randQB FP is much more
accurate than an existing single-pass algorithm. And with data from a scenic
image and an information retrieval application, we have shown the advantages of
the proposed algorithms over the adaptive range finder algorithm for solving
the fixed-precision problem.Comment: 21 pages, 10 figure
Randomized low-rank Dynamic Mode Decomposition for motion detection
N. Benjamin Erichson acknowledges support from the UK Engineering and Physical Sciences Research Council (EPSRC).This paper introduces a fast algorithm for randomized computation of a low-rank Dynamic Mode Decomposition (DMD) of a matrix. Here we consider this matrix to represent the development of a spatial grid through time e.g. data from a static video source. DMD was originally introduced in the fluid mechanics community, but is also suitable for motion detection in video streams and its use for background subtraction has received little previous investigation. In this study we present a comprehensive evaluation of background subtraction, using the randomized DMD and compare the results with leading robust principal component analysis algorithms. The results are convincing and show the random DMD is an efficient and powerful approach for background modeling, allowing processing of high resolution videos in real-time. Supplementary materials include implementations of the algorithms in Python.PostprintPeer reviewe
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