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Register a new userImproved scaling of Time-Evolving Block-Decimation algorithm through Reduced-Rank Randomized Singular Value Decomposition
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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.
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An algorithm of the tensor renormalization group is proposed based on a randomized algorithm for singular value decomposition. Our algorithm is applicable to a broad range of two-dimensional classical models. In the case of a square lattice, its computational complexity and memory usage are proportional to the fifth and the third power of the bond dimension, respectively, whereas those of the conventional implementation are of the sixth and the fourth power. The oversampling parameter larger than the bond dimension is sufficient to reproduce the same result as full singular value decomposition even at the critical point of the two-dimensional Ising model.
Quaternion matrix approximation problems construct the approximated matrix via the quaternion singular value decomposition (SVD) by selecting some singular value decomposition (SVD) triplets of quaternion matrices. In applications such as color image processing and recognition problems, only a small number of dominant SVD triplets are selected, while in some applications such as quaternion total least squares problem, small SVD triplets (small singular values and associated singular vectors) and numerical rank with respect to a small threshold are required. In this paper, we propose a randomized quaternion SVD (verbrandsvdQ) method to compute a small number of SVD triplets of a large-scale quaternion matrix. Theoretical results are given about approximation errors and the corresponding algorithm adapts to the low-rank matrix approximation problem. When the restricted rank increases, it might lead to information loss of small SVD triplets. The blocked quaternion randomized SVD algorithm is then developed when the numerical rank and information about small singular values are required. For color face recognition problems, numerical results show good performance of the developed quaternion randomized SVD method for low-rank approximation of a large-scale quaternion matrix. The blocked randomized SVD algorithm is also shown to be more robust than unblocked method through several experiments, and approximation errors from the blocked scheme are very close to the optimal error obtained by truncating a full SVD.
Since being analyzed by Rokhlin, Szlam, and Tygert and popularized by Halko, Martinsson, and Tropp, randomized Simultaneous Power Iteration has become the method of choice for approximate singular value decomposition. It is more accurate than simpler sketching algorithms, yet still converges quickly for any matrix, independently of singular value gaps. After $tilde{O}(1/epsilon)$ iterations, it gives a low-rank approximation within $(1+epsilon)$ of optimal for spectral norm error. We give the first provable runtime improvement on Simultaneous Iteration: a simple randomized block Krylov method, closely related to the classic Block Lanczos algorithm, gives the same guarantees in just $tilde{O}(1/sqrt{epsilon})$ iterations and performs substantially better experimentally. Despite their long history, our analysis is the first of a Krylov subspace method that does not depend on singular value gaps, which are unreliable in practice. Furthermore, while it is a simple accuracy benchmark, even $(1+epsilon)$ error for spectral norm low-rank approximation does not imply that an algorithm returns high quality principal components, a major issue for data applications. We address this problem for the first time by showing that both Block Krylov Iteration and a minor modification of Simultaneous Iteration give nearly optimal PCA for any matrix. This result further justifies their strength over non-iterative sketching methods. Finally, we give insight beyond the worst case, justifying why both algorithms can run much faster in practice than predicted. We clarify how simple techniques can take advantage of common matrix properties to significantly improve runtime.
The hierarchical SVD provides a quasi-best low rank approximation of high dimensional data in the hierarchical Tucker framework. Similar to the SVD for matrices, it provides a fundamental but expensive tool for tensor computations. In the present work we examine generalizations of randomized matrix decomposition methods to higher order tensors in the framework of the hierarchical tensors representation. In particular we present and analyze a randomized algorithm for the calculation of the hierarchical SVD (HSVD) for the tensor train (TT) format.
We introduce PyParSVDfootnote{https://github.com/Romit-Maulik/PyParSVD}, a Python library that implements a streaming, distributed and randomized algorithm for the singular value decomposition. To demonstrate its effectiveness, we extract coherent structures from scientific data. Futhermore, we show weak scaling assessments on up to 256 nodes of the Theta machine at Argonne Leadership Computing Facility, demonstrating potential for large-scale data analyses of practical data sets.
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