Literature survey on low rank approximation of matrices

Кумар, Нареш; Schneider, Jan

doi:10.1080/03081087.2016.1267104

Cited by 117 publications

(27 citation statements)

References 147 publications

(254 reference statements)

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“…Clearly, each segment, except those on level L, have two child segments. In a nutshell, A ( ) considers the interaction at level between a segment and its interaction list, and A (ad) considers all the interactions between adjacent segments at level L. The key idea behind H-matrices is to approximate the nonzero blocks A ( ) by a low rank approximation (see [34] for a thorough review). This idea is depicted in Fig.…”

Section: H-matricesmentioning

confidence: 99%

A Multiscale Neural Network Based on Hierarchical Matrices

Fan¹,

Lin²,

Ying³

et al. 2019

Multiscale Model. Simul.

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In this work we introduce a new multiscale artificial neural network based on the structure of H-matrices. This network generalizes the latter to the nonlinear case by introducing a local deep neural network at each spatial scale. Numerical results indicate that the network is able to efficiently approximate discrete nonlinear maps obtained from discretized nonlinear partial differential equations, such as those arising from nonlinear Schrödinger equations and the Kohn-Sham density functional theory.

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Section: H-matricesmentioning

confidence: 99%

A Multiscale Neural Network Based on Hierarchical Matrices

Fan¹,

Lin²,

Ying³

et al. 2019

Multiscale Model. Simul.

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“…The bottleneck is computing its inverse which has complexity O(N 3 ). Similarly to other well-established machine learning algorithms which share this bottleneck, one could make use of approximations that would trade off accuracy for computational expenses [17]. We also note that the per iteration complexity scales linearly in N , due to the normalization step.…”

Section: Time Complexitymentioning

confidence: 93%

Label Propagation for Learning With Label Proportions

Poyiadzi

Santos-Rodríguez

Twomey

2018

2018 IEEE 28th International Workshop on Machine Learning for Signal Processing (MLSP)

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Learning with Label Proportions (LLP) is the problem of recovering the underlying true labels given a dataset when the data is presented in the form of bags. This paradigm is particularly suitable in contexts where providing individual labels is expensive and label aggregates are more easily obtained. In the healthcare domain, it is a burden for a patient to keep a detailed diary of their daily routines, but often they will be amenable to provide higher level summaries of daily behavior. We present a novel and efficient graph-based algorithm that encourages local smoothness and exploits the global structure of the data, while preserving the 'mass' of each bag.

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“…The focus of this paper is not the 'best heuristic' for the matrix case, but the extension to high-dimensional problems. We refer the reader to [53] for the review of matrix low-rank approximation algorithms.…”

Section: Algorithm 2 One Step Of the Matrix Cross Interpolation Algormentioning

confidence: 99%

Parallel cross interpolation for high-precision calculation of high-dimensional integrals

Dolgov

Savostyanov

2020

Computer Physics Communications

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We propose a parallel version of the cross interpolation algorithm and apply it to calculate high-dimensional integrals motivated by Ising model in quantum physics. In contrast to mainstream approaches, such as Monte Carlo and quasi Monte Carlo, the samples calculated by our algorithm are neither random nor form a regular lattice. Instead we calculate the given function along individual dimensions (modes) and use this data to reconstruct its behaviour in the whole domain. The positions of the calculated univariate fibers are chosen adaptively for the given function. The required evaluations can be executed in parallel both along each mode (variable) and over all modes.To demonstrate the efficiency of the proposed method, we apply it to compute highdimensional Ising susceptibility integrals, arising from asymptotic expansions for the spontaneous magnetisation in two-dimensional Ising model of ferromagnetism. We observe strong superlinear convergence of the proposed method, while the MC and qMC algorithms converge sublinearly. Using multiple precision arithmetic, we also observed exponential convergence of the proposed algorithm. Combining high-order convergence, almost perfect scalability up to hundreds of processes, and the same flexibility as MC and qMC, the proposed algorithm can be a new method of choice for problems involving high-dimensional integration, e.g. in statistics, probability, and quantum physics. ≈ −1 I I J J I I I J J J Figure 1: Cross interpolation for matrices. The full matrix A is approximated by a low-rank decompositionÃ based on a small number of columns and rows computed in A. Note that the approximation (1) is exact in the positions of computed rows and columns. One of the first examples of the latter was the parallel density matrix renormalization group (DMRG) algorithm [81] for ground state computations in quantum physics. In mathematical community this research direction started with dimension-parallel linear solver [30] and cross algorithms in HT format [44]. The main difficulty of parallelisation over dimension is the need of algorithmic modifications, since state of the arts tensor algorithms were designed in intrinsically sequential way. Ideally, such modifications should not compromise numerical stability or convergence for the sake of parallel efficiency. In this paper we develop a parallel version of the TT cross interpolation algorithm [74]. The parallel algorithm is adaptive and converges with the same rate as the sequential version, but involves only local communications with constant loading of processes, and demonstrates almost perfect scaling up to the ultimate partitioning where each process is responsible for a single direction (mode, variable). Moreover, further speedup can be achieved using OpenMP parallelisation of tensor algebra in each process.The rest of the paper is organised as follows. In Sec. 2 we recall the cross interpolation method for matrices and provide necessary definitions. In Sec. 3 we discuss how the matrix interpolation can be applied for high-dimensi...

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Literature survey on low rank approximation of matrices

Cited by 117 publications

References 147 publications

A Multiscale Neural Network Based on Hierarchical Matrices

A Multiscale Neural Network Based on Hierarchical Matrices

Label Propagation for Learning With Label Proportions

Parallel cross interpolation for high-precision calculation of high-dimensional integrals

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