Low-Rank Tensor Methods with Subspace Correction for Symmetric Eigenvalue Problems

Kreßner, Daniel; Steinlechner, Michael; Uschmajew, Andr ́ E

doi:10.1137/130949919

Cited by 54 publications

(70 citation statements)

References 36 publications

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“…This chapter introduces feasible solutions for several generic huge-scale dimensionality reduction and related optimization problems, whereby the involved optimized cost functions are approximated by suitable low-rank TT networks. In this way, a very large-scale optimization problem can be converted into a set of much smaller optimization sub-problems of the same kind [Cichocki, 2014, Holtz et al, 2012a, Kressner et al, 2014a, Lee and Cichocki, 2016b, Schollwöck, 2011, which can be solved using standard methods.…”

Section: Chaptermentioning

confidence: 99%

“…In particular, we focus on Symmetric Eigenvalue Decomposition (EVD/PCA) and Generalized Eigenvalue Decomposition (GEVD) , Hubig et al, 2015, Huckle and Waldherr, 2012, Kressner et al, 2014a, SVD [Lee and Cichocki, 2015], solutions of overdetermined and undetermined systems of linear algebraic equations [Dolgov andSavostyanov, 2014, Oseledets and, the MoorePenrose pseudo-inverse of structured matrices [Lee and Cichocki, 2016b], and LASSO regression problems [Lee and Cichocki, 2016a]. Tensor networks for extremely large-scale multi-block (multi-view) data are also discussed, especially TN models for orthogonal Canonical Correlation Analysis (CCA) and related Higher-Order Partial Least Squares (HOPLS) problems [Hou, 2017, Hou et al, 2016b, Zhao et al, 2011.…”

Section: Chaptermentioning

confidence: 99%

“…In this way, a large-scale task is replaced by easier-to-handle sub-tasks, whereby the aim is to extract a specific TT core or its slices from the whole TT network. For this purpose, the TT sub-trains can be defined as follows [Holtz et al, 2012a, Kressner et al, 2014a]…”

Section: Extraction Of a Single Core And A Single Slice For Als Algormentioning

confidence: 99%

“…It is important to mention here, that the orthogonalization of core tensors is an essential procedure in many algorithms for the TT formats (in order to reduce computational complexity of contraction of core tensors and improve robustness of the algorithms) [Dolgov, 2014, Kressner et al, 2014a, Oseledets, 2011a, Steinlechner, 2016a.…”

Section: Extraction Of a Single Core And A Single Slice For Als Algormentioning

confidence: 99%

“…Slice products 13) and all the cores to the right of G (n) are right-orthogonal (3.14) then the frame matrices have orthogonal columns , Kressner et al, 2014a, Pižorn and Verstraete, 2012, that is…”

Section: Extraction Of Two Neighboring Cores For Modified Als (Mals)mentioning

confidence: 99%

See 4 more Smart Citations

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives

Cichocki

Phan

Zhao

et al. 2017

FNT in Machine Learning

185

126

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Section: Chaptermentioning

confidence: 99%

Section: Chaptermentioning

confidence: 99%

Section: Extraction Of a Single Core And A Single Slice For Als Algormentioning

confidence: 99%

Section: Extraction Of a Single Core And A Single Slice For Als Algormentioning

confidence: 99%

Section: Extraction Of Two Neighboring Cores For Modified Als (Mals)mentioning

confidence: 99%

See 3 more Smart Citations

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives

Cichocki

Phan

Zhao

et al. 2017

FNT in Machine Learning

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126

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Robust tensor train component analysis

Zhang

2021

Numerical Linear Algebra App

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Summary Robust Principal Component Analysis plays a key role in various fields such as image and video processing, data mining, and hyperspectral data analysis. In this paper, we study the problem of robust tensor train (TT) principal component analysis from partial observations, which aims to decompose a given tensor into the low TT rank and sparse components. The decomposition of the proposed model is used to find the hidden factors and help alleviate the curse of dimensionality via a set of connected low‐rank tensors. A relaxation model is to minimize a weighted combination of the sum of nuclear norms of unfolding matrices of core tensors and the tensor ℓ1 norm. A proximal alternating direction method of multipliers is developed to solve the resulting model. Furthermore, we show that any cluster point of the convergent subsequence is a Karush‐Kuhn‐Tucker point of the proposed model under some conditions. Extensive numerical examples on both synthetic data and real‐world datasets are presented to demonstrate the effectiveness of the proposed approach.

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Approximation of Singularities by Quantized‐Tensor FEM

Kazeev

Schwab

2015

Proc Appl Math and Mech

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In d dimensions, first-order tensor-product finite-element (FE) approximations of the solutions of second-order elliptic problems are well known to converge algebraically, with rate at most 1/d in the energy norm and with respect to the number of degrees of freedom. On the other hand, FE methods of higher regularity may achieve exponential convergence, e.g. global spectral methods for analytic solutions and hp methods for solutions from certain countably normed spaces, which may exhibit singularities.In this note, we revisit, in one dimension, the tensor-structured approach to the h-FE approximation of singular functions. We outline a proof of the exponential convergence of such approximations represented in the quantized-tensor-train (QTT) format. Compared to special approximation techniques, such as hp, that approach is fully adaptive in the sense that it finds suitable approximation spaces algorithmically. The convergence is measured with respect to the number of parameters used to represent the solution, which is not the dimension of the first-order FE space, but depends only polylogarithmically on that. We demonstrate the convergence numerically for a simple model problem and find the rate to be approximately the same as for hp approximations.

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Low-Rank Tensor Methods with Subspace Correction for Symmetric Eigenvalue Problems

Cited by 54 publications

References 36 publications

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives

Robust tensor train component analysis

Approximation of Singularities by Quantized‐Tensor FEM

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