Hierarchical Singular Value Decomposition of Tensors
TLDR
This hierarchical SVD has properties like the matrix SVD (and collapses to the SVD in $d=2$), and it is proved that one can find low rank (almost) best approximations in a hierarchical format ($\mathcal{H}$-Tucker) which requires only $\ mathcal{O}((d-1)k^3+dnk)$ parameters.Abstract:
We define the hierarchical singular value decomposition (SVD) for tensors of order $d\geq2$. This hierarchical SVD has properties like the matrix SVD (and collapses to the SVD in $d=2$), and we prove these. In particular, one can find low rank (almost) best approximations in a hierarchical format ($\mathcal{H}$-Tucker) which requires only $\mathcal{O}((d-1)k^3+dnk)$ parameters, where $d$ is the order of the tensor, $n$ the size of the modes, and $k$ the (hierarchical) rank. The $\mathcal{H}$-Tucker format is a specialization of the Tucker format and it contains as a special case all (canonical) rank $k$ tensors. Based on this new concept of a hierarchical SVD we present algorithms for hierarchical tensor calculations allowing for a rigorous error analysis. The complexity of the truncation (finding lower rank approximations to hierarchical rank $k$ tensors) is in $\mathcal{O}((d-1)k^4+dnk^2)$ and the attainable accuracy is just 2-3 digits less than machine precision.read more
Citations
More filters
Journal ArticleDOI
Tensor-Train Decomposition
TL;DR: The new form gives a clear and convenient way to implement all basic operations efficiently, and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19-dimensional operator.
Journal ArticleDOI
A literature survey of low-rank tensor approximation techniques
TL;DR: This survey attempts to give a literature overview of current developments in low-rank tensor approximation, with an emphasis on function-related tensors.
Journal ArticleDOI
A New Scheme for the Tensor Representation
Wolfgang Hackbusch,Stefan Kühn +1 more
TL;DR: A truncation algorithm can be implemented which is based on the standard matrix singular value decomposition (SVD) method and is possible to apply standard Linear Algebra tools for performing arithmetical operations and for the computation of data-sparse approximations.
Journal ArticleDOI
Model Compression and Hardware Acceleration for Neural Networks: A Comprehensive Survey
TL;DR: This article reviews the mainstream compression approaches such as compact model, tensor decomposition, data quantization, and network sparsification, and answers the question of how to leverage these methods in the design of neural network accelerators and present the state-of-the-art hardware architectures.
Journal ArticleDOI
Convergence Rates for Greedy Algorithms in Reduced Basis Methods
Peter Binev,Albert Cohen,Wolfgang Dahmen,Ronald A. DeVore,Guergana Petrova,Przemysław Wojtaszczyk +5 more
TL;DR: The reduced basis method was introduced for the accurate online evaluation of solutions to a parameter dependent family of elliptic PDEs by determining a “good” n-dimensional space to be used in approximating the elements of a compact set $\mathcal{F}$ in a Hilbert space $\ mathscal{H}$.
References
More filters
Journal ArticleDOI
Tensor Decompositions and Applications
Tamara G. Kolda,Brett W. Bader +1 more
TL;DR: This survey provides an overview of higher-order tensor decompositions, their applications, and available software.
Journal ArticleDOI
Analysis of individual differences in multidimensional scaling via an n-way generalization of 'eckart-young' decomposition
J. Douglas Carroll,Jih-Jie Chang +1 more
TL;DR: In this paper, an individual differences model for multidimensional scaling is outlined in which individuals are assumed differentially to weight the several dimensions of a common "psychological space" and a corresponding method of analyzing similarities data is proposed, involving a generalization of Eckart-Young analysis to decomposition of three-way (or higher-way) tables.
Journal ArticleDOI
A Multilinear Singular Value Decomposition
TL;DR: There is a strong analogy between several properties of the matrix and the higher-order tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, first-order perturbation effects, etc., are analyzed.
Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-model factor analysis
TL;DR: It is shown that an extension of Cattell's principle of rotation to Proportional Profiles (PP) offers a basis for determining explanatory factors for three-way or higher order multi-mode data.
Journal ArticleDOI
The multiconfiguration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wavepackets
TL;DR: In this article, a review of the multiconfiguration time-dependent Hartree (MCTDH) method for propagating wavepackets is given, and the formal derivation, numerical implementation, and performance of the method are detailed.