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Fast Tucker Rank Reduction for Non-Negative Tensors Using Mean-Field Approximation [article]

Kazu Ghalamkari, Mahito Sugiyama
2021 arXiv   pre-print
We present an efficient low-rank approximation algorithm for non-negative tensors.  ...  We empirically demonstrate that our algorithm is faster than the existing non-negative Tucker rank reduction methods and achieves competitive or better approximation of given tensors.  ...  In this paper, we present a fast low-Tucker-rank approximation method for non-negative tensors.  ... 
arXiv:2103.02898v3 fatcat:z36vobmhkbapbmzbiryhmskpc4

Tensor Approximation for Multidimensional and Multivariate Data [chapter]

Renato Pajarola, Susanne K. Suter, Rafael Ballester-Ripoll, Haiyan Yang
2021 Mathematics and Visualization  
Initially proposed as an extension of the concept of matrix rank for 3 and more dimensions, tensor decomposition methods have found applications in a remarkably wide range of disciplines.  ...  Furthermore, we will include a first outlook on porting these techniques to multivariate data such as vector and tensor fields.  ...  Also we acknowledge the Johns Hopkins Turbulence Database http://turbulence.pha.jhu.edu/ for the data used in Fig. 16 as well as their forced MHD simulation data http://turbulence.pha.jhu.edu/Forced_  ... 
doi:10.1007/978-3-030-56215-1_4 fatcat:wytporlfkbgnzhfhbxmalxylpy

Fast tensor method for summation of long-range potentials on 3D lattices with defects

Venera Khoromskaia, Boris N. Khoromskij
2015 Numerical Linear Algebra with Applications  
In this paper, we present a method for fast summation of long-range potentials on 3D lattices with multiple defects and having non-rectangular geometries, based on rank-structured tensor representations  ...  For the reduced higher-order SVD approximation to a sum of canonical/Tucker tensors, we prove the stable error bounds in the relative norm in terms of discarded singular values of the side matrices.  ...  of far fields.  ... 
doi:10.1002/nla.2023 fatcat:mymug5arijhpxo2ahzd7fvsb34

Tucker tensor method for fast grid-based summation of long-range potentials on 3D lattices with defects [article]

Venera Khoromskaia, Boris N. Khoromskij
2015 arXiv   pre-print
In this paper, we present a method for fast summation of long-range potentials on 3D lattices with multiple defects and having non-rectangular geometries, based on rank-structured tensor representations  ...  For the reduced higher-order SVD approximation to a sum of canonical/Tucker tensors, we prove the stable error bounds in the relative norm in terms of discarded singular values of the side matrices.  ...  of far fields.  ... 
arXiv:1411.1994v2 fatcat:aj4xjj6n3jfwhb56aa33yu5awu

Ubiquitous Nature of the Reduced Higher Order SVD in Tensor-Based Scientific Computing

Venera Khoromskaia, Boris N. Khoromskij
2022 Frontiers in Applied Mathematics and Statistics  
This allows to apply the RHOSVD rank-reduction techniques to non-regular functional data with many singularities, for example, to the rank-structured computation of the collective multi-particle interaction  ...  basic ingredient in tensor computations for real-life problems.  ...  Tucker-to-Canonical Transform In the rank reduction scheme for the canonical rank-R tensors, we use successively the canonical-to-Tucker (C2T) transform and then the Tucker-to-canonical (T2C) tensor approximation  ... 
doi:10.3389/fams.2022.826988 fatcat:brhr6solpve4rhpxziggxch55y

Tensor Computation: A New Framework for High-Dimensional Problems in EDA

Zheng Zhang, Kim Batselier, Haotian Liu, Luca Daniel, Ngai Wong
2017 IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems  
open EDA problems where the use of tensor computation could be of advantage.  ...  This paper presents "tensor computation" as an alternative general framework for the development of efficient EDA algorithms and tools.  ...  In some cases, the given tensor may have some special properties such as symmetry or non-negativeness. These properties need to be preserved in their decomposed forms for specific applications. B.  ... 
doi:10.1109/tcad.2016.2618879 fatcat:4li26hkadvex5c3xs3eb2ijwk4

Range-separated tensor formats for numerical modeling of many-particle interaction potentials [article]

Peter Benner, Venera Khoromskaia, Boris N. Khoromskij
2016 arXiv   pre-print
The main idea of the RS tensor format is the independent grid-based low-rank representation of the localized and global parts in the target tensor which allows the efficient numerical approximation of  ...  We introduce and analyze the new range-separated (RS) canonical/Tucker tensor format which aims for numerical modeling of the 3D long-range interaction potentials in multi-particle systems.  ...  The Canonical-to-Tucker algorithm can be easily modified to use an ε-truncation stopping criterion.  ... 
arXiv:1606.09218v3 fatcat:54ktxocdxvc5zegbfocltxtjvi

Tucker-tensor algorithm for large-scale Kohn-Sham density functional theory calculations

Phani Motamarri, Vikram Gavini, Thomas Blesgen
2016 Physical review B  
In this work, we propose a systematic way of computing a low-rank globally-adapted localized Tucker-tensor basis for solving the Kohn-Sham DFT problem.  ...  In every iteration of the self-consistent field procedure of the Kohn-Sham DFT problem, we construct an additive separable approximation of the Kohn-Sham Hamiltonian.  ...  Hackbusch (MPI MIS, Leipzig) for the valuable comments and useful discussions. This work was performed in part under the auspices of AFOSR Grant No. FA9550-13-1-0113. V.  ... 
doi:10.1103/physrevb.93.125104 fatcat:4rleiamufnhgzeijiuterdyadq

Extension of PCA to Higher Order Data Structures: An Introduction to Tensors, Tensor Decompositions, and Tensor PCA [article]

Ali Zare, Alp Ozdemir, Mark A. Iwen, Selin Aviyente
2018 arXiv   pre-print
In particular, we present tensor methods that aim to solve three important challenges typically addressed by PCA: dimensionality reduction, i.e. low-rank tensor approximation, supervised learning, i.e.  ...  learning linear subspaces for feature extraction, and robust low-rank tensor recovery.  ...  Currently, PCA is commonly used to address three major problems in data science: 1) Dimensionality reduction for large and high dimensional data sets, i.e. low-rank subspace approximation [2] ; 2) Subspace  ... 
arXiv:1803.00704v2 fatcat:4pahe67v7fd23gbbkbiipxzj6q

Approximation of the electron density of Aluminium clusters in tensor-product format

T. Blesgen, V. Gavini, V. Khoromskaia
2012 Journal of Computational Physics  
This shows good promise for resolving the electronic structure of materials using tensor-structured techniques.  ...  As main result, the rank of the Tucker approximation for the accurate representation of the electron density is small and only weakly dependent on the system size for the systems studied here.  ...  This problem is circumvented with the Kohn-Sham approach (KSDFT) where the ground state properties are computed by solving for the wave-functions of a non-interacting system of electrons in a mean field  ... 
doi:10.1016/j.jcp.2011.12.009 fatcat:kzhdbzevkzcmxf76nqdq2orvle

Tensor Numerical Methods for High-dimensional PDEs: Basic Theory and Initial Applications [article]

Boris N. Khoromskij
2014 arXiv   pre-print
Combined with the traditional numerical schemes, these novel tools establish a new promising approach for solving multidimensional integral and differential equations using low-parametric rank-structured  ...  Numerical tests demonstrate the benefits of the rank-structured tensor approximation on the aforementioned examples of multidimensional PDEs.  ...  Khoromskaia (MPI MiS Leipzig) for useful comments on Section 2 and §3.1 which have led to the substantial improvement of the revised manuscript.  ... 
arXiv:1408.4053v1 fatcat:fyhlbkfrgzbp7awgzall3ywbpe

Tensor-based methods for numerical homogenization from high-resolution images

L. Giraldi, A. Nouy, G. Legrain, P. Cartraud
2013 Computer Methods in Applied Mechanics and Engineering  
It includes the tensor approximations in suitable tensor formats of fields of material properties or indicator functions of multiple material phases recovered from segmented images.  ...  We present a complete numerical strategy based on tensor approximation techniques for the solution of numerical homogenization problems with geometrical data coming from high resolution images.  ...  A priori definition of an approximation on a tensor subset Let us consider a tensor subset S (e.g. rank-1 tensors set, Tucker tensors set...).  ... 
doi:10.1016/j.cma.2012.10.012 fatcat:vdw2pvimxbafbhtjxeop2xovhe

Orthogonal tucker decomposition using factor priors for 2D+3D facial expression recognition

Yunfang Fu, Qiuqi Ruan, Ziyan Luo, Gaoyun An, Yi Jin
2021 IET Biometrics  
As a powerful technique, Tucker decomposition on the basis of the low rank approximation is often used to extract the useful information from the constructed 4D tensor composed of 3D face scans and 2D  ...  To recover the missing information, a framework for tensor completion (TC) will be embedded naturally.  ...  The reason for this phenomenon is the same because the rank reduction step in Algorithm 1 is used here.  ... 
doi:10.1049/bme2.12035 fatcat:kp466d7igjb5pdqnrpm27vvm4q

Clutter suppression in ultrasound: performance evaluation and review of low-rank and sparse matrix decomposition methods

Naiyuan Zhang, Md Ashikuzzaman, Hassan Rivaz
2020 BioMedical Engineering OnLine  
A potential solution to these issues is using decomposition into low-rank and sparse matrices (DLSM) framework.  ...  Many other algorithms under DLSM avoid full SVD and use approximated SVD or SVD-free ideas which may have better performance with higher robustness and less computing time.  ...  The authors also wish to acknowledge the LRSLibrary which has been developed by Andrews Sobral, for collecting and sharing the open-source DLSM algorithms.  ... 
doi:10.1186/s12938-020-00778-z pmid:32466753 fatcat:aewabqpjjfgozdgrcaylhly2eq

Tensor Decompositions for Integral Histogram Compression and Look-Up

Rafael Ballester-Ripoll, Renato Pajarola
2018 IEEE Transactions on Visualization and Computer Graphics  
Index Terms-integral histogram, tensor approximation, higher-order decompositions !  ...  To this end we propose an incremental tensor decomposition algorithm that allows us to compress integral histograms of hundreds of gigabytes.  ...  Schittny from the Institute of Anatomy at University of Bern for the Lung.  ... 
doi:10.1109/tvcg.2018.2802521 pmid:29994512 fatcat:4hnavb4jffcjrjjgzjojgob3u4
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