scholarly journals An Introduction to Hierarchical (H-) Rank and TT-Rank of Tensors with Examples

2011 ◽  
Vol 11 (3) ◽  
pp. 291-304 ◽  
Author(s):  
Lars Grasedyck ◽  
Wolfgang Hackbusch
Keyword(s):  
Sparse Representation ◽  
Higher Order ◽  
The Other ◽  
Matrix Rank ◽  
The Matrix ◽  
Rank One ◽  
Higher Order Tensors

Abstract We review two similar concepts of hierarchical rank of tensors (which extend the matrix rank to higher order tensors): the TT-rank and the H-rank (hierarchical or H-Tucker rank). Based on this notion of rank, one can define a data-sparse representation of tensors involving O(dnk + dk^3) data for order d tensors with mode sizes n and rank k. Simple examples underline the differences and similarities between the different formats and ranks. Finally, we derive rank bounds for tensors in one of the formats based on the ranks in the other format.

scholarly journals Fast Estimation of Approximate Matrix Ranks Using Spectral Densities

2017 ◽  
Vol 29 (5) ◽  
pp. 1317-1351 ◽  
Author(s):  
Shashanka Ubaru ◽  
Yousef Saad ◽  
Abd-Krim Seghouane
Keyword(s):  
Spectral Density ◽  
Large Data ◽  
Lanczos Algorithm ◽  
Spectral Densities ◽  
Matrix Rank ◽  
Density Distributions ◽  
Fast Estimation ◽  
The Matrix ◽  
Rank One ◽  
Probability Density Distributions

Many machine learning and data-related applications require the knowledge of approximate ranks of large data matrices at hand. This letter presents two computationally inexpensive techniques to estimate the approximate ranks of such matrices. These techniques exploit approximate spectral densities, popular in physics, which are probability density distributions that measure the likelihood of finding eigenvalues of the matrix at a given point on the real line. Integrating the spectral density over an interval gives the eigenvalue count of the matrix in that interval. Therefore, the rank can be approximated by integrating the spectral density over a carefully selected interval. Two different approaches are discussed to estimate the approximate rank, one based on Chebyshev polynomials and the other based on the Lanczos algorithm. In order to obtain the appropriate interval, it is necessary to locate a gap between the eigenvalues that correspond to noise and the relevant eigenvalues that contribute to the matrix rank. A method for locating this gap and selecting the interval of integration is proposed based on the plot of the spectral density. Numerical experiments illustrate the performance of these techniques on matrices from typical applications.

scholarly journals Scaled Coupled Norms and Coupled Higher-Order Tensor Completion

2020 ◽  
Vol 32 (2) ◽  
pp. 447-484
Author(s):  
Kishan Wimalawarne ◽  
Makoto Yamada ◽  
Hiroshi Mamitsuka
Keyword(s):  
Real Data ◽  
Higher Order ◽  
Low Rank ◽  
Tensor Completion ◽  
Coupled Modes ◽  
Coupled Mode ◽  
The Matrix ◽  
Tensor Norms ◽  
Higher Order Tensors ◽  
Risk Bounds

Recently, a set of tensor norms known as coupled norms has been proposed as a convex solution to coupled tensor completion. Coupled norms have been designed by combining low-rank inducing tensor norms with the matrix trace norm. Though coupled norms have shown good performances, they have two major limitations: they do not have a method to control the regularization of coupled modes and uncoupled modes, and they are not optimal for couplings among higher-order tensors. In this letter, we propose a method that scales the regularization of coupled components against uncoupled components to properly induce the low-rankness on the coupled mode. We also propose coupled norms for higher-order tensors by combining the square norm to coupled norms. Using the excess risk-bound analysis, we demonstrate that our proposed methods lead to lower risk bounds compared to existing coupled norms. We demonstrate the robustness of our methods through simulation and real-data experiments.

Pembobotan Berdasarkan Tingkat Kesamaan Semantik pada Metode Fuzzy Semi-Supervised Co-Clustering untuk Pengelompokkan Dokumen Teks

2014 ◽  
Vol 6 (2) ◽  
pp. 46-51
Author(s):  
Galang Amanda Dwi P. ◽  
Gregorius Edwadr ◽  
Agus Zainal Arifin
Keyword(s):  
Supervised Learning ◽  
Semantic Similarity ◽  
The Other ◽  
Classification Result ◽  
Document Similarity ◽  
The Matrix ◽  
Index Terms ◽  
Membership Value ◽  
Degree Of Similarity

Nowadays, a large number of information can not be reached by the reader because of the misclassification of text-based documents. The misclassified data can also make the readers obtain the wrong information. The method which is proposed by this paper is aiming to classify the documents into the correct group.  Each document will have a membership value in several different classes. The method will be used to find the degree of similarity between the two documents is the semantic similarity. In fact, there is no document that doesn’t have a relationship with the other but their relationship might be close to 0. This method calculates the similarity between two documents by taking into account the level of similarity of words and their synonyms. After all inter-document similarity values obtained, a matrix will be created. The matrix is then used as a semi-supervised factor. The output of this method is the value of the membership of each document, which must be one of the greatest membership value for each document which indicates where the documents are grouped. Classification result computed by the method shows a good value which is 90 %. Index Terms - Fuzzy co-clustering, Heuristic, Semantica Similiarity, Semi-supervised learning.

scholarly journals Voigt Transform and Umbral Image

2020 ◽  
Vol 25 (3) ◽  
pp. 49
Author(s):  
Silvia Licciardi ◽  
Rosa Maria Pidatella ◽  
Marcello Artioli ◽  
Giuseppe Dattoli
Keyword(s):  
Spectroscopic Studies ◽  
Higher Order ◽  
Point Of View ◽  
The Other ◽  
Pivotal Role ◽  
Umbral Calculus ◽  
Laguerre Functions ◽  
Hermite And Laguerre Functions ◽  
Higher Order Functions ◽  
Voigt Function

In this paper, we show that the use of methods of an operational nature, such as umbral calculus, allows achieving a double target: on one side, the study of the Voigt function, which plays a pivotal role in spectroscopic studies and in other applications, according to a new point of view, and on the other, the introduction of a Voigt transform and its possible use. Furthermore, by the same method, we point out that the Hermite and Laguerre functions, extension of the corresponding polynomials to negative and/or real indices, can be expressed through a definition in a straightforward and unified fashion. It is illustrated how the techniques that we are going to suggest provide an easy derivation of the relevant properties along with generalizations to higher order functions.

Matrix-Mineral Relationships-A. Morphologist's Viewpoint

1979 ◽  
Vol 58 (2_suppl) ◽  
pp. 922-929 ◽  
Author(s):  
M.U. Nylen
Keyword(s):  
Organic Material ◽  
Crucial Role ◽  
Crystal Surface ◽  
Organic Matrix ◽  
The Other ◽  
Enamel Matrix ◽  
Initial Cell ◽  
Cell Product ◽  
Ultrastructural Morphology ◽  
The Matrix

The literature on the ultrastructural morphology of the enamel matrix and its relationship to the crystals is reviewed. Two morphological entities of the matrix are discussed: One is the so-called stippled material which may be the initial cell product; the other, variously described as fibrillar, lamellar, tubular or helical, is thought by many to play a crucial role in nucleation and orientation of the crystals. A number of observations, however, suggest that the latter structures form secondarily to the crystals and that in reality they represent organic material adsorbed to the crystal surface and maintained as independent structures upon removal of the mineral. The need for additional studies is stressed including systematic studies of interactions between constituents of the organic matrix and the apatite crystals.

A Yield Criterion of Porous Materials With Anisotropy Caused by Geometry or Distribution of Pores

1991 ◽  
Vol 113 (4) ◽  
pp. 425-429 ◽  
Author(s):  
T. Hisatsune ◽  
T. Tabata ◽  
S. Masaki
Keyword(s):  
Velocity Field ◽  
Porous Materials ◽  
Upper Bound ◽  
Volume Fraction ◽  
Yield Criterion ◽  
Longitudinal Direction ◽  
Axisymmetric Deformation ◽  
The Other ◽  
The Matrix ◽  
Spherical Cells

Axisymmetric deformation of anisotropic porous materials caused by geometry of pores or by distribution of pores is analyzed. Two models of the materials are proposed: one consists of spherical cells each of which has a concentric ellipsoidal pore; and the other consists of ellipsoidal cells each of which has a concentric spherical pore. The velocity field in the matrix is assumed and the upper bound approach is attempted. Yield criteria are expressed as ellipses on the σm σ3 plane which are longer in longitudinal direction with increasing anisotropy and smaller with increasing volume fraction of the pore. Furthermore, the axes rotate about the origin at an angle α from the σm-axis, while the axis for isotropic porous materials is on the σm-axis.

Wounded by reality: emotional deprivation and unresolved trauma in psychoanalytical couple’s therapy

2021 ◽  
Vol 15 (2) ◽  
pp. 253-268
Author(s):  
Hélène Béïnoglou
Keyword(s):  
Romantic Relationships ◽  
Emotional Experience ◽  
Therapeutic Process ◽  
Clinical Work ◽  
The Other ◽  
The Matrix ◽  
Emotional Deprivation ◽  
Unresolved Trauma

In this article, I will focus on highly conflictual couples with extensive emotional deprivation and unresolved trauma, which prevents them from developing healthy romantic relationships and overcoming the challenges entailed in any intimate attachment. I will describe how everyday interactions are experienced as threatening or even lethal movements between the partners. The question which arises in the psychoanalytical therapeutic process is how to help the couple tolerate the sensory reminders of the unresolved trauma as a necessary precursor to any process of symbolisation. In order to provide a safe enough therapeutic attachment bond, extensive time is dedicated to the emotional experience of self and the other in the here-and-now of the session, which validates the emotional experience of the couple as well as contains it. The therapy focuses on the transferential and countertransferential movements inspired by the matrix of the victim, abuser, and uninvolved witness (Davies & Frawley, 1994) to elaborate the intertwining of the unresolved trauma with the couple’s form of attachment. In order to illustrate my argument, I present two examples: one from a fictional narration and another from my clinical work.

Concise Equations for Rotor Dynamics Analysis

1995 ◽  
Author(s):  
W. J. Chen
Keyword(s):  
Equations Of Motion ◽  
Rotor Dynamics ◽  
Memory Storage ◽  
The Other ◽  
Dynamics Analysis ◽  
Real Domain ◽  
The Matrix ◽  
System Equations ◽  
Ordering Algorithms ◽  
Matrix Bandwidth

Abstract Concise equations for rotor dynamics analysis are presented. Two coordinate ordering methods are introduced in the element equations of motion. One is in the real domain and the other is in the complex domain. The two proposed ordering algorithms lead to more compact element matrices. A station numbering technique is also proposed for the system equations during the assembly process. This numbering technique can minimize the matrix bandwidth, the memory storage and can increase the computational efficiency.

2D Structural Acoustic Analysis Using the FEM/FMBEM with Different Coupled Element Types

2017 ◽  
Vol 42 (1) ◽  
pp. 37-48 ◽  
Author(s):  
Leilei Chen ◽  
Wenchang Zhao ◽  
Cheng Liu ◽  
Haibo Chen
Keyword(s):  
Acoustic Analysis ◽  
Interaction Analysis ◽  
Fast Multipole Method ◽  
Boundary Elements ◽  
Higher Order ◽  
Boundary Integral ◽  
The Matrix ◽  
Coupling Approach ◽  
Element Type ◽  
Relative Errors

Abstract A FEM-BEM coupling approach is used for acoustic fluid-structure interaction analysis. The FEM is used to model the structure and the BEM is used to model the exterior acoustic domain. The aim of this work is to improve the computational efficiency and accuracy of the conventional FEM-BEM coupling approach. The fast multipole method (FMM) is applied to accelerating the matrix-vector products in BEM. The Burton-Miller formulation is used to overcome the fictitious eigen-frequency problem when using a single Helmholtz boundary integral equation for exterior acoustic problems. The continuous higher order boundary elements and discontinuous higher order boundary elements for 2D problem are developed in this work to achieve higher accuracy in the coupling analysis. The performance for coupled element types is compared via a simple example with analytical solution, and the optimal element type is obtained. Numerical examples are presented to show the relative errors of different coupled element types.

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