43,583 research outputs found
Tensor Networks for Big Data Analytics and Large-Scale Optimization Problems
In this paper we review basic and emerging models and associated algorithms
for large-scale tensor networks, especially Tensor Train (TT) decompositions
using novel mathematical and graphical representations. We discus the concept
of tensorization (i.e., creating very high-order tensors from lower-order
original data) and super compression of data achieved via quantized tensor
train (QTT) networks. The purpose of a tensorization and quantization is to
achieve, via low-rank tensor approximations "super" compression, and
meaningful, compact representation of structured data. The main objective of
this paper is to show how tensor networks can be used to solve a wide class of
big data optimization problems (that are far from tractable by classical
numerical methods) by applying tensorization and performing all operations
using relatively small size matrices and tensors and applying iteratively
optimized and approximative tensor contractions.
Keywords: Tensor networks, tensor train (TT) decompositions, matrix product
states (MPS), matrix product operators (MPO), basic tensor operations,
tensorization, distributed representation od data optimization problems for
very large-scale problems: generalized eigenvalue decomposition (GEVD),
PCA/SVD, canonical correlation analysis (CCA).Comment: arXiv admin note: text overlap with arXiv:1403.204
Core-Core Dynamics in Spin Vortex Pairs
We investigate magnetic nano-pillars, in which two thin ferromagnetic
nanoparticles are separated by a nanometer thin nonmagnetic spacer and can be
set into stable spin vortex-pair configurations. The 16 ground states of the
vortex-pair system are characterized by parallel or antiparallel chirality and
parallel or antiparallel core-core alignment. We detect and differentiate these
individual vortex-pair states experimentally and analyze their dynamics
analytically and numerically. Of particular interest is the limit of strong
core-core coupling, which we find can dominate the spin dynamics in the system.
We observe that the 0.2 GHz gyrational resonance modes of the individual
vortices are replaced with 2-6 GHz range collective rotational and vibrational
core-core resonances in the configurations where the cores form a bound pair.
These results demonstrate new opportunities in producing and manipulating spin
states on the nanoscale and may prove useful for new types of ultra-dense
storage devices where the information is stored as multiple vortex-core
configurations
Tensor-based dynamic mode decomposition
Dynamic mode decomposition (DMD) is a recently developed tool for the
analysis of the behavior of complex dynamical systems. In this paper, we will
propose an extension of DMD that exploits low-rank tensor decompositions of
potentially high-dimensional data sets to compute the corresponding DMD modes
and eigenvalues. The goal is to reduce the computational complexity and also
the amount of memory required to store the data in order to mitigate the curse
of dimensionality. The efficiency of these tensor-based methods will be
illustrated with the aid of several different fluid dynamics problems such as
the von K\'arm\'an vortex street and the simulation of two merging vortices
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