4 research outputs found
Efficient computation of highly oscillatory integrals by using QTT tensor approximation
We propose a new method for the efficient approximation of a class of highly
oscillatory weighted integrals where the oscillatory function depends on the
frequency parameter , typically varying in a large interval. Our
approach is based, for fixed but arbitrary oscillator, on the pre-computation
and low-parametric approximation of certain -dependent prototype
functions whose evaluation leads in a straightforward way to recover the target
integral. The difficulty that arises is that these prototype functions consist
of oscillatory integrals and are itself oscillatory which makes them both
difficult to evaluate and to approximate. Here we use the quantized-tensor
train (QTT) approximation method for functional -vectors of logarithmic
complexity in in combination with a cross-approximation scheme for TT
tensors. This allows the accurate approximation and efficient storage of these
functions in the wide range of grid and frequency parameters. Numerical
examples illustrate the efficiency of the QTT-based numerical integration
scheme on various examples in one and several spatial dimensions.Comment: 20 page
Computing the density of states for optical spectra by low-rank and QTT tensor approximation
In this paper, we introduce a new interpolation scheme to approximate the
density of states (DOS) for a class of rank-structured matrices with
application to the Tamm-Dancoff approximation (TDA) of the Bethe-Salpeter
equation (BSE). The presented approach for approximating the DOS is based on
two main techniques. First, we propose an economical method for calculating the
traces of parametric matrix resolvents at interpolation points by taking
advantage of the block-diagonal plus low-rank matrix structure described in [6,
3] for the BSE/TDA problem. Second, we show that a regularized or smoothed DOS
discretized on a fine grid of size can be accurately represented by a low
rank quantized tensor train (QTT) tensor that can be determined through a least
squares fitting procedure. The latter provides good approximation properties
for strictly oscillating DOS functions with multiple gaps, and requires
asymptotically much fewer () functional calls compared with the full
grid size . This approach allows us to overcome the computational
difficulties of the traditional schemes by avoiding both the need of stochastic
sampling and interpolation by problem independent functions like polynomials
etc. Numerical tests indicate that the QTT approach yields accurate recovery of
DOS associated with problems that contain relatively large spectral gaps. The
QTT tensor rank only weakly depends on the size of a molecular system which
paves the way for treating large-scale spectral problems.Comment: 26 pages, 25 figure
Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
Part 2 of this monograph builds on the introduction to tensor networks and
their operations presented in Part 1. It focuses on tensor network models for
super-compressed higher-order representation of data/parameters and related
cost functions, while providing an outline of their applications in machine
learning and data analytics. A particular emphasis is on the tensor train (TT)
and Hierarchical Tucker (HT) decompositions, and their physically meaningful
interpretations which reflect the scalability of the tensor network approach.
Through a graphical approach, we also elucidate how, by virtue of the
underlying low-rank tensor approximations and sophisticated contractions of
core tensors, tensor networks have the ability to perform distributed
computations on otherwise prohibitively large volumes of data/parameters,
thereby alleviating or even eliminating the curse of dimensionality. The
usefulness of this concept is illustrated over a number of applied areas,
including generalized regression and classification (support tensor machines,
canonical correlation analysis, higher order partial least squares),
generalized eigenvalue decomposition, Riemannian optimization, and in the
optimization of deep neural networks. Part 1 and Part 2 of this work can be
used either as stand-alone separate texts, or indeed as a conjoint
comprehensive review of the exciting field of low-rank tensor networks and
tensor decompositions.Comment: 232 page
Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
Part 2 of this monograph builds on the introduction to tensor networks and
their operations presented in Part 1. It focuses on tensor network models for
super-compressed higher-order representation of data/parameters and related
cost functions, while providing an outline of their applications in machine
learning and data analytics. A particular emphasis is on the tensor train (TT)
and Hierarchical Tucker (HT) decompositions, and their physically meaningful
interpretations which reflect the scalability of the tensor network approach.
Through a graphical approach, we also elucidate how, by virtue of the
underlying low-rank tensor approximations and sophisticated contractions of
core tensors, tensor networks have the ability to perform distributed
computations on otherwise prohibitively large volumes of data/parameters,
thereby alleviating or even eliminating the curse of dimensionality. The
usefulness of this concept is illustrated over a number of applied areas,
including generalized regression and classification (support tensor machines,
canonical correlation analysis, higher order partial least squares),
generalized eigenvalue decomposition, Riemannian optimization, and in the
optimization of deep neural networks. Part 1 and Part 2 of this work can be
used either as stand-alone separate texts, or indeed as a conjoint
comprehensive review of the exciting field of low-rank tensor networks and
tensor decompositions.Comment: 232 page