7,422 research outputs found
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
Dictionary-based Tensor Canonical Polyadic Decomposition
To ensure interpretability of extracted sources in tensor decomposition, we
introduce in this paper a dictionary-based tensor canonical polyadic
decomposition which enforces one factor to belong exactly to a known
dictionary. A new formulation of sparse coding is proposed which enables high
dimensional tensors dictionary-based canonical polyadic decomposition. The
benefits of using a dictionary in tensor decomposition models are explored both
in terms of parameter identifiability and estimation accuracy. Performances of
the proposed algorithms are evaluated on the decomposition of simulated data
and the unmixing of hyperspectral images
The condition number of join decompositions
The join set of a finite collection of smooth embedded submanifolds of a
mutual vector space is defined as their Minkowski sum. Join decompositions
generalize some ubiquitous decompositions in multilinear algebra, namely tensor
rank, Waring, partially symmetric rank and block term decompositions. This
paper examines the numerical sensitivity of join decompositions to
perturbations; specifically, we consider the condition number for general join
decompositions. It is characterized as a distance to a set of ill-posed points
in a supplementary product of Grassmannians. We prove that this condition
number can be computed efficiently as the smallest singular value of an
auxiliary matrix. For some special join sets, we characterized the behavior of
sequences in the join set converging to the latter's boundary points. Finally,
we specialize our discussion to the tensor rank and Waring decompositions and
provide several numerical experiments confirming the key results
Renormalization of tensor-network states
We have discussed the tensor-network representation of classical statistical
or interacting quantum lattice models, and given a comprehensive introduction
to the numerical methods we recently proposed for studying the tensor-network
states/models in two dimensions. A second renormalization scheme is introduced
to take into account the environment contribution in the calculation of the
partition function of classical tensor network models or the expectation values
of quantum tensor network states. It improves significantly the accuracy of the
coarse grained tensor renormalization group method. In the study of the quantum
tensor-network states, we point out that the renormalization effect of the
environment can be efficiently and accurately described by the bond vector.
This, combined with the imaginary time evolution of the wavefunction, provides
an accurate projection method to determine the tensor-network wavfunction. It
reduces significantly the truncation error and enable a tensor-network state
with a large bond dimension, which is difficult to be accessed by other
methods, to be accurately determined.Comment: 18 pages 23 figures, minor changes, references adde
Tensor decomposition with generalized lasso penalties
We present an approach for penalized tensor decomposition (PTD) that
estimates smoothly varying latent factors in multi-way data. This generalizes
existing work on sparse tensor decomposition and penalized matrix
decompositions, in a manner parallel to the generalized lasso for regression
and smoothing problems. Our approach presents many nontrivial challenges at the
intersection of modeling and computation, which are studied in detail. An
efficient coordinate-wise optimization algorithm for (PTD) is presented, and
its convergence properties are characterized. The method is applied both to
simulated data and real data on flu hospitalizations in Texas. These results
show that our penalized tensor decomposition can offer major improvements on
existing methods for analyzing multi-way data that exhibit smooth spatial or
temporal features
Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure
The numerical solution of partial differential equations on high-dimensional
domains gives rise to computationally challenging linear systems. When using
standard discretization techniques, the size of the linear system grows
exponentially with the number of dimensions, making the use of classic
iterative solvers infeasible. During the last few years, low-rank tensor
approaches have been developed that allow to mitigate this curse of
dimensionality by exploiting the underlying structure of the linear operator.
In this work, we focus on tensors represented in the Tucker and tensor train
formats. We propose two preconditioned gradient methods on the corresponding
low-rank tensor manifolds: A Riemannian version of the preconditioned
Richardson method as well as an approximate Newton scheme based on the
Riemannian Hessian. For the latter, considerable attention is given to the
efficient solution of the resulting Newton equation. In numerical experiments,
we compare the efficiency of our Riemannian algorithms with other established
tensor-based approaches such as a truncated preconditioned Richardson method
and the alternating linear scheme. The results show that our approximate
Riemannian Newton scheme is significantly faster in cases when the application
of the linear operator is expensive.Comment: 24 pages, 8 figure
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