2,898 research outputs found
Tensor completion in hierarchical tensor representations
Compressed sensing extends from the recovery of sparse vectors from
undersampled measurements via efficient algorithms to the recovery of matrices
of low rank from incomplete information. Here we consider a further extension
to the reconstruction of tensors of low multi-linear rank in recently
introduced hierarchical tensor formats from a small number of measurements.
Hierarchical tensors are a flexible generalization of the well-known Tucker
representation, which have the advantage that the number of degrees of freedom
of a low rank tensor does not scale exponentially with the order of the tensor.
While corresponding tensor decompositions can be computed efficiently via
successive applications of (matrix) singular value decompositions, some
important properties of the singular value decomposition do not extend from the
matrix to the tensor case. This results in major computational and theoretical
difficulties in designing and analyzing algorithms for low rank tensor
recovery. For instance, a canonical analogue of the tensor nuclear norm is
NP-hard to compute in general, which is in stark contrast to the matrix case.
In this book chapter we consider versions of iterative hard thresholding
schemes adapted to hierarchical tensor formats. A variant builds on methods
from Riemannian optimization and uses a retraction mapping from the tangent
space of the manifold of low rank tensors back to this manifold. We provide
first partial convergence results based on a tensor version of the restricted
isometry property (TRIP) of the measurement map. Moreover, an estimate of the
number of measurements is provided that ensures the TRIP of a given tensor rank
with high probability for Gaussian measurement maps.Comment: revised version, to be published in Compressed Sensing and Its
Applications (edited by H. Boche, R. Calderbank, G. Kutyniok, J. Vybiral
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
Tensor Ring Decomposition with Rank Minimization on Latent Space: An Efficient Approach for Tensor Completion
In tensor completion tasks, the traditional low-rank tensor decomposition
models suffer from the laborious model selection problem due to their high
model sensitivity. In particular, for tensor ring (TR) decomposition, the
number of model possibilities grows exponentially with the tensor order, which
makes it rather challenging to find the optimal TR decomposition. In this
paper, by exploiting the low-rank structure of the TR latent space, we propose
a novel tensor completion method which is robust to model selection. In
contrast to imposing the low-rank constraint on the data space, we introduce
nuclear norm regularization on the latent TR factors, resulting in the
optimization step using singular value decomposition (SVD) being performed at a
much smaller scale. By leveraging the alternating direction method of
multipliers (ADMM) scheme, the latent TR factors with optimal rank and the
recovered tensor can be obtained simultaneously. Our proposed algorithm is
shown to effectively alleviate the burden of TR-rank selection, thereby greatly
reducing the computational cost. The extensive experimental results on both
synthetic and real-world data demonstrate the superior performance and
efficiency of the proposed approach against the state-of-the-art algorithms
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