28,203 research outputs found
Decomposition of geometric perturbations
For an infinitesimal deformation of a Riemannian manifold, we prove that the
scalar, vector, and tensor modes in decompositions of perturbations of the
metric tensor, the scalar curvature, the Ricci tensor, and the Einstein tensor
decouple if and only if the manifold is Einstein. Four-dimensional space-time
satisfying the condition of the theorem is homogeneous and isotropic.
Cosmological applications are discussed.Comment: 7 page
Empirical Evaluation of Four Tensor Decomposition Algorithms
Higher-order tensor decompositions are analogous to the familiar Singular Value Decomposition (SVD), but they transcend the limitations of matrices (second-order tensors). SVD is a powerful tool that has achieved impressive results in information retrieval, collaborative filtering, computational linguistics, computational vision, and other fields. However, SVD is limited to two-dimensional arrays of data (two modes), and many potential applications have three or more modes, which require higher-order tensor decompositions. This paper evaluates four algorithms for higher-order tensor decomposition: Higher-Order Singular Value Decomposition (HO-SVD), Higher-Order Orthogonal Iteration (HOOI), Slice Projection (SP), and Multislice Projection (MP). We measure the time (elapsed run time), space (RAM and disk space requirements), and fit (tensor reconstruction accuracy) of the four algorithms, under a variety of conditions. We find that standard implementations of HO-SVD and HOOI do not scale up to larger tensors, due to increasing RAM requirements. We recommend HOOI for tensors that are small enough for the available RAM and MP for larger tensors
Tensor Decompositions for Signal Processing Applications From Two-way to Multiway Component Analysis
The widespread use of multi-sensor technology and the emergence of big
datasets has highlighted the limitations of standard flat-view matrix models
and the necessity to move towards more versatile data analysis tools. We show
that higher-order tensors (i.e., multiway arrays) enable such a fundamental
paradigm shift towards models that are essentially polynomial and whose
uniqueness, unlike the matrix methods, is guaranteed under verymild and natural
conditions. Benefiting fromthe power ofmultilinear algebra as theirmathematical
backbone, data analysis techniques using tensor decompositions are shown to
have great flexibility in the choice of constraints that match data properties,
and to find more general latent components in the data than matrix-based
methods. A comprehensive introduction to tensor decompositions is provided from
a signal processing perspective, starting from the algebraic foundations, via
basic Canonical Polyadic and Tucker models, through to advanced cause-effect
and multi-view data analysis schemes. We show that tensor decompositions enable
natural generalizations of some commonly used signal processing paradigms, such
as canonical correlation and subspace techniques, signal separation, linear
regression, feature extraction and classification. We also cover computational
aspects, and point out how ideas from compressed sensing and scientific
computing may be used for addressing the otherwise unmanageable storage and
manipulation problems associated with big datasets. The concepts are supported
by illustrative real world case studies illuminating the benefits of the tensor
framework, as efficient and promising tools for modern signal processing, data
analysis and machine learning applications; these benefits also extend to
vector/matrix data through tensorization. Keywords: ICA, NMF, CPD, Tucker
decomposition, HOSVD, tensor networks, Tensor Train
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
Max vs Min: Tensor Decomposition and ICA with nearly Linear Sample Complexity
We present a simple, general technique for reducing the sample complexity of
matrix and tensor decomposition algorithms applied to distributions. We use the
technique to give a polynomial-time algorithm for standard ICA with sample
complexity nearly linear in the dimension, thereby improving substantially on
previous bounds. The analysis is based on properties of random polynomials,
namely the spacings of an ensemble of polynomials. Our technique also applies
to other applications of tensor decompositions, including spherical Gaussian
mixture models
Mixed Tensors of the General Linear Supergroup
We describe the image of the canonical tensor functor from Deligne's
interpolating category to attached to the
standard representation. This implies explicit tensor product decompositions
between any two projective modules and any two Kostant modules of ,
covering the decomposition between any two irreducible
-representations. We also obtain character and dimension formulas. For
we classify the mixed tensors with non-vanishing superdimension. For
we characterize the maximally atypical mixed tensors and show some
applications regarding tensor products.Comment: v3: Improved exposition, corrected minor mistakes v2: shortened and
revised version. Comments welcom
Multidimensional Data Analysis Based on Block Convolutional Tensor Decomposition
Tensor decompositions are powerful tools for analyzing multi-dimensional data
in their original format. Besides tensor decompositions like Tucker and CP,
Tensor SVD (t-SVD) which is based on the t-product of tensors is another
extension of SVD to tensors that recently developed and has found numerous
applications in analyzing high dimensional data. This paper offers a new
insight into the t-Product and shows that this product is a block convolution
of two tensors with periodic boundary conditions. Based on this viewpoint, we
propose a new tensor-tensor product called the based
on Block convolution with reflective boundary conditions. Using a tensor
framework, this product can be easily extended to tensors of arbitrary order.
Additionally, we introduce a tensor decomposition based on our
for arbitrary order tensors. Compared to t-SVD, our
new decomposition has lower complexity, and experiments show that it yields
higher-quality results in applications such as classification and compression
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