1,162 research outputs found
Operator Ideals arising from Generating Sequences
In this note, we will discuss how to relate an operator ideal on Banach
spaces to the sequential structures it defines. Concrete examples of ideals of
compact, weakly compact, completely continuous, Banach-Saks and weakly
Banach-Saks operators will be demonstrated.Comment: 17 pages, for the Proceedings of International Conference on Algebra
2010, World Scientific. (The International Conference on Algebra in honor of
the 70th birthday of Professor Shum Kar Ping was held by Universitas Gadjah
Mada (UGM)in Yogyakarta, Indonesia on October 7-10, 2010.
Symmetric Tensor Decomposition by an Iterative Eigendecomposition Algorithm
We present an iterative algorithm, called the symmetric tensor eigen-rank-one
iterative decomposition (STEROID), for decomposing a symmetric tensor into a
real linear combination of symmetric rank-1 unit-norm outer factors using only
eigendecompositions and least-squares fitting. Originally designed for a
symmetric tensor with an order being a power of two, STEROID is shown to be
applicable to any order through an innovative tensor embedding technique.
Numerical examples demonstrate the high efficiency and accuracy of the proposed
scheme even for large scale problems. Furthermore, we show how STEROID readily
solves a problem in nonlinear block-structured system identification and
nonlinear state-space identification
A constructive arbitrary-degree Kronecker product decomposition of tensors
We propose the tensor Kronecker product singular value decomposition~(TKPSVD)
that decomposes a real -way tensor into a linear combination
of tensor Kronecker products with an arbitrary number of factors
. We generalize the matrix Kronecker product to
tensors such that each factor in the TKPSVD is a -way
tensor. The algorithm relies on reshaping and permuting the original tensor
into a -way tensor, after which a polyadic decomposition with orthogonal
rank-1 terms is computed. We prove that for many different structured tensors,
the Kronecker product factors
are guaranteed to inherit this structure. In addition, we introduce the new
notion of general symmetric tensors, which includes many different structures
such as symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors.Comment: Rewrote the paper completely and generalized everything to tensor
A Constructive Algorithm for Decomposing a Tensor into a Finite Sum of Orthonormal Rank-1 Terms
We propose a constructive algorithm that decomposes an arbitrary real tensor
into a finite sum of orthonormal rank-1 outer products. The algorithm, named
TTr1SVD, works by converting the tensor into a tensor-train rank-1 (TTr1)
series via the singular value decomposition (SVD). TTr1SVD naturally
generalizes the SVD to the tensor regime with properties such as uniqueness for
a fixed order of indices, orthogonal rank-1 outer product terms, and easy
truncation error quantification. Using an outer product column table it also
allows, for the first time, a complete characterization of all tensors
orthogonal with the original tensor. Incidentally, this leads to a strikingly
simple constructive proof showing that the maximum rank of a real tensor over the real field is 3. We also derive a conversion of the
TTr1 decomposition into a Tucker decomposition with a sparse core tensor.
Numerical examples illustrate each of the favorable properties of the TTr1
decomposition.Comment: Added subsection on orthogonal complement tensors. Added constructive
proof of maximal CP-rank of a 2x2x2 tensor. Added perturbation of singular
values result. Added conversion of the TTr1 decomposition to the Tucker
decomposition. Added example that demonstrates how the rank behaves when
subtracting rank-1 terms. Added example with exponential decaying singular
value
Tensor Network alternating linear scheme for MIMO Volterra system identification
This article introduces two Tensor Network-based iterative algorithms for the
identification of high-order discrete-time nonlinear multiple-input
multiple-output (MIMO) Volterra systems. The system identification problem is
rewritten in terms of a Volterra tensor, which is never explicitly constructed,
thus avoiding the curse of dimensionality. It is shown how each iteration of
the two identification algorithms involves solving a linear system of low
computational complexity. The proposed algorithms are guaranteed to
monotonically converge and numerical stability is ensured through the use of
orthogonal matrix factorizations. The performance and accuracy of the two
identification algorithms are illustrated by numerical experiments, where
accurate degree-10 MIMO Volterra models are identified in about 1 second in
Matlab on a standard desktop pc
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