Journal
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
Volume 30, Issue 4, Pages 1709-1734Publisher
SIAM PUBLICATIONS
DOI: 10.1137/070711621
Keywords
multilinear algebra; singular value decomposition; tensor decomposition; low rank approximation
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Funding
- NSF [DMS-0510131, DMS-0528492]
- Minnesota Supercomputing Institute
- Direct For Mathematical & Physical Scien [0810938] Funding Source: National Science Foundation
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It is known that a higher order tensor does not necessarily have an optimal low rank approximation, and that a tensor might not be orthogonally decomposable (i.e., admit a tensor SVD). We provide several sufficient conditions which lead to the failure of the tensor SVD, and characterize the existence of the tensor SVD with respect to the higher order SVD (HOSVD). In the face of these difficulties to generalize standard results known in the matrix case to tensors, we consider the low rank orthogonal approximation of tensors. The existence of an optimal approximation is theoretically guaranteed under certain conditions, and this optimal approximation yields a tensor decomposition where the diagonal of the core is maximized. We present an algorithm to compute this approximation and analyze its convergence behavior. Numerical experiments indicate a linear convergence rate for this algorithm.
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