Approximation of $2^d\times2^d$ Matrices Using Tensor Decomposition

Authors:
I. V. Oseledets
Affiliations:
ivan.oseledets@gmail.com
Venue:
SIAM Journal on Matrix Analysis and Applications
Year:
2010

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Abstract

A new method for structured representation of matrices and vectors is presented. The method is based on the representation of a matrix as a $d$-dimensional tensor and applying the TT-decomposition proposed recently. It turned out that for many important cases the number of parameters to represent an $n\times n$ matrix falls down to $\mathcal{O}(\log^{\alpha}n)$, giving a logarithmic storage. It is shown that this format can be used not only for storage reduction, but also for linear algebra operations. Possible applications include differential and integral equations, and data and image compression.