Efficient and Stable Decompositions for Tensors in Many Dimensions

Decompositions of d-dimensional tensors are crucial either in structure recovery problems and merely for a compact representation of tensors. However, the well-known decompositions have serious drawbacks: the Tucker decompositions suffer from exponential dependence on the dimensionality d while fixed-rank canonical decompositions are not stable. In this talk we present new decompositions [1] that are stable and have the same number of representation parameters as canonical decompositions for the same tensor. Moreover, the new format possesses nice stability properties of the SVD (as opposed to the canonical format) and is convenient for basic operations with tensors. This work is joint with Ivan Oseledets. [1] I.V.Oseledets and E.E.Tyrtyshnikov, Breaking the curse of dimensionality, or how to use SVD in many dimensions, Research Report 09-03, Kowloon Tong, Hong Kong: ICM HKBU, 2009 (www.math.hkbu.edu.hk/ICM/pdf/09-03.pdf).