Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions

@article{Oseledets2009BreakingTC,
  title={Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions},
  author={I. Oseledets and E. Tyrtyshnikov},
  journal={SIAM J. Sci. Comput.},
  year={2009},
  volume={31},
  pages={3744-3759}
}
  • I. Oseledets, E. Tyrtyshnikov
  • Published 2009
  • Mathematics, Computer Science
  • SIAM J. Sci. Comput.
  • For $d$-dimensional tensors with possibly large $d>3$, an hierarchical data structure, called the Tree-Tucker format, is presented as an alternative to the canonical decomposition. It has asymptotically the same (and often even smaller) number of representation parameters and viable stability properties. The approach involves a recursive construction described by a tree with the leafs corresponding to the Tucker decompositions of three-dimensional tensors, and is based on a sequence of SVDs for… CONTINUE READING
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    References

    SHOWING 1-10 OF 21 REFERENCES
    Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time
    • 137
    • PDF
    Efficient MATLAB Computations with Sparse and Factored Tensors
    • 381
    • PDF
    On the Best Rank-1 and Rank-(R1 , R2, ... , RN) Approximation of Higher-Order Tensors
    • 1,259
    Tensor Decompositions and Applications
    • 6,061
    • PDF
    On the Best Rank-1 and Rank-(
    • 183
    • Highly Influential
    Numerical operator calculus in higher dimensions
    • 263
    • Highly Influential
    • PDF
    Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem
    • 747
    • PDF
    Mosaic-Skeleton approximations
    • 186
    • PDF
    Approximation with Kronecker Products
    • 337
    • PDF
    Algorithms for Numerical Analysis in High Dimensions
    • 334
    • PDF