Tensor Summary

Why use tensors:

  • natural data representation
  • get better compression when constructing a tensor from a vector or matrix, and efficient operation on the compressed tensor formats (e.g. canonical, Tucker, TT formats)

Problems:

  • No library to support fast tensor operations and tensor decompositions [source]
  • Dimensionality curse (Need to be more clear)
    • Space
    • Running time

Tensor Decompositions:

  • CP Decomposition:
    • The decomposition of tensor T is unique (up to scaling and permutation) if none of the vector pairs are co-linear.
    • Matrix decomposition (e.g. SVD) is not unique.
    • Algorithm: CP-ALS, CP-APR
  • Tucker Decomposition:
  • tensor power method:
  • Tensor Train: [Paper]
  • Hierarchical Tucker: [Paper]

Tensor Decomposition Applications:

  • Healthcare:
  • Deep Learning:
  • Machine Learning:
    • design learning algorithms for estimating parameters of latent variable models like Hidden Markov Model, Mixture of Gaussians and Latent Dirichlet Allocation, community models, probabilistic Context-Free-Grammars, and two-layer neural networks. [source]
    • Tensor methods are very competitive for unsupervised learning of large-scale probabilistic latent variable models, as opposed to traditional methods such as expectation maximization (EM) or Markov chain Monte Carlo (MCMC). The main gain is in terms of computation: (i) tensor methods are embarrassingly parallel and scalable to  large problems, (ii) they can build on efficient linear algebraic libraries, but are much more powerful and informative compared to matrix methods. On the other hand, tensor methods are not sample efficient, meaning they require more samples than EM to reach the same level of accuracy (assuming computation is not an issue). Improving statistical efficiency of spectral methods is an ongoing research topic. [source]

  • Data compression

Build tensors:

  • Build tensors from algorithm property, then do tensor decomposition
  • Build tensors from applications nature, then do tensor approximation
  • Build tensors from vectors or matrices, then do tensor approximation for data compression
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