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A Tensor is an 𝑑-way array

 

 

 

 

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If you need more information, these references are helpful:�

  1. Kolda, T. G., & Bader, B. W. (2009). Tensor decompositions and applications. SIAM review, 51(3), 455-500.
  2. Matlab Tensor Toolbox: https://www.tensortoolbox.org/

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The following slides contain material from Tammy Kolda, Charles Van Loan and Lars Elden

Tensor Decompositions

 

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What is a Tensor

 

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Tensor

Definition:

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Notations

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Tensor Unfolding

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A Common Framework for Tensor Computations

Tensor Unfolding

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Illustration of row-wise and column-wise unfolding (flattening, matricizing) of a third-order tensor

Andrzej Cichocki, Rafal Zdunek, and Shun'ichi Amari. Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source (2009)

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SVD Expansion

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Tensor Decompositions

CP Model: Sum of d-way outer products, useful for interpretation

Tucker Model: Project onto high-variance subspaces to reduce dimensionality

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The Kronecker Product

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The Kronecker Product

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All Possible Entry- Entry Products

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Kronecker Products of Kronecker Products

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You have seen them before

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Block matrix

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Block Matrices: Addition

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Block Matrix: Multiplication

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Block Matrix: Transposition

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A block matrix can have block matrix entries

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Reshape

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Reshape

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SVD

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SVD

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The Tucker product

Definition:

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The Tucker product representation

Before…

Now…

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The Higher-Order SVD (HOSVD)

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The HOSVD of a matrix

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The HOSVD of a matrix