Higher-order singular value decomposition (HOSVD) is a generalization of the well-known singular value decomposition (SVD) for matrices. HOSVD defines a decomposition for tensors of higher dimension (greater than two). This decomposition enables reducing the rank of the tensors, which can be used as a method for data compression. In this Demonstration, color images are interpreted as third-order tensors. The third dimension corresponds to RGB channels and is not rank reduced. You can vary the ranks of the first and second dimensions, the

pixels.

For a matrix, the column and row ranks are equal, and a unique matrix rank can be defined. For higher-dimensional tensors this does not hold, but a rank for each of its dimensions can be defined. This concept is called the

-rank. For a detailed description of this rank concept, see [1].

For matrices, a low-rank approximation is optimal. An

-rank approximation of a tensor is not necessarily optimal. The higher-order orthogonal iteration (HOOI) is an optimization algorithm that improves the low-rank approximation. For HOOI, see [2].

**Explanation of the Controls**"compression rate" slider: adjusts the ranks of the first and second dimensions 1–128; 128 means full-rank HOSVD (no compression), 1 means rank-(1,1,3)-HOSVD (maximum compression in this Demonstration)

"optimization" checkbox: controls whether the HOOI optimization algorithm (with 10 iterations) is executed or not

"test image" buttons: choose one of three different test images from

*Mathematica*'s

ExampleData[1] L. De Lathauwer, B. De Moor, and J. Vandewalle, "A Multilinear Singular Value Decomposition,"

*SIAM Journal on Matrix Analysis and Applications*,

**21**(4), 2000 pp. 1253–1278.

doi:10.1137/S0895479896305696.

[2] L. De Lathauwer and J. Vandewalle, "Dimensionality Reduction in Higher-Order Signal Processing and Rank-(R1,R2,...,RN) Reduction in Multilinear Algebra,"

*Linear Algebra and Its Applications*,

**391**, 2004 pp. 31–55.

doi:10.1016/j.laa.2004.01.016.