A locally convergent Jacobi iteration for the tensor singular value problem

Abstract

Multi-linear functionals or tensors are useful in study and analysis multi-dimensional signal and system. Tensor approximation, which has various applications in signal processing and system theory, can be achieved by generalizing the notion of singular values and singular vectors of matrices to tensor. In this paper, we showed local convergence of a parallelizable numerical method (based on the Jacobi iteration) for obtaining the singular values and singular vectors of a tensor.

A Fréchet differentiation

Let $\mathbb {X}$ and $\mathbb {Y}$ be Banach spaces. The Fréchet derivative of an operator $F:\mathbb {X}\rightarrow \mathbb {Y}$ at x is the unique linear operator $D_{F(x)}:\mathbb {X}\rightarrow \mathbb {Y}$ such that

$$\begin{aligned} \lim _{\Vert h\Vert \rightarrow 0} \frac{F(x+h)-F(x)-D_{F(x)}(h)}{\Vert h\Vert } =0. \end{aligned}$$

For details, see Ok (2007).

Double (Fréchet) differentiation of F at x in directions, first in y and then z, is the unique bilinear operator $D^2_{F(x)}(y,z)$ given by

$$\begin{aligned} D^2_{F(x)}(y,z) := D_{D_{F(x)}(y)}(z). \end{aligned}$$

Note that $D^2_{F(x)}(y,z) = D^2_{F(x)}(z,y)$ (Cartan 1971, Theorem 5.1.1).