Trace maximization algorithm for the approximate tensor diagonalization

TL;DR

A Jacobi-type algorithm for the approximate diagonalization of tensors of order d ≥3 via tensor trace maximization via tensor trace maximization is developed and it is shown that both versions of the algorithm converge to the stationary points of the corresponding objective functions.

Abstract

In this paper we develop a Jacobi-type algorithm for the approximate diagonalization of tensors of order $d\geq3$ via tensor trace maximization. For a general tensor this is an alternating least squares algorithm and the rotation matrices are chosen in each mode one-by-one to maximize the tensor trace. On the other hand, for symmetric tensors we discuss a structure-preserving variant of this algorithm where in each iteration the same rotation is applied in all modes. We show that both versions of the algorithm converge to the stationary points of the corresponding objective functions.

Figures (7)

• Figure 1: Convergence of Algorithm \ref{['agm:jacobi']} for different values of $\eta$ on tensors of order $3$ and $4$ that are diagonalizable using orthogonal transformations.
• Figure 2: Convergence of Algorithm \ref{['agm:jacobi']} for different values of $\eta$ on random tensors of order $3$ and $6$.
• Figure 3: Portion of the number of microiterations within one iteration for different values of $\eta$ on random tensors of order $3$ and $6$.
• Figure 4: Trace maximization compared to the maximization of the squares of the diagonal elements for two random tensors during the first $10$ iterations.
• Figure 5: Comparison of different initialization strategies for Algorithm \ref{['agm:jacobi']} for a random tensor of order $4$ during the first $10$ iterations.

Theorems & Definitions (5)

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