Tensor PCA from basis in tensor space

TL;DR

The core of the proposed approach is the derivation of a basis in tensor space from a real self-adjoint tensor operator, thus reducing the problem of deriving a basis to an eigenvalue problem.

Abstract

The aim of this paper is to present a mathematical framework for tensor PCA. The proposed approach is able to overcome the limitations of previous methods that extract a low dimensional subspace by iteratively solving an optimization problem. The core of the proposed approach is the derivation of a basis in tensor space from a real self-adjoint tensor operator, thus reducing the problem of deriving a basis to an eigenvalue problem. Three different cases have been studied to derive: i) a basis from a self-adjoint tensor operator; ii) a rank-1 basis; iii) a basis in a subspace. In particular, the equivalence between eigenvalue equation for a real self-adjoint tensor operator and standard matrix eigenvalue equation has been proven. For all the three cases considered, a subspace approach has been adopted to derive a tensor PCA. Experiments on image datasets validate the proposed mathematical framework.

Figures (15)

• Figure 1: The behavior of eigenvalues as obtained from TinyImageNet dataset.
• Figure 2: Some basis elements from TinyImageNet dataset corresponding to the highest eigenvalues.
• Figure 3: Several images taken from the dataset TinyImageNet.
• Figure 4: The same images of Fig. \ref{['fig:TINY']} reconstructed by (\ref{['eq4.1.1']}).
• Figure 5: The same images of Fig. \ref{['fig:TINY']} reconstructed with truncated basis.