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.

针对张量 PCA 问题，本文提出了一种数学框架，通过从实自伴张量算子中导出张量空间的基础来减少问题的导出；研究了三种不同的情况来导出：从自伴张量算子中的基础，一个秩 - 1 的基础，以及一个子空间中的基础，并采用子空间方法导出张量 PCA。实验验证了该数学框架的有效性。