Tensor PCA from basis in tensor space
This provides a more efficient method for tensor PCA, which is incremental as it builds on existing approaches but offers a new mathematical formulation.
The paper tackles the problem of tensor PCA by proposing a mathematical framework that reduces basis derivation to an eigenvalue problem, overcoming limitations of iterative optimization methods, and validates it with experiments on image datasets.
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.