  Recently, introducing Tensor Decomposition (TD) methods into unsupervised
feature selection (UFS) has been a rising research point. A tensor structure is
beneficial for mining the relations between different modes and helps relieve
the computation burden. However, while existing methods exploit TD to minimize
the reconstruction error of a data tensor, they don't fully utilize the
interpretable and discriminative information in the factor matrices. Moreover,
most methods require domain knowledge to perform feature selection. To solve
the above problems, we develop two Sparse Tensor Principal Component Analysis
(STPCA) models that utilize the projection directions in the factor matrices to
perform UFS. The first model extends Tucker Decomposition to a multiview sparse
regression form and is transformed into several alternatively solved convex
subproblems. The second model formulates a sparse version of the family of
Tensor Singular Value Decomposition (T-SVDs) and is transformed into individual
convex subproblems. For both models, we prove the optimal solution of each
subproblem falls onto the Hermitian Positive Semidefinite Cone (HPSD).
Accordingly, we design two fast algorithms based on HPSD projection and prove
their convergence. According to the experimental results on two original
synthetic datasets (Orbit and Array Signal) and five real-world datasets, the
two proposed methods are suitable for handling different data tensor scenarios
and outperform the state-of-the-art UFS methods.
