STOD: Sparse Tensor Train Optimization via Orthogonal Decomposition for High-Dimensional Learning

This paper proposes a novel Tensor Train (TT)-based tensor-on-tensor regression optimization framework for variable selection based on mode-1 hyperslice sparsity. The formulation incorporates an l2,0-regularized term on the first TT-core while imposing Stiefel manifold constraints on the remaining M−1 TT-cores. Leveraging the property that the group sparsity of the first core is equivalent to the hyperslice sparsity of the global structure, we establish theoretical guarantees for the uniform variable-selection consistency of the proposed model. To efficiently solve the proposed model, we design an alternating iterative algorithm equipped with a preconditioned metric and prove its convergence to a critical point. Extensive numerical experiments on both synthetic and real-world datasets demonstrate that the numerical solutions generated by our algorithm exhibit exact support recovery in practice, tightly aligning with our theoretical analysis.