Computational and Statistical Guarantees for Tensor-on-Tensor Regression With Tensor Train Decomposition

Recently, a tensor-on-tensor (ToT) regression model has been proposed to generalize tensor recovery, encompassing scenarios like scalar-on-tensor regression and tensor-on-vector regression. However, the exponential growth in tensor complexity poses challenges for storage and computation in ToT regression. To overcome this hurdle, tensor decompositions have been introduced, with the tensor train (TT)-based ToT model proving efficient in practice due to reduced memory requirements, enhanced computational efficiency, and decreased sampling complexity. Despite these practical benefits, a disparity exists between theoretical analysis and real-world performance. In this paper, we delve into the theoretical and algorithmic aspects of the TT-based ToT regression model. Assuming the regression operator satisfies the restricted isometry property (RIP), we conduct an error analysis for the solution to a constrained least-squares optimization problem. This analysis includes upper error bound and minimax lower bound, revealing that such error bounds polynomially depend on the order $N+M$. To efficiently find solutions meeting such error bounds, we propose two optimization algorithms: the iterative hard thresholding (IHT) algorithm (employing gradient descent with TT-singular value decomposition (TT-SVD)) and the factorization approach using the Riemannian gradient descent (RGD) algorithm. When RIP is satisfied, spectral initialization facilitates proper initialization, and we establish the linear convergence rate of both IHT and RGD. Notably, compared to the IHT, which optimizes the entire tensor in each iteration while maintaining the TT structure through TT-SVD and poses a challenge for storage memory in practice, the RGD optimizes factors in the so-called left-orthogonal TT format, enforcing orthonormality among most of the factors, over the Stiefel manifold, thereby reducing the storage complexity of the IHT. However, this reduction in storage memory comes at a cost: the recovery of RGD is worse than that of IHT, while the error bounds of both algorithms depend on $N+M$ polynomially. Experimental validation substantiates the validity of our theoretical findings.