Q-less QR decomposition in inner product spaces

Tensor computation is intensive and difficult. Invariably, a vital component is the truncation of tensors, so as to control the memory and associated computational requirements. Various tensor toolboxes have been designed for such a purpose, in addition to transforming tensors between different formats. In this paper, we propose a simple Q-less QR truncation technique for tensors {x_(i)} with x_(i)∈R^{n₁×⋯×n_d} in the simple and natural Kronecker product form. It generalizes the QR decomposition with column pivoting, adapting the well-known Gram-Schmidt orthogonalization process. The main difficulty lies in the fact that linear combinations of tensors cannot be computed or stored explicitly. All computations have to be performed on the coefficients α_i in an arbitrary tensor v=∑_iα_ix_(i). The orthonormal Q factor in the QR decomposition X≡[x₍₁₎,⋯,x_(p)]=QR cannot be computed but expressed as XR^-1 when required. The resulting algorithm has an O(p²dn) computational complexity, with n=maxn_i. Some illustrative examples in the numerical solution of tensor linear equations are presented.