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)}{x(i)} with x(i)∈Rn1×⋯×ndx(i)∈Rn1×⋯×nd 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 αiαi in an arbitrary tensor v=∑iαix(i)v=∑iαix(i). The orthonormal Q   factor in the QR decomposition X≡[x(1),⋯,x(p)]=QRX≡[x(1),⋯,x(p)]=QR cannot be computed but expressed as XR−1XR−1 when required. The resulting algorithm has an O(p2dn)O(p2dn) computational complexity, with n=max⁡nin=max⁡ni. Some illustrative examples in the numerical solution of tensor linear equations are presented.