On some orthogonalization schemes in Tensor Train format

In the framework of tensor spaces, we consider orthogonalization kernels to generate an orthogonal basis of a tensor subspace from a set of linearly independent tensors. In particular, we investigate numerically the loss of orthogonality of six orthogonalization methods, namely Classical and Modified Gram-Schmidt with (CGS2, MGS2) and without (CGS, MGS) re-orthogonalization, the Gram approach, and the Householder transformation. To tackle the curse of dimensionality, we represent tensor with low rank approximation using the Tensor Train (TT) formalism, and we introduce recompression steps in the standard algorithm outline through the TT-rounding method at a prescribed accuracy. After describing the algorithm structure and properties, we illustrate numerically that the theoretical bounds for the loss of orthogonality in the classical matrix computation round-off analysis results are maintained, with the unit round-off replaced by the TT-rounding accuracy. The computational analysis for each orthogonalization kernel in terms of the memory requirement and the computational complexity measured as a function of the number of TT-rounding, which happens to be the computational most expensive operation, completes the study.