  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 experimentally study 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 overcome the curse of
dimensionality, we represent tensors with a low-rank approximation using the
Tensor Train (TT) formalism. In addition, we introduce recompression steps in
the standard algorithm outline through the TT-rounding method at a prescribed
accuracy. After describing the structure and properties of the algorithms, we
illustrate their loss of orthogonality with numerical experiments. The
theoretical bounds from the classical matrix computation round-off analysis,
obtained over several decades, seem to be maintained, with the unit round-off
replaced by the TT-rounding accuracy. The computational analysis for each
orthogonalization kernel in terms of the memory requirements and the
computational complexity measured as a function of the number of TT-rounding,
which happens to be the most computationally expensive operation, completes the
study.
