Uniqueness of Tensor Train Decomposition with Linear Dependencies

With the increase in measurement/sensing technologies, the collected data are intrinsically multidimensional in a large number of applications. This can be interpreted as a growth of the dimensionality/order of the associated tensor. There exists therefore a crucial need to derive equivalent and alternative models of a high-order tensor as a graph of low-order tensors. In this work we consider a “ train ” graph, i.e., a $Q$ -order tensor will be represented as a Tensor Train (TT) composed of $Q$ - 2 3-order core tensors and two core matrices. In this context, it has been shown that a canonical rank- $R$ CPD model can always be represented exactly by a TT model whose cores are canonical rank- $R$ CPD. This model is called TT-CPD. We generalize this equivalence to the PARALIND model in order to take into account potential linear dependencies in factors. We derive and discuss here uniqueness conditions for the case of the TT-PARALIND model.

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