Is there a pseudoinverse for tensors of finitary rank?

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Question

Matrices estimated from data are not always invertible. Fortunately a ‘nearest inverse’ can be calculated with the Moore-Penrose inverse. Recently I have been taking inverses of tensors of rank greater than two that are calculated from real-world data, but I am finding that they are often non-invertible.

Is there a pseudoinverse for finitary rank tensors in a similar sense to the Moore-Penrose pseudoinverse?