2. Exact tensor decompositions
You'd think that the usual matrix decompositions could easily be extended to tensors, but this is not the case. In fact, there are major differences as soon as we go from order 2 to order 3, which both make the use of tensors interesting and raise new theoretical and algorithmic difficulties.
2.1 2.1 Tensor rank, polyadic canonical decomposition (CP)
In Example 3, we saw a simple form of tensor, which is very useful in applications, as it represents functions with separate variables. Decomposing a tensor into a sum of simple tensors leads to the first proposition:
Proposition 4: CP decomposition (CPD). Any tensor
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