Belangrijkste concepten
The author presents a new, constructive uniqueness theorem for tensor decomposition that applies to order 3 tensors of format n×n×p and can prove uniqueness of decomposition for generic tensors up to rank r = 4n/3 as soon as p ≥4. This leads to the first efficient algorithm for overcomplete decomposition of generic tensors.
Samenvatting
The paper presents a new uniqueness theorem and an efficient decomposition algorithm for overcomplete tensor decomposition.
Key highlights:
- The uniqueness theorem applies to order 3 tensors of format n×n×p and can prove uniqueness of decomposition for generic tensors up to rank r = 4n/3 as soon as p ≥4.
- This is an improvement over Kruskal's uniqueness theorem, which can only prove uniqueness up to rank n+1 for tensors of format n×n×4.
- The uniqueness theorem has an algorithmic proof, leading to an efficient decomposition algorithm.
- The algorithm can efficiently decompose generic tensors in the overcomplete regime (n ≤r ≤4n/3), which was not possible prior to this work.
- The algorithm relies on the method of commuting extensions pioneered by Strassen, as well as the classical Jennrich algorithm for undercomplete tensor decomposition.
The author also provides a genericity result showing that the conditions of the uniqueness theorem are generically satisfied for tensors in the range n ≤r ≤4n/3.