We propose a new sufficient condition for verifying whether generic rank-r
complex tensors of arbitrary order admit a unique decomposition as a linear
combination of rank-1 tensors. A practical algorithm is proposed for verifying
this condition, with which it was established that in all spaces of dimension
less than 15000, with a few known exceptions, listed in the paper, generic
identifiability holds for ranks up to one less than the generic rank of the
space. This is the largest possible rank value for which generic
identifiability can hold, except for spaces with a perfect shape. The algorithm
can also verify the identifiability of a given specific rank-r decomposition,
provided that it can be shown to correspond to a nonsingular point of the r-th
order secant variety. For sufficiently small rank, which nevertheless improves
upon the known bounds for specific identifiability, some local equations of
this variety are known, allowing us to verify this property. As a particular
example of our approach, we prove the identifiability of a specific 5x5x5
tensor of rank 7, which cannot be handled by the conditions recently provided
in [I. Domanov and L. De Lathauwer, On the Uniqueness of the Canonical Polyadic
Decomposition of third-order tensors--Part II: Uniqueness of the overall
decomposition, SIAM J. Matrix Anal. Appl. 34(3), 2013]. Finally, we also
present a surprising new class of weakly-defective Segre varieties that
nevertheless turns out to admit a generically unique decomposition.

我们提出了一种新的充分条件，用于验证任意阶的通用秩为 r 的复合张量是否能够唯一地分解为一组秩一张量的线性组合；我们还提出了一种实用的算法来验证这个条件，并建立了关于此类张量通用可识别性的性质；此外，我们还证明了一类弱缺陷 Segre 变量的通用唯一分解性质。