Let X⊂Pr be an integral and non-degenerate variety. Set
n:=dim(X). We prove that if the (k+n−1)-secant variety of X has (the
expected) dimension (k+n−1)(n+1)−1<r and X is not uniruled by lines, then
X is not k-weakly defective and hence the k-secant variety satisfies
identifiability, i.e. a general element of it is in the linear span of a unique
S⊂X with ♯(S)=k. We apply this result to many Segre-Veronese
varieties and to the identifiability of Gaussian mixtures G1,d. If X is
the Segre embedding of a multiprojective space we prove identifiability for the
k-secant variety (assuming that the (k+n−1)-secant variety has dimension
(k+n−1)(n+1)−1<r, this is a known result in many cases), beating several
bounds on the identifiability of tensors.Comment: 12 page