On complex and real identifiability of tensors

We report about the state of the art on complex and real generic identifiability of tensors, we describe some of our recent results obtained in [6] and we present perspectives on the subject.Comment: To appear on Rivista di Matematica dell'Universit\`a di Parma, Volume 8, Number 2, 2017, pages 367-37

Convergence analysis of Riemannian Gauss-Newton methods and its connection with the geometric condition number

We obtain estimates of the multiplicative constants appearing in local convergence results of the Riemannian Gauss-Newton method for least squares problems on manifolds and relate them to the geometric condition number of [P. B\"urgisser and F. Cucker, Condition: The Geometry of Numerical Algorithms, 2013]

Tensor decomposition of polarized seismic waves

National audienceIn antenna array processing, tensor decompositions allow to jointly estimate sources and their location. But these rechniques can be used only if data are recorded as a function of at least three diversities, which are usually time, space and space translation. The approach presented therein is based on polarization diversity, a very attractive alternative when the antenna array does not enjoy space invariance. Then we derive Cramér-Rao bounds in this context, by resorting to differentiation conventions for real-complex mixed variables.En traitement d'antenne, les décompositions tensorielles permettent d'estimer conjointement les sources et de les localiser. Pour que ces dernières puissent être utilisées, il faut que les données présentent au moins trois diversités, qui sont habituellement le temps, l'espace, et la translation dans l'espace. L'approche présentée ici est basée sur la diversité de polarisation, une alternative très attractive lorsque l'antenne de jouit pas d'invariance spatiale. Nous dérivons ensuite les bornes de Cramér-Rao dans ce contexte, en nous appuyant sur des conventions de différentiation de variables mixtes réelles et complexes

On the dimension of contact loci and the identifiability of tensors

Let $X\subset \mathbb{P}^r$ 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 $S\subset X$ with $\sharp (S) =k$. We apply this result to many Segre-Veronese varieties and to the identifiability of Gaussian mixtures $G_{1,d}$. 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