A point p in a projective space is h -identifiable via a variety X if there is a unique way to write p as a linear combination of h points of X . Identifiability is important both in algebraic geometry and in applications. In this paper we propose an entirely new approach to study identifiability, connecting it to the notion of secant defect for any smooth projective variety. In this way we are able to improve the known bounds on identifiability and produce new identifiability statements. In particular, we give optimal bounds for some Segre and Segre–Veronese varieties and provide the first identifiability statements for Grassmann varieties.

A projective variety X ⊂ P N is h-identifiable if the generic element in its h-secant variety uniquely determines h points on X. In this paper we propose an entirely new approach to study identifiability, connecting it to the notion of secant defect. In this way we are able to improve all known bounds on identifiability. In particular we give optimal bounds for some Segre and Segre-Veronese varieties and provide the first identifiability statements for Grassmann varieties.

Comon's conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen's conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon's conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties.

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We consider the inverse problem for the polynomial map that sends an ‐tuple of quadratic forms in variables to the sum of their th powers. This map captures the moment problem for mixtures of centered ‐variate Gaussians. In the first nontrivial case , we show that for any , this map is generically one‐to‐one (up to permutations of and third roots of unity) in two ranges: for and for , thus proving generic identifiability for mixtures of centered Gaussians from their (exact) moments of degree at most . The first result is obtained by the explicit geometry of the tangential contact locus of the variety of sums of cubes of quadratic forms, as described by Chiantini and Ottaviani [SIAM J. Matrix Anal. Appl. 33 (2012), no. 3, 1018–1037], while the second result is accomplished using the link between secant nondefectivity with identifiability, proved by Casarotti and Mella [J. Eur. Math. Soc. (JEMS) (2022)]. The latter approach also generalizes to sums of th powers of ‐forms for and .