Motivated by the study of decompositions of tensors as Hadamard products (i.e., coefficient-wise products) of low-rank tensors, we introduce the notion of Hadamard rank of a given point with respect to a projective variety: if it exists, it is the smallest number of points in the variety such that the given point is equal to their Hadamard product. We prove that if the variety $X$ is not contained in a coordinate hyperplane or a binomial hypersurface, then the generic point has a finite $X$-Hadamard-rank. Although the Hadamard rank might not be well defined for special points, we prove that the general Hadamard rank with respect to secant varieties of toric varieties is finite and the maximum Hadamard rank for points with no coordinates equal to zero is at most twice the generic rank. In particular, we focus on Hadamard ranks with respect to secant varieties of toric varieties since they provide a geometric framework in which Hadamard decompositions of tensors can be interpreted. Finally, we give a lower bound to the dimension of Hadamard products of secant varieties of toric varieties: this allows us to deduce the general Hadamard rank with respect to secant varieties of several Segre-Veronese varieties.

翻译：受张量分解为低秩张量Hadamard积（即逐系数乘积）研究的启发，我们引入了关于射影簇的给定点的Hadamard秩概念：若存在，则它是簇中使得给定点等于这些点Hadamard积的最小点数。我们证明，若簇$X$不包含于坐标超平面或二项式超曲面中，则一般点具有有限的$X$-Hadamard秩。尽管特殊点的Hadamard秩可能未定义，但我们证明了关于环面簇割线簇的一般Hadamard秩是有限的，且坐标非零点的最大Hadamard秩至多为一般秩的两倍。特别地，我们聚焦于环面簇割线簇的Hadamard秩，因为它们为张量Hadamard分解提供了可解释的几何框架。最后，我们给出了环面簇割线簇Hadamard积维数的下界：这使得我们能够推导出关于若干Segre-Veronese簇割线簇的一般Hadamard秩。