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Hadamard ranks of algebraic varieties

Dario Antolini, Guido Montúfar, Alessandro Oneto

TL;DR

This work introduces the Hadamard-$X$-rank, a multiplicative analogue of $X$-rank, to study decompositions of points in projective space as Hadamard products of points (or secant points) on a variety $X\subset\mathbb{P}^N$. It develops a tropical-geometric framework to characterize when the generic Hadamard-rank is finite, notably showing finiteness for non-concise/ binomial-containing cases and providing a complete tropical classification for infinite generic ranks. The paper then derives dimension bounds for Hadamard products of secant varieties of toric (in particular Segre-Veronese) varieties, giving a general lower bound $\dim\sigma_R(X) \le \dim\sigma_{\mathbf{r}}(X)$ with $R=\sum_k(r_k-1)+1$, and specializes these results to Veronese and Segre-Veronese cases, yielding explicit formulas for generic Hadamard ranks in many settings. These results offer a geometric framework for Hadamard decompositions of tensors and have potential implications for algebraic statistics, RBM-like models, and tensor-rank theory, including questions of defectivity, identifiability, and explicit equations of the models.

Abstract

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.

Hadamard ranks of algebraic varieties

TL;DR

This work introduces the Hadamard--rank, a multiplicative analogue of -rank, to study decompositions of points in projective space as Hadamard products of points (or secant points) on a variety . It develops a tropical-geometric framework to characterize when the generic Hadamard-rank is finite, notably showing finiteness for non-concise/ binomial-containing cases and providing a complete tropical classification for infinite generic ranks. The paper then derives dimension bounds for Hadamard products of secant varieties of toric (in particular Segre-Veronese) varieties, giving a general lower bound with , and specializes these results to Veronese and Segre-Veronese cases, yielding explicit formulas for generic Hadamard ranks in many settings. These results offer a geometric framework for Hadamard decompositions of tensors and have potential implications for algebraic statistics, RBM-like models, and tensor-rank theory, including questions of defectivity, identifiability, and explicit equations of the models.

Abstract

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 is not contained in a coordinate hyperplane or a binomial hypersurface, then the generic point has a finite -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.

Paper Structure

This paper contains 26 sections, 25 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Hadamard ranks of algebraic varieties
  3. Additive decompositions and secant varieties of algebraic varieties
  4. Multiplicative decompositions and Hadamard powers of algebraic varieties
  5. On the finiteness of Hadamard ranks
  6. On the finiteness of maximum $X$-Hadamard-ranks
  7. On the (in)finiteness of the generic $X$-Hadamard-rank
  8. Hadamard-idempotent varieties and tropical geometry
  9. Structure Theorem of Tropical Varieties
  10. Hadamard-idempotent varieties and linear tropical varieties
  11. Dimensions of Hadamard products of secant varieties
  12. Expected dimensions
  13. A useful parametrization
  14. Parametrizations of $r$th secant varieties
  15. Parametrization of Hadamard products of secant varieties
  16. ...and 11 more sections

Key Result

Lemma 3.3

Let $L \subset \mathbb{P}^N$ be a line such that $L \cap \Delta_{N-2} = \emptyset$. Then:

Theorems & Definitions (83)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Remark 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Definition 2.10
  • ...and 73 more