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Warped geometries of Segre-Veronese manifolds

Simon Jacobsson, Lars Swijsen, Joeri Van der Veken, Nick Vannieuwenhoven

TL;DR

This work develops and analyzes alpha-warped geometries for Segre--Veronese manifolds, providing closed-form exponential and logarithmic maps and explicit intrinsic distances that enable efficient Riemannian computations on partially symmetric rank-1 tensors. By showing that a normal Riemannian covering relates the Segre--Veronese manifold to a warped pre-Segre--Veronese product, the authors derive practical geodesic formulas, compatibility criteria, and a linear-time matching algorithm to obtain geodesics between rank-1 tensors. They establish geodesic connectivity criteria dependent on the warping parameter $\alpha$, compute sectional curvatures, and illustrate a concrete application to Fréchet means for tensor averaging. The numerical experiments, implemented in Julia and integrated with Manifolds.jl, demonstrate robust intrinsic averaging and efficient computation, highlighting the practical impact for tensor decomposition and aggregation tasks in high-dimensional data.

Abstract

Segre-Veronese manifolds are smooth submanifolds of tensors comprising the partially symmetric rank-1 tensors. We investigate a one-parameter family of warped geometries of Segre-Veronese manifolds, which includes the standard Euclidean geometry. This parameter controls by how much spherical tangent directions are weighted relative to radial tangent directions. We present closed expressions for the exponential map, the logarithmic map, and the intrinsic distance on these warped Segre-Veronese manifolds, which can be computed efficiently numerically. It is shown that Segre-Veronese manifolds are not geodesically connected in the Euclidean geometry, while they are for some values of the warping parameter. The benefits of geodesics connectedness may outweigh using the Euclidean geometry in certain applications. One such application is presented: numerically computing the Riemannian center of mass for averaging rank-1 tensors.

Warped geometries of Segre-Veronese manifolds

TL;DR

This work develops and analyzes alpha-warped geometries for Segre--Veronese manifolds, providing closed-form exponential and logarithmic maps and explicit intrinsic distances that enable efficient Riemannian computations on partially symmetric rank-1 tensors. By showing that a normal Riemannian covering relates the Segre--Veronese manifold to a warped pre-Segre--Veronese product, the authors derive practical geodesic formulas, compatibility criteria, and a linear-time matching algorithm to obtain geodesics between rank-1 tensors. They establish geodesic connectivity criteria dependent on the warping parameter , compute sectional curvatures, and illustrate a concrete application to Fréchet means for tensor averaging. The numerical experiments, implemented in Julia and integrated with Manifolds.jl, demonstrate robust intrinsic averaging and efficient computation, highlighting the practical impact for tensor decomposition and aggregation tasks in high-dimensional data.

Abstract

Segre-Veronese manifolds are smooth submanifolds of tensors comprising the partially symmetric rank-1 tensors. We investigate a one-parameter family of warped geometries of Segre-Veronese manifolds, which includes the standard Euclidean geometry. This parameter controls by how much spherical tangent directions are weighted relative to radial tangent directions. We present closed expressions for the exponential map, the logarithmic map, and the intrinsic distance on these warped Segre-Veronese manifolds, which can be computed efficiently numerically. It is shown that Segre-Veronese manifolds are not geodesically connected in the Euclidean geometry, while they are for some values of the warping parameter. The benefits of geodesics connectedness may outweigh using the Euclidean geometry in certain applications. One such application is presented: numerically computing the Riemannian center of mass for averaging rank-1 tensors.
Paper Structure (25 sections, 23 theorems, 82 equations, 2 figures)

This paper contains 25 sections, 23 theorems, 82 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Applications of geodesics
  3. Contributions
  4. Outline
  5. alpha-warped geometries for Segre--Veronese manifolds
  6. Warped geometries of Rn*
  7. The alpha-warped Segre--Veronese submanifold
  8. The alpha-warped pre-Segre--Veronese manifold
  9. The exponential map
  10. The logarithmic map
  11. Compatibility
  12. Geodesics $\Longrightarrow$ compatibility
  13. Compatibility $\Longrightarrow$ geodesics
  14. Pre-Segre--Veronese manifolds
  15. Upper bound
  16. ...and 10 more sections

Key Result

Lemma 2.2

\newlabellem_riemannian_covering0 The map is a normal Riemannian covering where the isometric deck transforms are of the form $\imath_\sigma(\lambda, \mathbf{u}_1,\ldots,\mathbf{u}_d) = (\lambda,\sigma_1\mathbf{u}_1,\ldots,\sigma_d \mathbf{u}_d)$ with $\sigma_i\in\set{-1, 1}$ and $\sigma_1^{k_1}\cdots\sigma_d^{k_d} = 1$.

Figures (2)

  • Figure 1: Geodesics in the $\alpha$-warped geometry of the punctured plane $\mathcal{S}^{(1)}$ for $\alpha = 0.05$ (the red paths curving towards the origin, i.e., acceleration vectors pointing toward the origin), $\frac{1}{4}$, $\frac{1}{2}$, $\frac{3}{4}$, $1$ (the purple straight lines between $(0,1)$ and $(1,0)$), $\frac{5}{4}$, $\frac{3}{2}$, $\frac{7}{4}$, and $1.95$ (the blue paths curving away from the origin, i.e., acceleration vectors pointing away from $(0,0)$). Essentially, the left panel shows evaluations of the exponential map, and the right panel shows evaluations of the logarithmic map.
  • Figure 1: The mean relative distance from \ref{['eqn_mean_rel_dist']} between the reference tensor rank decomposition and the approximate decompositions computed from noisy samples of this tensor rank decomposition using a single ZIPTF CST2023, the consensus C-ZIPTF CST2023, and the Fréchet F-ZIPTF in several $\alpha$-warped geometries of the Segre manifold (this article) on the decomposition problem setup described in \ref{['sec_applications']}. In the figure's horizontal axis, "S" refers to a single ZIPTF, "C" to C-ZIPTF, and the numerical values $\alpha=\frac{1}{\sqrt{5}}, \dots, \frac{1}{\sqrt{2}}, 1, \sqrt{2}, \dots, \sqrt{5},$ correspond to F-ZIPTF where the Fréchet mean was computed in the $\alpha$-warped geometry of the Segre manifold.

Theorems & Definitions (39)

  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proof 1
  • Proposition 3.1: Exponential map of $\mathcal{P}_\alpha^\mathbf{k}$
  • Proof 2
  • Corollary 3.2
  • Corollary 3.3: The exponential map of $\mathcal{S}_\alpha^\mathbf{k}$
  • Corollary 3.4: Injectivity radius
  • ...and 29 more