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Towards sparse optimization over convex loops: Equivalence of Square Root Velocity distance and Wasserstein-Fisher-Rao

Giacomo Cristinelli, José A. Iglesias

Abstract

The Wasserstein-Fisher-Rao (WFR) distance on $S^{2}$ has recently been shown to coincide with a classical elastic distance between $S^{2}$-immersions in the theory of Riemannian shape analysis. While this correspondence holds in dimension $2$, the analogous statement fails in general on $S^{1}$ and, in the case of convex curves, it cannot be derived from existing two-dimensional arguments. In this paper, we establish that for convex absolutely continuous immersions of $S^{1}$ in the plane, the shape distance induced by the square root velocity transformation (SRVT) is indeed equivalent to the WFR distance acting on their associated length measures. The proof exploits a monotonicity principle for optimal transport on the universal cover of the circle, which in turn guarantees the existence of an optimal reparametrization achieving the SRVT infimum and enables a one-dimensional unbalanced optimal transport reformulation. Motivated by this equivalence, we further investigate the role of sparsity in shape optimization problems formulated in terms of length measures and regularized by the WFR distance. We study linear optimization over the corresponding balls, for which we prove a finiteness result when the reference measure is discrete, and propose a convex, positively one-homogeneous regularizer suitable for conditional gradient algorithms.

Towards sparse optimization over convex loops: Equivalence of Square Root Velocity distance and Wasserstein-Fisher-Rao

Abstract

The Wasserstein-Fisher-Rao (WFR) distance on has recently been shown to coincide with a classical elastic distance between -immersions in the theory of Riemannian shape analysis. While this correspondence holds in dimension , the analogous statement fails in general on and, in the case of convex curves, it cannot be derived from existing two-dimensional arguments. In this paper, we establish that for convex absolutely continuous immersions of in the plane, the shape distance induced by the square root velocity transformation (SRVT) is indeed equivalent to the WFR distance acting on their associated length measures. The proof exploits a monotonicity principle for optimal transport on the universal cover of the circle, which in turn guarantees the existence of an optimal reparametrization achieving the SRVT infimum and enables a one-dimensional unbalanced optimal transport reformulation. Motivated by this equivalence, we further investigate the role of sparsity in shape optimization problems formulated in terms of length measures and regularized by the WFR distance. We study linear optimization over the corresponding balls, for which we prove a finiteness result when the reference measure is discrete, and propose a convex, positively one-homogeneous regularizer suitable for conditional gradient algorithms.
Paper Structure (14 sections, 30 theorems, 133 equations, 3 figures)

This paper contains 14 sections, 30 theorems, 133 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Square root velocity transformation
  4. Length measure for convex planar curves
  5. Wasserstein-Fisher-Rao
  6. Logarithmic entropy transport viewpoint
  7. The SRVT distance is Wasserstein-Fisher-Rao
  8. Admissibility of length measures
  9. Lift to the universal cover and one-dimensional optimal transport
  10. Equivalence result
  11. Towards optimization over length measures
  12. Linear optimization over WFR balls
  13. A convex one-homogeneous regularizer
  14. Extension of the local optimality theory to extended-valued costs

Key Result

Theorem 2.5

The set of curves in $\accentset{\diamond}{\mathcal{Q}}$ is in one-to-one correspondence to $\mathcal{M}^0(S^1)$.

Figures (3)

  • Figure 1: An example of a loop $c\in\mathcal{I}$ on the left, and its length measure $\mu_c$ on the right.
  • Figure 2: Two loops with the same corresponding length measure.
  • Figure 3: An example of the local modification construction, where the left and right picture depicts the support of $\zeta^*$, and $\zeta_x$, respectively. The supports of $\tau_r$ on the left and of $\bar{\tau}_r$ on the right are highlighted in cyan. The supports of $\tau_l$ on the left and of $\bar{\tau}_l$ on the right are highlighted in brown.

Theorems & Definitions (74)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • proof
  • ...and 64 more