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An Interpolating Distance between Optimal Transport and Fisher-Rao
Lenaic Chizat, Bernhard Schmitzer, Gabriel Peyré, François-Xavier Vialard
TL;DR
The paper addresses how to compare non-negative measures with unequal masses while retaining transport geometry. It introduces a convex dynamical metric combining transport with Fisher–Rao growth, proves existence and selected uniqueness results for geodesics, and demonstrates image interpolation. The analysis also characterizes localized transport behavior and regularized Fisher–Rao behavior, subject to stated smoothness and nondegeneracy limitations.
Problem
Optimal transport is restricted to equal-mass measures, motivating a metric for unbalanced measures and phenomena involving mass creation or destruction.
Method
The paper adds a source term to the Benamou–Brenier continuity equation and combines its Fisher–Rao penalty with the transport term in a convex variational formulation.
Results
The paper proves geodesic existence, obtains generalized transport and Fisher–Rao as limiting models, and proves travelling-Dirac results under suitable conditions.
Takeaways & Limitations
The distance behaves as a spatially localized version of optimal transport and as a Fisher–Rao regularization that is stable under perturbation.
Takeaways & Limitations
The uniqueness proof applies only when the optimality certificate is nondegenerate, while a necessary optimality condition for the relevant function space remains beyond scope.
Abstract
from arXiv · showhide
This paper defines a new transport metric over the space of non-negative measures. This metric interpolates between the quadratic Wasserstein and the Fisher-Rao metrics and generalizes optimal transport to measures with different masses. It is defined as a generalization of the dynamical formulation of optimal transport of Benamou and Brenier, by introducing a source term in the continuity equation. The influence of this source term is measured using the Fisher-Rao metric, and is averaged with the transportation term. This gives rise to a convex variational problem defining our metric. Our first contribution is a proof of the existence of geodesics (i.e. solutions to this variational problem). We then show that (generalized) optimal transport and Fisher-Rao metrics are obtained as limiting cases of our metric. Our last theoretical contribution is a proof that geodesics between mixtures of sufficiently close Diracs are made of translating mixtures of Diracs. Lastly, we propose a numerical scheme making use of first order proximal splitting methods and we show an application of this new distance to image interpolation.
1 Introduction
The paper introduces a convex dynamical distance for non-negative measures that relaxes equal-mass optimal transport by penalizing mass creation and destruction, interpolating between Wasserstein and Fisher–Rao geometries.
- Optimal transport background: Optimal transport induces Wasserstein metrics between probability measures but is restricted to measures with equal mass.This restriction limits applications involving unnormalized measures or mass creation and destruction.
- Model: The model generalizes the Benamou–Brenier continuity equation by adding a source term ζ alongside transport momentum ω.The resulting constraint is ∂tρ + ∇·ω = ζ with prescribed initial and final measures.
- Model: The source term is penalized through a growth energy based on the rate g = ζ/ρ, while transport retains the kinetic-energy term based on velocity v.This separates spatial motion from local growth or decay within one dynamical formulation.
- Model: Reparametrization invariance and a Riemannian interpretation motivate using the Fisher–Rao metric for the source penalty.Fisher–Rao is identified as the unique reparametrization-invariant Riemannian metric in the stated smooth-density setting.
- Contributions: The interpolating distance extends to non-negative measures as a convex, positively 1-homogeneous, lower-semicontinuous variational functional.The paper focuses on the p = 2, q = 2 case, described as the only Riemannian-like metric in this family.
- Contributions: The paper connects the model to partial optimal transport, studies limiting models and atomic geodesics, and presents numerical image-interpolation applications.It also reports existence and uniqueness results for travelling Diracs and a numerical scheme for image interpolation.
2 Definition and Existence of Geodesics
The paper defines the interpolating distance through a convex measure-valued continuity equation with transport and source terms, then proves strong duality, attainment, metric properties, and sufficient uniqueness conditions.
- Existence: Fenchel–Rockafellar duality proves strong duality and attainment of the primal variational problem, establishing existence of geodesics.The dual formulation uses convex conjugates and the continuity constraint.
- Definition: The variational problem minimizes a convex functional Dδ over measure triplets μ = (ρ, ω, ζ) satisfying the continuity equation with source and endpoint constraints.The formulation also imposes homogeneous Neumann boundary conditions and allows time disintegration under stated absolute-continuity assumptions.
- Definition: The admissible set is nonempty, including linear interpolation with zero momentum and a source equal to the time derivative.For equal-mass endpoints, an admissible path with ζ = 0 also exists.
- Definition: The interpolating distance WFδ is defined on non-negative measures by the infimum of Dδ over admissible paths.Its homogeneous integrand makes the definition independent of the chosen dominating reference measure.
- Properties: The distance admits bounds including the Fisher–Rao metric as a tight first bound, with the second bound attained in the worst case of disjoint supports.The Fisher–Rao bound corresponds to imposing ω = 0.
- Properties: WFδ is a metric, with symmetry from time reversal and the triangle inequality from concatenating admissible paths.Equivalent characterizations arise through time rescaling and disintegration in time.
- Uniqueness: Convex-analysis optimality certificates provide a sufficient condition for geodesic uniqueness, including the travelling-Dirac analysis.The certificate condition is not necessary, because optimal certificates may lack the required smoothness.
3 Interpolation Properties
The parameter δ controls the spatial scale at which transport and mass creation or removal are balanced. As δ tends to infinity or zero, the metric recovers generalized optimal transport or Fisher–Rao behavior, respectively.
- Parameter rescaling: Changing δ changes the scale at which mass creation and removal intervene to match the endpoint measures.Smaller δ makes creation and removal act at a finer scale.
- Parameter rescaling: Under spatial scaling by s, minimizing triplets for WFδ map to minimizing triplets for WFsδ between the scaled measures.The spatial transformation rescales the transport component and the metric parameter together.
- Parameter rescaling: For s < 1, spatial contraction decreases the distance, with strict inequalities when the transport component is nonzero.The monotonicity result compares both the variational functional and the induced distance.
- Limit models: The interpolating functional separates into a Benamou–Brenier transport term and a Fisher–Rao term weighted by δ2.The two component functionals are independent of δ, while δ2 determines their relative contribution.
- Limit models: Fisher–Rao geodesics are unique, have an explicit expression, and arise from an isometric injection into L2(dν).The proof uses absolute continuity with respect to ν = ρ0 + ρ1, convex duality, and uniqueness of L2 geodesics.
- Limit models: As δn → +∞, minimizers converge weakly up to subsequences to minimizers of generalized Benamou–Brenier transport; as δn → 0, they converge to Fisher–Rao minimizers.These are the two limiting models identified by the paper’s main theorem for varying δ.
4 Explicit Geodesics
The section derives explicit geodesics for inflating measures, single Dirac transport, and atomic mixtures, using optimality certificates to establish existence and uniqueness in supported regimes.
- General framework: An optimality certificate can establish that a candidate geodesic is unique, while the paper notes this condition is sufficient but not necessary.The certificate’s regularity may be stronger than required because the infimum can be attained in a larger function space.
- No transport case: For proportional endpoint measures, the geodesic uses no transport and is the Fisher-Rao geodesic.All minimizers have zero transport component, yielding uniqueness in this case.
- Transport of one Dirac to another: For two Diracs separated by less than πδ, the unique geodesic is a travelling Dirac; at or beyond πδ, the behavior changes.At the cut locus there are infinitely many geodesics, whereas beyond it the unique geodesic is Fisher-Rao.
- Matching atomic measures: For suitable atomic mixtures, geodesics decompose into travelling Diracs or Fisher-Rao geodesics for individual pairs, with uniqueness under the theorem’s mass conditions.The construction combines componentwise geodesics between paired Diracs; Figure 1 illustrates additional decreasing and increasing stationary Diracs.
- Transport of one Dirac to another: At the cut locus |x1−x0| = πδ, travelling-Dirac geodesics remain optimal, infinitely many geodesics can be constructed, and the cost reaches the Fisher-Rao upper bound.The distance for separations at least πδ equals the Fisher-Rao value, and beyond the cut locus only the Fisher-Rao geodesic remains.
- Transport of one Dirac to another: In arbitrary dimension, the single-Dirac construction extends by solving the problem on the line spanned by x1−x0.The corresponding certificate is C1, and the cut-locus and beyond-cut-locus behaviors extend as well.
5 Numerical Results
The numerical section develops a staggered-grid proximal-splitting implementation for computing geodesics and evaluates it on synthetic and biological image-interpolation problems. The experiments compare Wasserstein, Fisher–Rao, partial transport, and WFδ behaviors, including cases with unequal masses.
- 5 Numerical Results: The implementation uses first-order proximal splitting, specifically a Douglas–Rachford algorithm, to solve the discretized transport-with-sources problem.The discrete problem separates the functional, continuity constraint, and interpolation constraint; proximal operators and affine projections make these components computationally tractable.
- 5.1 Discretization: The discretization uses centered and staggered grids over space-time, with the source ζ kept on the centered grid because it is not differentiated.Separate centered and staggered variables are linked through a midpoint interpolation constraint.
- 5.3 Experiments: In the Gaussian-bump experiment, W2 splits a rightmost bump and transports part of it across the interval, while source-based models alter density through local mass variation.The experiment uses two Gaussian densities with masses 1 and 2, discretized with N = 256 spatial samples and T = 11 time samples.
- 5.3 Experiments: In the 2D ring experiment, partial transport and WFδ attenuate tangential motion relative to W2, producing behavior described as more consistent with the intuitive solution.The comparison uses dFR, W2, partial optimal transport with 2δ = 0.2, and WFδ with πδ = 0.4.
- 5.3 Experiments: In brain-image interpolation with varying mass, WFδ locally adapts growth and yields a velocity field more consistent with tissue evolution than W2, though transport artifacts remain.The data comprise segmented images of the same young brain at different times; WFδ uses πδ = 0.2.
Conclusion
The paper introduces a distance interpolating between optimal transport and Fisher–Rao for unbalanced measures and analyzes its theoretical and numerical behavior. It establishes geodesic properties, characterizes atomic-measure geodesics, and demonstrates image-interpolation applications.
- Conclusion: The proposed distance is the only Riemannian-like metric in a family of convex homogeneous dynamic minimization problems that interpolates between optimal transport and Fisher–Rao.
- Conclusion: Varying the parameter δ yields theoretical information about limiting models and the behavior of geodesics for atomic measures.
- Conclusion: The paper proves existence of geodesics and uniqueness of travelling-Dirac solutions under suitable conditions.
- Conclusion: A numerical scheme based on proximal splitting is detailed, with applications to image interpolation.