<p>The Wasserstein space of probability measures is known for its intricate Riemannian structure, which underpins the Wasserstein geometry and enables gradient flow algorithms. However, the Wasserstein geometry may not be suitable for certain tasks or data modalities. Motivated by scenarios where the global structure of the data needs to be preserved, this work initiates the study of gradient flows and Riemannian structure in the Gromov-Wasserstein (GW) geometry, which is particularly suited for such purposes. We focus on the inner product GW (IGW) distance between distributions on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>, which preserves the angles within the data and serves as a convenient initial setting due to its analytic tractability. Given a functional <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textsf{F}:\mathcal {P}_2(\mathbb {R}^d)\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">F</mi> <mo>:</mo> <msub> <mi mathvariant="script">P</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> to optimize and an initial distribution <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\rho _0\in \mathcal {P}_2(\mathbb {R}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ρ</mi> <mn>0</mn> </msub> <mo>∈</mo> <msub> <mi mathvariant="script">P</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we present an implicit IGW minimizing movement scheme that generates a sequence of distributions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\{\rho _i\}_{i=0}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>ρ</mi> <mi>i</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>n</mi> </msubsup> </math></EquationSource> </InlineEquation>, which are close in IGW and aligned in the 2-Wasserstein sense. Taking the time step to zero, we prove that the (piecewise constant interpolation of the) discrete solution converges to an IGW generalized minimizing movement (GMM) <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(\rho _t)_t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> that follows the continuity equation with a velocity field <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$v_t\in L^2(\rho _t;\mathbb {R}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>v</mi> <mi>t</mi> </msub> <mo>∈</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mi>t</mi> </msub> <mo>;</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, specified by a global transformation of the Wasserstein gradient of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textsf{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">F</mi> </math></EquationSource> </InlineEquation> (viz., the gradient of its first variation). The transformation is given by a mobility operator that modifies the Wasserstein gradient to encode not only local information, but also global structure, as expected for the IGW gradient flow. Our gradient flow analysis leads us to identify the Riemannian structure that gives rise to the intrinsic IGW geometry, using which we establish a Benamou-Brenier-like formula for IGW. We conclude with a formal derivation, akin to the Otto calculus, of the IGW gradient as the inverse mobility acting on the Wasserstein gradient. Numerical experiments demonstrating the global nature of IGW interpolations are provided to complement the theory.</p>

Gradient Flows and Riemannian Structure in the Gromov-Wasserstein Geometry

期刊论文
发表日期：

2025年7月8日

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Zhengxin Zhang,
Ziv Goldfeld,
Kristjan Greenewald,
Youssef Mroueh,
Bharath K. Sriperumbudur

摘要

The Wasserstein space of probability measures is known for its intricate Riemannian structure, which underpins the Wasserstein geometry and enables gradient flow algorithms. However, the Wasserstein geometry may not be suitable for certain tasks or data modalities. Motivated by scenarios where the global structure of the data needs to be preserved, this work initiates the study of gradient flows and Riemannian structure in the Gromov-Wasserstein (GW) geometry, which is particularly suited for such purposes. We focus on the inner product GW (IGW) distance between distributions on $\mathbb {R}^d$ $R^{d}$ , which preserves the angles within the data and serves as a convenient initial setting due to its analytic tractability. Given a functional $\textsf{F}:\mathcal {P}_2(\mathbb {R}^d)\rightarrow \mathbb {R}$ $F : P_{2} (R^{d}) \to R$ to optimize and an initial distribution $\rho _0\in \mathcal {P}_2(\mathbb {R}^d)$ $ρ_{0} \in P_{2} (R^{d})$ , we present an implicit IGW minimizing movement scheme that generates a sequence of distributions $\{\rho _i\}_{i=0}^n$ ${ρ_{i}}_{i = 0}^{n}$ , which are close in IGW and aligned in the 2-Wasserstein sense. Taking the time step to zero, we prove that the (piecewise constant interpolation of the) discrete solution converges to an IGW generalized minimizing movement (GMM) $(\rho _t)_t$ ${(ρ_{t})}_{t}$ that follows the continuity equation with a velocity field $v_t\in L^2(\rho _t;\mathbb {R}^d)$ $v_{t} \in L^{2} (ρ_{t}; R^{d})$ , specified by a global transformation of the Wasserstein gradient of $\textsf{F}$ $F$ (viz., the gradient of its first variation). The transformation is given by a mobility operator that modifies the Wasserstein gradient to encode not only local information, but also global structure, as expected for the IGW gradient flow. Our gradient flow analysis leads us to identify the Riemannian structure that gives rise to the intrinsic IGW geometry, using which we establish a Benamou-Brenier-like formula for IGW. We conclude with a formal derivation, akin to the Otto calculus, of the IGW gradient as the inverse mobility acting on the Wasserstein gradient. Numerical experiments demonstrating the global nature of IGW interpolations are provided to complement the theory.