Sobolev Metrics Research Articles

Completeness and Geodesic Distance Properties for Fractional Sobolev Metrics on Spaces of Immersed Curves

We investigate the geometry of the space of immersed closed curves equipped with reparametrization-invariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractional-order q∈[0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q\\in [0,\\infty )$$\\end{document}. We establish the critical Sobolev index on the metric for several key geometric properties. Our first main result shows that the Riemannian metric induces a metric space structure if and only if q>1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q>1/2$$\\end{document}. Our second main result shows that the metric is geodesically complete (i.e., the geodesic equation is globally well posed) if q>3/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q>3/2$$\\end{document}, whereas if q<3/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q<3/2$$\\end{document} then finite-time blowup may occur. The geodesic completeness for q>3/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q>3/2$$\\end{document} is obtained by proving metric completeness of the space of Hq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^q$$\\end{document}-immersed curves with the distance induced by the Riemannian metric.

Sobolev metrics on spaces of manifold valued curves

We study completeness properties of reparametrization invariant Sobolev metrics of order $n\ge 2$ on the space of manifold valued open and closed immersed curves. In particular, for several important cases of metrics, we show that Sobolev immersions are metrically and geodesically complete (thus the geodesic equation is globally well-posed). These results were previously known only for closed curves with values in Euclidean space. For the class constant coefficient Sobolev metrics on open curves, we show that they are metrically incomplete, and that this incompleteness only arises from curves that vanish completely (unlike "local" failures that occur in lower order metrics).

Displacement smoothness of entropic optimal transport

The function that maps a family of probability measures to the solution of the dual entropic optimal transport problem is known as the Schr¨odinger map. We prove that when the cost function is Ck+1 with k ∈ ℕ* then this map is Lipschitz continuous from the L2-Wasserstein space to the space of Ck functions. Our result holds on compact domains and covers the multi-marginal case. We also include regularity results under negative Sobolev metrics weaker than Wasserstein under stronger smoothness assumptions on the cost. As applications, we prove displacement smoothness of the entropic optimal transport cost and the well-posedness of certain Wasserstein gradient flows involving this functional, including the Sinkhorn divergence and a multi-species system.

MDSSN: An end-to-end deep network on triangle mesh parameterization

Mesh data plays a crucial role in a wide range of applications worldwide within the field of 3D computer vision. In contrast to traditional Euclidean space arrangements, meshes encompass spatial information, including edges, faces, angles, and graph structures that embed face information within the mesh data. Nevertheless, conventional deep learning frameworks such as convolutional neural networks (CNNs) have struggled to achieve significant advancements in handling meshes. This paper proposes a simple mesh computation framework called Mesh Decomposition Second-order Sobolev Network (MDSSN) to model triangle meshes and represent their shape. The construction of face-based Riemannian and edge-based Riemannian graphs is motivated by the remarkable representation capabilities of edges and faces in mesh data, achieved through a graph composition mechanism. Furthermore, we design end-to-end operators, including the Sobolev-filter block, pooling, and unpooling blocks, which draw inspiration from traditional deep learning frameworks for modeling the mesh structure. Importantly, the design of the mesh-filter block incorporates the second-order Sobolev metric. To address common challenges in mesh classification and segmentation tasks, we have designed dedicated classification and segmentation modules. We utilize a joint loss function that integrates losses from adjacent faces and categories. We evaluate the performance of the MDSSN in mesh classification and segmentation tasks, focusing on its ability to represent 3D shapes. Experimental results demonstrate that MDSSN achieves superior performance compared to other state-of-the-art approaches.

Elastic Shape Analysis of Surfaces with Second-Order Sobolev Metrics: A Comprehensive Numerical Framework

This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic distances between parametrized or unparametrized immersed surfaces represented as 3D meshes. Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on shape populations, and for computing parallel transport along paths of surfaces. Our proposed approach fundamentally relies on a relaxed variational formulation for the geodesic matching problem via the use of varifold fidelity terms, which enable us to enforce reparametrization independence when computing geodesics between unparametrized surfaces, while also yielding versatile algorithms that allow us to compare surfaces with varying sampling or mesh structures. Importantly, we demonstrate how our relaxed variational framework can be extended to tackle partially observed data. The different benefits of our numerical pipeline are illustrated over various examples, synthetic and real.

A Sobolev rough path extension theorem via regularity structures

We show that every ℝd-valued Sobolev path with regularity a and integrability p can be lifted to a Sobolev rough path provided 1/2 &gt; α &gt; 1/p&gt; ⋁ 1/3. The novelty of our approach is its use of ideas underlying Hairer’s reconstruction theorem generalized to a framework allowing for Sobolev models and Sobolev modelled distributions. Moreover, we show that the corresponding lifting map is locally Lipschitz continuous with respect to the inhomogeneous Sobolev metric.

A New Variational Model for Shape Graph Registration with Partial Matching Constraints

This paper introduces a new extension of Riemannian elastic curve matching to a general class of geometric structures, which we call (weighted) shape graphs, that allows for shape registration with partial matching constraints and topological inconsistencies. Weighted shape graphs are the union of an arbitrary number of component curves in Euclidean space with potential connectivity constraints between some of their boundary points, together with a weight function defined on each component curve. The framework of higher-order invariant Sobolev metrics is particularly well suited for constructing notions of distances and geodesics between unparametrized curves. The main difficulty in adapting this framework to the setting of shape graphs is the absence of topological consistency, which typically results in an inadequate search for an exact matching between two shape graphs. We overcome this hurdle by defining an inexact variational formulation of the matching problem between (weighted) shape graphs of any underlying topology, relying on the convenient measure representation given by varifolds to relax the exact matching constraint. We then prove the existence of minimizers to this variational problem when we choose Sobolev metrics of sufficient regularity and a total variation (TV) regularization on the weight function. We propose a numerical optimization approach which adapts the smoothed fast iterative shrinkage-thresholding algorithm (SFISTA) to deal with $TV$ norm minimization and allows us to reduce the matching problem to solving a sequence of smooth unconstrained minimization problems. We finally illustrate the capabilities of our new model through several examples showcasing its ability to tackle partially observed and topologically varying data.

Smooth Perturbations of the Functional Calculus and Applications to Riemannian Geometry on Spaces of Metrics

We show for a certain class of operators A and holomorphic functions f that the functional calculus $$A\mapsto f(A)$$ is holomorphic. Using this result we are able to prove that fractional Laplacians $$(1+\Delta ^g)^p$$ depend real analytically on the metric g in suitable Sobolev topologies. As an application we obtain local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics.

Constrained willmore surfaces

Smooth curves and surfaces can be characterized as minimizers of squared curvature bending energies subject to constraints. In the univariate case with an isometry (length) constraint this leads to classic non-linear splines. For surfaces, isometry is too rigid a constraint and instead one asks for minimizers of the Willmore (squared mean curvature) energy subject to a conformality constraint. We present an efficient algorithm for (conformally) constrained Willmore surfaces using triangle meshes of arbitrary topology with or without boundary. Our conformal class constraint is based on the discrete notion of conformal equivalence of triangle meshes. The resulting non-linear constrained optimization problem can be solved efficiently using the competitive gradient descent method together with appropriate Sobolev metrics. The surfaces can be represented either through point positions or differential coordinates. The latter enable the realization of abstract metric surfaces without an initial immersion. A versatile toolkit for extrinsic conformal geometry processing, suitable for the construction and manipulation of smooth surfaces, results through the inclusion of additional point, area, and volume constraints.

A Numerical Framework for Elastic Surface Matching, Comparison, and Interpolation

Surface comparison and matching is a challenging problem in computer vision. While reparametrization-invariant Sobolev metrics provide meaningful elastic distances and point correspondences via the geodesic boundary value problem, solving this problem numerically tends to be difficult. Square root normal fields (SRNF) considerably simplify the computation of certain elastic distances between parametrized surfaces. Yet they leave open the issue of finding optimal reparametrizations, which induce elastic distances between unparametrized surfaces. This issue has concentrated much effort in recent years and led to the development of several numerical frameworks. In this paper, we take an alternative approach which bypasses the direct estimation of reparametrizations: we relax the geodesic boundary constraint using an auxiliary parametrization-blind varifold fidelity metric. This reformulation has several notable benefits. By avoiding altogether the need for reparametrizations, it provides the flexibility to deal with simplicial meshes of arbitrary topologies and sampling patterns. Moreover, the problem lends itself to a coarse-to-fine multi-resolution implementation, which makes the algorithm scalable to large meshes. Furthermore, this approach extends readily to higher-order feature maps such as square root curvature fields and is also able to include surface textures in the matching problem. We demonstrate these advantages on several examples, synthetic and real.

Suitable Spaces for Shape Optimization

The differential-geometric structure of the manifold of smooth shapes is applied to the theory of shape optimization problems. In particular, a Riemannian shape gradient with respect to the first Sobolev metric and the Steklov–Poincaré metric are defined. Moreover, the covariant derivative associated with the first Sobolev metric is deduced in this paper. The explicit expression of the covariant derivative leads to a definition of the Riemannian shape Hessian with respect to the first Sobolev metric. In this paper, we give a brief overview of various optimization techniques based on the gradients and the Hessian. Since the space of smooth shapes limits the application of the optimization techniques, this paper extends the definition of smooth shapes to H^{1/2}-shapes, which arise naturally in shape optimization problems. We define a diffeological structure on the new space of H^{1/2}-shapes. This can be seen as a first step towards the formulation of optimization techniques on diffeological spaces.

Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics

The group \({\text {Diff}}({\mathcal {M}})\) of diffeomorphisms of a closed manifold \({\mathcal {M}}\) is naturally equipped with various right-invariant Sobolev norms \(W^{s,p}\). Recent work showed that for sufficiently weak norms, the geodesic distance collapses completely (namely, when \(sp\le \dim {\mathcal {M}}\) and \(s<1\)). But when there is no collapse, what kind of metric space is obtained? In particular, does it have a finite or infinite diameter? This is the question we study in this paper. We show that the diameter is infinite for strong enough norms, when \((s-1)p\ge \dim {\mathcal {M}}\), and that for spheres the diameter is finite when \((s-1)p<1\). In particular, this gives a full characterization of the diameter of \({\text {Diff}}(S^1)\). In addition, we show that for \({\text {Diff}}_c({\mathbb {R}}^n)\), if the diameter is not zero, it is infinite.

Enhanced identification of significant regulators of gene expression

BackgroundDiseases like cancer will lead to changes in gene expression, and it is relevant to identify key regulatory genes that can be linked directly to these changes. This can be done by computing a Regulatory Impact Factor (RIF) score for relevant regulators. However, this computation is based on estimating correlated patterns of gene expression, often Pearson correlation, and an assumption about a set of specific regulators, normally transcription factors. This study explores alternative measures of correlation, using the Fisher and Sobolev metrics, and an extended set of regulators, including epigenetic regulators and long non-coding RNAs (lncRNAs). Data on prostate cancer have been used to explore the effect of these modifications.ResultsA tool for computation of RIF scores with alternative correlation measures and extended sets of regulators was developed and tested on gene expression data for prostate cancer. The study showed that the Fisher and Sobolev metrics lead to improved identification of well-documented regulators of gene expression in prostate cancer, and the sets of identified key regulators showed improved overlap with previously defined gene sets of relevance to cancer. The extended set of regulators lead to identification of several interesting candidates for further studies, including lncRNAs. Several key processes were identified as important, including spindle assembly and the epithelial-mesenchymal transition (EMT).ConclusionsThe study has shown that using alternative metrics of correlation can improve the performance of tools based on correlation of gene expression in genomic data. The Fisher and Sobolev metrics should be considered also in other correlation-based applications.

Geodesic completeness of the $$H^{3/2}$$ metric on $$\mathrm {Diff}(S^{1})$$

Of concern is the study of the long-time existence of solutions to the Euler–Arnold equation of the right-invariant \(H^{\frac{3}{2}}\)-metric on the diffeomorphism group of the circle. In previous work by Escher and Kolev it has been shown that this equation admits long-time solutions if the order s of the metric is greater than \(\frac{3}{2}\), but the behaviour for the critical Sobolev index \(s=\frac{3}{2}\) has been left open. In this article we fill this gap by proving the analogous result also for the boundary case. We show that the behaviour is the same for all Sobolev metrics of order \(\frac{3}{2}\) regardless of lower-order terms.

Semi-invariant Riemannian metrics in hydrodynamics

Many models in mathematical physics are given as non-linear partial differential equation of hydrodynamic type; the incompressible Euler, KdV, and Camassa–Holm equations are well-studied examples. A beautiful approach to well-posedness is to go from the Eulerian to a Lagrangian description. Geometrically it corresponds to a geodesic initial value problem on the infinite-dimensional group of diffeomorphisms with a right invariant Riemannian metric. By establishing regularity properties of the Riemannian spray one can then obtain local, and sometimes global, existence and uniqueness results. There are, however, many hydrodynamic-type equations, notably shallow water models and compressible Euler equations, where the underlying infinite-dimensional Riemannian structure is not fully right invariant, but still semi-invariant with respect to the subgroup of volume preserving diffeomorphisms. Here we study such metrics. For semi-invariant metrics of Sobolev H^k-type we give local and some global well-posedness results for the geodesic initial value problem. We also give results in the presence of a potential functional (corresponding to the fluid’s internal energy). Our study reveals many pitfalls in going from fully right invariant to semi-invariant Sobolev metrics; the regularity requirements, for example, are higher. Nevertheless the key results, such as no loss or gain in regularity along geodesics, can be adopted.

Fractional Sobolev metrics on spaces of immersions

We prove that the geodesic equations of all Sobolev metrics of fractional order one and higher on spaces of diffeomorphisms and, more generally, immersions are locally well posed. This result builds on the recently established real analytic dependence of fractional Laplacians on the underlying Riemannian metric. It extends several previous results and applies to a wide range of variational partial differential equations, including the well-known Euler–Arnold equations on diffeomorphism groups as well as the geodesic equations on spaces of manifold-valued curves and surfaces.

Simplifying Transforms for General Elastic Metrics on the Space of Plane Curves.

In the shape analysis approach to computer vision problems, one treats shapes as points in an infinite-dimensional Riemannian manifold, thereby facilitating algorithms for statistical calculations such as geodesic distance between shapes and averaging of a collection of shapes. The performance of these algorithms depends heavily on the choice of the Riemannian metric. In the setting of plane curve shapes, attention has largely been focused on a two-parameter family of first order Sobolev metrics, referred to as elastic metrics. They are particularly useful due to the existence of simplifying coordinate transformations for particular parameter values, such as the well-known square-root velocity transform. In this paper, we extend the transformations appearing in the existing literature to a family of isometries, which take any elastic metric to the flat L 2 metric. We also extend the transforms to treat piecewise linear curves and demonstrate the existence of optimal matchings over the diffeomorphism group in this setting. We conclude the paper with multiple examples of shape geodesics for open and closed curves. We also show the benefits of our approach in a simple classification experiment.

Vanishing Distance Phenomena and the Geometric Approach to SQG

In this article we study the induced geodesic distance of fractional order Sobolev metrics on the groups of (volume preserving) diffeomorphisms and symplectomorphisms. The interest in these geometries is fueled by the observation that they allow for a geometric interpretation for prominent partial differential equations in the field of fluid dynamics. These include in particular the modified Constantin–Lax–Majda and surface quasi-geostrophic equations. The main result of this article shows that both of these equations stem from a Riemannian metric with vanishing geodesic distance.

Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups

We study the geodesic distance induced by right-invariant metrics on the group $\operatorname{Diff}_\text{c}(M)$ of compactly supported diffeomorphisms, for various Sobolev norms $W^{s,p}$. Our main result is that the geodesic distance vanishes identically on every connected component whenever $s<\min\{n/p,1\}$, where $n$ is the dimension of $M$. We also show that previous results imply that whenever $s > n/p$ or $s \ge 1$, the geodesic distance is always positive. In particular, when $n\ge 2$, the geodesic distance vanishes if and only if $s<1$ in the Riemannian case $p=2$, contrary to a conjecture made in Bauer et al. [BBHM13].

Minimizing acceleration on the group of diffeomorphisms and its relaxation

We study a second-order variational problem on the group of diffeomorphisms of the interval [0, 1] endowed with a right-invariant Sobolev metric of order 2, which consists in the minimization of the acceleration. We compute the relaxation of the problem which involves the so-called Fisher–Rao functional, a convex functional on the space of measures. This relaxation enables the derivation of several optimality conditions and, in particular, a sufficient condition which guarantees that a given path of the initial problem is also a minimizer of the relaxed one. Based on these sufficient conditions, the main result is that, when the value of the (minimized) functional is small enough, the minimizers are classical, that is the defect measure vanishes.