1 Introduction↩︎

The geometric setting of optimal transport relies on the differential geometry of diffeomorphism groups. In particular, the problem of moving one mass to another by a diffeomorphism while minimizing a certain cost can be understood as a construction of geodesics for an appropriate metric in the space of normalized densities; see, for example, Benamou and Brenier [1] or Villani [2] and references therein. Many applications include similar problems where one needs to measure the proximity between different shapes; see, for example, Michor and Mumford [3] or Younes [4]. However, the diffeomorphism action by itself does not allow one to change the total mass of the density. One arrives at the problem of constructing a natural extension of the action which would allow one to connect in the most economical way densities of different total masses, thus leading to the theory of unbalanced optimal transport [5]. Furthermore, other applications, for instance to the theory of spin liquids, require distances between densities that are not scalar- but vector- or matrix-valued. The latter problems are the main subject of this work.

There are several obstructions even to properly posing the problem of the mass transport for vector/matrix densities. Namely, a unique diffeomorphism might not allow one to move simultaneously all coordinate densities in a required way. Moreover, another major difficulty is the positivity constraint for each coordinate density. This constraint is a serious obstacle for many previously suggested scenarios for admissible paths between densities.

In this paper we resolve these difficulties by proposing the gauge-theoretic approach to the problem. Furthermore, we regard the vector densities as half-densities, with the \({\rm SO}(k)\)-gauge transformations acting on them along with diffeomorphisms, which resolves both the positivity and action transitivity issues. The corresponding group acting on densities is enlarged from diffeomorphisms to a semidirect product group. Such groups differ from one problem to another, and they are described below in the various cases of balanced and unbalanced vector and matrix densities.

Next, we propose a tight connection to Poisson geometry and demonstrate how the momentum map helps compute the horizontal distributions and prove the Riemannian submersion properties. Finally, it turned out that natural Bures-type metrics on the acting groups and vector/matrix densities lead to a familiar form of the geodesic equations. The geodesic equations turn out to be various ramifications of the Burgers equations of fully compressible fluids.

Overview of the main results↩︎

Here is a quick overview of the main results, first for the vector half-densities and then for the matrix densities.

Vector-valued optimal transport

To present a sample of the results we start with a manifold \(M\) of any dimension \(n\) and consider the space of normalized vector-valued half-densities \[\mathrm{VHProb} =\{ \mathbf{v} \in C^\infty(M,\mathbb{R}^k ) \otimes \operatorname{HProb}(M)~|~ \int_M \lvert \mathbf{v} \rvert^2 = 1 ~{\rm and ~} \mathbf{v}\not=0 ~{\rm on ~} M\}\] where \(k\) and \(n\) are unrelated. These half-densities can be represented as \(\mathbf{v}={v}\otimes \sqrt\varrho\in \mathrm{VHProb}\) with \(\int_M\varrho=1\) and \(\varrho > 0\). The group acting on half-densities in \(\mathrm{VHProb}\) is the semidirect product \(\widetilde{G}= \operatorname{Diff}(M)\ltimes C^\infty(M,\operatorname{SO}(k))\ni (\varphi, A)\), where the diffeomorphism \(\varphi\) acts on half-densities \(\mathbf{v}\) by the pullback with its inverse and with the square root of its Jacobian appearing as a factor, while the \({\rm SO}(k)\)-current \(A\) on \(M\) is applied point-wise to the obtained vector-valued half-density. This action turns out to be transitive and preserves positivity of the densities, see Section 3.1.

One can introduce natural (weak) Riemannnian metrics on both the group \(\widetilde{G}\) and the density space \(\mathrm{VHProb}\), and the natural projection becomes a Riemannian submersion (Theorem 8). In particular, the vector Wasserstein distance \(\mathit{WV}(\mathbf{v}_0,\mathbf{v}_1)\) between densities \(\mathbf{v}_0,\mathbf{v}_1 \in \mathrm{VHProb}\) is \[\mathit{WV}^2(\mathbf{v}_0,\mathbf{v}_1) = \inf_{u,a,\mathbf{v}}\int_0^1 \int_M \left( \lvert u\rvert^2 + \operatorname{tr}(aa^\top) \right) \lvert\mathbf{v}\rvert^2 \, dt\,,\] over time-dependent vector fields \(u(t)\in \mathfrak{X}(M)\), vector half-densities \(\mathbf{v}(t) \in \mathrm{VHProb}\), and matrix valued functions \(a(t) \in C^\infty(M, \mathfrak{so}(k))\) related by the constraints \[\dot{\mathbf{v}} = -L_u \mathbf{v} + a \mathbf{v} ,\quad \mathbf{v}(0) = \mathbf{v}_0, \quad \mathbf{v}(1) = \mathbf{v}_1,\] see Definition 3. Essentially, the distance between any two vector densities increases whenever more mass has been moved and more mass was necessary to “rotate” in the gauge group to relate their vector values.

The corresponding geodesic equations are given by the following:

These geodesic equations become natural extensions of the classical Burgers equation for the velocity \(u\) and transported scalar or gauge field \(a\). However, due to complexity of the horizontal distribution in the group \(\operatorname{Diff}(M)\ltimes C^\infty(M,\operatorname{SO}(k))\) intertwining different components, one cannot split out completely the velocity part and describe solutions to these seemingly simple equations solely via gradient fields as in the classical scalar optimal transport.

Analogous equations describe the unbalanced vector transport, but with the \({\rm Conf}(k)\)-gauge group of linear conformal transformations replacing \(\operatorname{SO}(k)\), see Section 4.2. Since we are interested in the local questions and the corresponding geodesic equations, everywhere in the paper we assume that vector half-densities on \(M\) under consideration are homotopy equivalent and that the corresponding purely topological obstructions for their global gauge equivalence vanish, as we discuss below in Remark 12.

Matrix-valued optimal transport

Similar metrics can be defined in the matrix density case. Namely, we consider the space of normalized matrix densities on \(M\) (of any dimension \(n\)) \[\mathrm{MProb}:=\{\mathbf{S} \in \Omega^n(M, {\rm Sym}_+(k))~|~ \, \int_M \operatorname{tr}(\mathbf{S})=1 \}\] of unit total mass with values in positive-definite quadratic forms of unit trace in \({\mathbb{R}}^k\). Here the \({\rm Sym}_+(k)\) stands for the cone of positive-definite quadratic forms of \(k\) variables. The scalar density associated with \(\mathbf{S}\) is \(\operatorname{tr}(\mathbf{S})\), and the total mass is normalized: \(\int_M \operatorname{tr}(\mathbf{S})=1\).

The semidirect product group acting on this space is \[\widehat{PG}={\rm Diff}(M)\ltimes C^\infty(M, {\rm PGL}(k, \mathbb{R})),\] with possibly different \(k\) and \(n\). The group action is the composition of the diffeomorphism action and a pointwise “rescaling” as follows: For an element \[(\varphi, A)\in {\rm Diff}(M)\ltimes C^\infty(M, {\rm PGL}(k, \mathbb{R}))\] the action on a matrix density \(\mathbf{S}\) is \((\varphi, A)\cdot \mathbf{S} = \pi\circ A (\varphi_*\mathbf{S}) A^\top\), where \(\varphi_*\) acts on each matrix element thought of as a top-form, \(A\) acts by pointwise change of basis in the quadratic form, and the pointwise rescaling \(\pi\) takes positive-definite quadratic forms to those of unit trace. Similarly to the above, the action of the group \(\widehat{PG}\) on matrix densities \(\mathrm{MProb}\) of unit total masses is transitive.

As above, let now \(M\) be a Riemannian manifold with metric \((\cdot\,, \cdot)\). We fix an initial matrix density \(\mathbf{S}_0 \in \mathrm{MProb}\). A matrix analogue of the Bures-type \(L^2\)-metric on the group leads to the following definition of the matrix Wasserstein distance \(\mathit{WM}(\mathbf{S}_0,\mathbf{S}_1)\) between two matrix densities \(\mathbf{S}_0,\mathbf{S}_1 \in \mathrm{MProb}\): \[\mathit{WM}^2(\mathbf{S}_0, \mathbf{S}_1) = \inf_{u,a,\mathbf{S}}\int_0^1 \int_M \left( \lvert u\rvert^2 + \operatorname{tr}(aa^\top) \right) \operatorname{tr}(\mathbf{S}) \, dt\,,\] where the infimum is taken over time-dependent vector fields \(u(t)\in \mathfrak{X}(M)\), matrix densities \(\mathbf{S}(t) \in \mathrm{MProb}\), and matrix valued functions \(a(t) \in C^\infty(M, \mathfrak{gl}(k))\) related by the constraints \[\dot{\mathbf{S}} = -L_u\mathbf{S} + a\mathbf{S} + \mathbf{S} a^\top ,\quad \mathbf{S}(0) = \mathbf{S}_0, \quad \mathbf{S}(1) = \mathbf{S}_1, \quad \operatorname{tr}(a\mathbf{S})=0\] Here again this metric naturally extends the classical Wasserstein metric by penalizing both whenever some mass is to be moved over \(M\) and whenever two matrix valued masses need an additional transformation in the fibers to identify them. An appropriate metric on the group \(\widehat{PG}\) projects to the above one via Riemannian submersion. By introducing factorization \(\mathbf{S} = S\otimes\varrho\) where \(\operatorname{tr}S = 1\) one can write the corresponding reduced geodesic equations on (the cotangent space of) the group \(\widehat{PG}\) as follows:

Note that the geodesic equations turn out to be rather explicit and can be regarded as a Burgers-type intertwined version of compressible chiral fluids. As before, the difficulty of solving them explicitly is in the complexity of the horisontal distribution for the corresponding Riemannian submersion. We discuss the equations in more detail in Section 5.2, as well as an unbalanced version of this transport in Section 5.1. It is worth mentioning that unlike the vector case, all matrix densities on \(M\) are homotopy equivalent since the space of positive-definite quadratic forms is contractible. One might still encounter topological obstructions for their gauge equivalence which we assume vanishing, see Remark 12.

Notations Throughout the paper we adopt the following notations: \(\mathrm{Prob}\) and \(\mathrm{HProb}\) stand respectively for normalized densities and half-densities, while for non-normalized ones we reserve \(\mathrm{Dens}\) and \(\mathrm{HDens}\). The vector and matrix objects are marked by V or M, so that, e.g., normalized vector half-densities are \(\mathrm{VHProb}\), while all matrix densities are \(\mathrm{MDens}\).

Motivation One of the most straightforward applications of optimal transport of vector densities includes colored medical imaging, for example multi parametric magnetic resonance imaging where the color channels correspond to different types of imaging data [6]. By making a colored picture of an organ, one can see “how far” it is from its healthier shape, size, and color. Medical imaging is also relevant for matrix-valued densities, for example, for diffusion tensor imaging where the imaging data consists of point-wise symmetric matrices that describe the diffusion of water in the tissue.

As another application we mentioned the theory of quantum spin liquids in condensed matter physics. It describes a phase of matter that can be formed by interacting quantum spins in certain magnetic materials. This state is intuitively described as a “liquid” of disordered spins, much in the way liquid water is in a disordered state compared to crystalline ice. In this paper we propose a possible measure of the distance between any two such vector-valued states. We discuss several other motivations in the last section.

The paper is organized as follows. In section 2 we give background on the geometric setting of classical optimal transport. In section 3 we introduce vector half-densities and their corresponding transport maps (based on a semi-direct product), and we give a geometric description of the corresponding transport problem, including the geodesic equations. In section 4 we first review the framework of unbalanced optimal transport, and then discuss how to generalize it to vector half-densities. Finally, in section 5 we present a universal point-of-view, which deals with matrix densities and encapsulates all the other frameworks. In section 6 we discuss various open questions and further directions. In appendices we describe how the moment map manifests itself in optimal transport (appendix 7) and demostrate the geodesic equations for an alternative metric on matrix densities (appendix 8).

Acknowledgements

The authors are indebted to Sergei Tabachnikov, Anton Izosimov and François-Xavier Vialard, as well as other participants of the workshops “Math en plein air", for fruitful discussions. B.K. was partially supported by an NSERC Discovery Grant. K.M. was supported by the Swedish Research Council (grant number 2022-03453), the Knut and Alice Wallenberg Foundation (grant numbers WAF2019.0201), and the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine.

2 Optimal transport and semidirect products↩︎

2.1 Geometric setting of scalar optimal transport↩︎

Let \(\mu\) be a smooth reference volume form (or density) of unit total mass on a manifold \(M\), and consider the projection \(\pi : \operatorname{Diff}(M) \to \operatorname{Prob}(M)\) of diffeomorphisms onto the space \(\operatorname{Prob}(M)\) of smooth probability (i.e. normalized) densities on \(M\). The diffeomorphism group \(\operatorname{Diff}(M)\) is fibered over \(\operatorname{Prob}(M)\) by means of this projection \(\pi\) as follows: the fiber over a volume form \(\varrho\) consists of all diffeomorphisms \(\eta\) that push \(\mu\) to \(\varrho\), \(\eta_*\mu=\varrho\). In other words, two diffeomorphisms \(\eta_1\) and \(\eta_2\) belong to the same fiber if and only if \(\eta_1=\eta_2\circ\varphi\) for some diffeomorphism \(\varphi\) preserving the volume form \(\mu\). Diffeomorphisms from \(\operatorname{Diff}(M)\) act transitively on smooth densities, according to the Moser theorem.

Suppose now that \(M\) is a compact \(n\)-dimensional Riemannian manifold with metric \((~,~)\). Then one can define (weak) Riemannian metrics on both the diffeomorphism group \(\operatorname{Diff}(M)\) and the density space \(\operatorname{Prob}(M)\). Namely, at each point \(\eta\in\operatorname{Diff}(M)\) of the diffeomorphism group is defined in the following straightforward way: given \(X,Y\in \mathfrak{X}(M)\), the inner product of two vectors \(X\circ\eta,Y\circ\eta\in T_\eta\operatorname{Diff}(M)\) is \[\label{eq:diff95metric} \langle\langle X\circ\eta,Y\circ\eta\rangle\rangle_{\eta} =\int_M \left(X\circ\eta(x),Y\circ\eta(x)\right)\,\mu(x).\tag{1}\] This metric is right-invariant when restricted to the subgroup \(\operatorname{Diff}_\mu(M)\) of volume-preserving diffeomorphisms, although it is not right-invariant on the whole group \(\operatorname{Diff}(M)\). It is worth mentioning that for a flat manifold \(M\) this metric is a flat metric on \(\operatorname{Diff}(M)\): a neighborhood of the identity \(e\in \operatorname{Diff}(M)\) with the metric \(\langle\langle~,~\rangle\rangle_{\operatorname{Diff}}\) is isometric to a neighborhood in the pre-Hilbert space of smooth “vector-functions” \(\eta: M\to M\) with the \(L^2\) inner product \(\langle~,~\rangle_{L^2(M)}\).

On the other hand, on the density space \(\operatorname{Prob}(M)\) there also exists a natural metric inspired by the following optimal mass transport problem: find a (sufficiently regular) map \(\eta:M\to M\) that pushes the measure \(\mu\) forward to \(\varrho\) and attains the minimum of the \(L^2\)-cost functional \(\int_M \operatorname{dist}^2(x,\eta(x))\mu\) among all such maps, where \(\operatorname{dist}\) is the Riemannian distance on \(M\). The minimal cost of transport defines the \(L^2\) Wasserstein distance \({\operatorname{Dist}}\) on densities \(\operatorname{Prob}(M)\): \[{\operatorname{Dist}}^2(\mu, \varrho) :=\inf_\eta\Big\{\int_M \operatorname{dist}^2(x,\eta(x))\mu~ |~\eta_*\mu=\varrho\Big\}\,.\] One can see that the Wasserstein distance is formally generated by the (weak) Riemannian metric on the space \(\operatorname{Prob}(M)\) of smooth densities: \[\label{eq:dens95metric} \langle\dot{\varrho},\dot{\varrho}\rangle_\varrho = \int_M |{\nabla {\theta}}|^2 \varrho \quad\text{for } \dot{\rho} + {\rm div}_\mu({\rho} \nabla {\theta})=0,\tag{2}\] where \(\dot{\rho} \in C^\infty(M)\) is a tangent vector to \({\rm Prob}(M)\) at the point \(\varrho=\rho\mu\). (The gradient vector field \(\nabla{\theta}\) has the minimal \(L^2\) energy among all vector fields moving the volume form \(\varrho\) in the direction \(\dot{\varrho}\) due to the Hodge decomposition.) Thus both \(\operatorname{Diff}(M)\) and \({\rm Prob}(M)\) can be regarded as infinite-dimensional Riemannian manifolds.

Recall that for two Riemannian manifolds \(P\) and \(B\) a Riemannian submersion \(\pi: P\to B\) is a mapping onto \(B\) that has maximal rank and preserves lengths of horizontal tangent vectors to \(P\). For a bundle \(P\to B\) this means that on \(P\) there is a distribution of horizontal spaces that are orthogonal to fibers and projected isometrically to the tangent spaces to \(B\).

2.2 Geodesics in the scalar transport↩︎

One of the main problems in optimal mass transport is to describe geodesics on the space \(\operatorname{Prob}(M)\) of densities, that is, on the base of the bundle \(\operatorname{Diff}(M)\to \operatorname{Prob}(M)\). However, it is easier to start instead with the description of geodesics on \(\operatorname{Diff}(M)\) and then restrict to horizontal geodesics, which correspond to geodesics on the base.

Define the (inviscid) Burgers equation as the evolution equation \[\label{eq:burgers} \partial_t u+\nabla_u u=0\,\tag{3}\] for a vector field \(u\) on \(M\), where \(\nabla_u u\) stands for the covariant derivative on \(M\). Consider the flow \((t,x)\mapsto \eta(t,x)\) corresponding to velocity field \(u\): \[\partial_t \eta (t,x) = u(t,\eta(t,x)), \qquad \eta(0,x)=x.\]

2.3 Semi-direct products↩︎

Let \(G\) be a Lie group acting from the left on another Lie group \(L\) such that the action map \(g\cdot L\to L\) is an automorphism: \(g\cdot (\ell_1 \ell_2) = (g\cdot \ell_1)(g\cdot \ell_2)\). Then we can form the semi-direct product group \(G\ltimes L\) with product \[(g_1,\ell_1)\cdot (g_2, \ell_2) = \big(g_1g_2, \ell_1(g_1\cdot \ell_2)\big).\] Assume further that both \(G\) and \(L\) act from the left on the vector space \(V\). An action of \(G\ltimes L\) on \(V\) is then given by \((g,\ell)\cdot v = \ell \cdot (g\cdot v)\). Assume that this action is transitive.

3 Mass transport of vector half-densities↩︎

3.1 The group action on vector half-densities↩︎

Now we consider the corresponding setup for vector densities and describe a similar semi-direct product group acting on them in a transitive way.

Given a compact orientable manifold \(M\) of dimension \(n\) define the semi-direct product group \(\widetilde{G}=G\ltimes N\) where \(G= \operatorname{Diff}(M)\) and \(N = C^\infty(M,\operatorname{SO}(k))\).³ We define its action on the space \[\mathrm{VHProb} =\{ \mathbf{v} \in C^\infty(M,\mathbb{R}^k ) \otimes \operatorname{HProb}(M)~\Big|~ \int_M \lvert \mathbf{v} \rvert^2 = 1 ~{\rm and ~} \mathbf{v}\not=0 ~{\rm on ~} M\}\] of normalized vector-valued half-densities. (Half-densities mean top degree forms which under the action of diffeomorphisms are getting multiplied by the square root of the Jacobian, as we discuss below.) They can be represented as \(\mathbf{v}={v}\otimes \sqrt\varrho\in \mathrm{VHProb}\) with \(\int_M\varrho=1\) and \(\varrho > 0\). Here the tensor product is over \(C^\infty(M, \mathbb{R})\), and one can also assume that \({v}(x)\in S^{k-1}\subset \mathbb{R}^k\) for all \(x\in M\).

The action of \(\varphi\in \operatorname{Diff}(M)\) on \(\mathbf{v} \in \mathrm{VHProb}\) is the component-wise half-density pushforward: \[\varphi\cdot \mathbf{v} = \varphi_*\mathbf{v} = {v}\circ\varphi^{-1} \sqrt{\varphi_*\varrho}\] The corresponding infinitesimal action of \(u\in \mathfrak{X}(M)\) is the component-wise Lie derivative: \(u\cdot \mathbf{v} = -L_u \mathbf{v}\). In particular, relative to the representation \(\mathbf{v} = v\otimes \sqrt{\varrho}\), we have \[L_u (v\otimes \sqrt{\varrho}) = (L_u v) \otimes \sqrt{\varrho} + v\otimes \frac{1}{2}\frac{L_u\varrho}{\sqrt{\varrho}}= (L_u v + \frac{L_u\varrho}{2\varrho}v )\otimes\sqrt{\varrho}.\]

The action of the gauge transformation \(A \in N\) on \(\mathbf{v}\in \mathrm{VHProb}\) is pointwise: \[(A\cdot \mathbf{v})(x) = A(x)\mathbf{v}(x).\] The corresponding infinitesimal action is \((a\cdot \mathbf{v})(x) = a(x)\mathbf{v}(x)\) for \(a \in T_e N = C^\infty(M,\mathfrak{so}(k))\).

The combined infinitesimal action of \((u,a) \in T_{(id,e)}(G\ltimes N)\) is \[\label{eq:inf95action95on95v} (u,a)\cdot \mathbf{v} = -L_u \mathbf{v} + a \mathbf{v} .\tag{4}\]

Note that the fibers of the corresponding fibration \(\widetilde{G}\to \mathrm{VHProb}\) depend on the choice of the reference density \(\mathbf{v}\in \mathrm{VHProb}\) and are given by its stabilizer. For instance, for a vector density \(\mathbf{v}={v}\otimes \sqrt\varrho\) whose \(v\)-components are all the same in a certain coordinate system in \(\mathbb{R}^k\), its stabilizer is the subgroup \(\mathrm{Stab}_\mathbf{v}:=\operatorname{Diff}_\mu(M) \ltimes C^\infty(M, \operatorname{SO}(k-1))\). Indeed, it consists of those diffeomorphisms \(\varphi\) which preserve density \(\varrho=\lvert\mathbf{v}\rvert^2\) and at each point of \(x\in M\) the subgroup \({\rm SO}(k-1)\subset {\rm SO}(k)\) fixes a given nonzero vector \({v}(x)\).

Consider now the principal bundle \(\pi\colon G\ltimes N \to \mathrm{VHProb}\) given by the action on a fixed element \(\mathbf{v}_0\in \mathrm{VHProb}\), i.e., \(\pi(\varphi,A) = A\, \varphi_* \mathbf{v}_0\). This fixed element corresponds, as we shall see, to the initial vector half-density in the transport problem.

3.2 Riemannian metrics and submersion↩︎

Now we define a Riemannian metric on \(\widetilde{G}= G\ltimes N\).

This is a natural combination of the non-invariant \(L^2\) metric on \(\mathrm{Diff}(M)\) and the Bures metric on matrices. The first term of this metric evaluates how much mass has been moved, while the second term measures how much mass was necessary to rotate in the gauge group to relate their vector values.⁴

Once the metric on the group is fixed, one can give a dynamical formulation of the corresponding optimal transport problem on the space of vector densities.

The corresponding horizontal distribution on the group is described as follows.

Note that this description of the horisontal distribution is much less explicit than the one for the group \(\operatorname{Diff}(M)\) with the \(L^2\)-metric fibered over scalar densities: in that case \(H_{\varphi}(G)\) is given by gradient vector fields \(\{\nabla\theta \mid {\it for ~all~} \,\theta\in C^\infty(M,\mathbb{R})\}\) and it does not depend on the base point. In the semidirect product setting above, the horisontal distribution can be understood as a continuous combination of gradients and, moreover, it depends on the base point via the direction vector \(v\), but not on the density part \(\varrho\).

We are proving this proposition below, but now note that such a description allows one to define a metric on vector densities inherited from the metric on the group via a Riemannian submersion. As its corollary we obtain the following key statement.

A proof of Proposition 7 on the description of horizontal spaces, implying the Riemannian submersion property, can be obtained from the following explicit form of the momentum map for that group.

Proof of Proposition 7 now immediately follows from the above description of the momentum map \(\Phi\) since, by construction, the horizontal distribution \(H_{(\varphi,A)}(G\ltimes N)\) is given by the condition \[\langle (u,a), \Phi(\mathbf{v},\boldsymbol{\theta}) \rangle = 0\] if and only if \((u,a)\cdot (\varphi,A)\) is vertical. \(\square\)

3.3 Geodesics and Hamiltonian equations↩︎

We start by deriving the coadjoint operator for the Lie algebra \(\tilde{\mathfrak{g}}=\mathfrak{g}\ltimes\mathfrak{n}\) of the group \(\widetilde{G}= \operatorname{Diff}(M)\ltimes C^\infty(M,\operatorname{SO}(k))\).

The geodesic equations for the Bures-type metric in Definition 2, as a Hamiltonian system on the Lie–Poisson space \(\tilde{\mathfrak{g}}^*\simeq\mathfrak{g}^*\oplus\mathfrak{n}^*\), is given by the following theorem.

4 Unbalanced mass transport↩︎

4.1 Various settings of the unbalanced transport↩︎

Unbalanced mass transport considers geodesics between densities of different total masses. Let \({\rm Dens}(M)=\Omega^n(M, {\mathbb{R}}_+)\) stand for all densities with arbitrary total masses on an \(n\)-dimensional manifold \(M\).

It became natural now to associate an unbalanced transport with consideration of the semidirect product group \[G=\operatorname{Diff}(M)\ltimes C^\infty(M, \mathbb{R}_+),\] see [11], Section 3.2.1. Then \((\varphi, \lambda)\in \operatorname{Diff}(M)\ltimes C^\infty(M, \mathbb{R}_+)\) acts on a density \(\mu \in {\rm Dens}(M)\) as follows: \[\mu\mapsto (\varphi^*\mu)\cdot \lambda^2.\] This action is transitive: one can get an arbitrary density from any initial one by choosing an appropriate factor \(\lambda\). (Here and in [11] one uses the multiplicative group of nonvanishing functions, but there is also an additive version: \((\varphi, f)\in \operatorname{Diff}(M)\ltimes C^\infty(M, \mathbb{R}),\) where \(\lambda:=\exp(f)\) and the action is \(\mu\mapsto \varphi^*\mu\cdot \exp(2f)\).)

One introduces a metric on the group \(G\), which projects (as a Riemannian submersion) to a metric on \({\rm Dens}(M)\), see details in [11]. The obtained metric on \({\rm Dens} (M)= \operatorname{Prob}(M) \times \mathbb{R}_+\) is a cone over the metric on the space of normalized densities \(\operatorname{Prob}(M)\) on \(M\) and quadratic scaling in \(\mathbb{R}_+\), see formula (3.22) in [11] (see also earlier papers [5], [12]).

4.2 Vector unbalanced transport↩︎

By replacing \({\rm SO}(k)\) with the conformal linear group \({\rm Conf}(k):={\rm SO}(k)\times {\mathbb{R}}_+\) (rotations and dilations) as the main gauge group, one can consider an unbalanced vector optimal transport. The standard unbalanced transport becomes a particular case of this more universal framework for \(k=1\).

Namely, in the unbalanced setting we define the semi-direct product group \({\boldsymbol{G}}= \operatorname{Diff}(M) \ltimes C^\infty(M, {\rm Conf}(k))\) It acts on the space of all (non-normalized) vector-valued half-densities \[\mathrm{{\boldsymbol{V}}HDens} =\{ \mathbf{v} \in C^\infty(M,\mathbb{R}^k ) \otimes \operatorname{HDens}(M)~|~ \mathbf{v}\not=0 ~{\rm on ~} M\}\] We still represent those densities as \(\mathbf{v}={v}\otimes \sqrt\varrho\in \mathrm{{\boldsymbol{V}}HDens}\) with \(v(x)\in S^{k-1}\) in a unique way. The combined group action is given by the same formula as before.

In the same way one can see that the action of the group \({\boldsymbol{G}}= \operatorname{Diff}(M)\ltimes C^\infty(M,\operatorname{Conf}(k))\) half-densities in \(\mathrm{{\boldsymbol{V}}HDens}\) is transitive.

The corresponding combined infinitesimal action of \((u,a) \in T_{(id,e)}\boldsymbol{G}\) is given by the same formula \[(u,a)\cdot \mathbf{v} = -L_u \mathbf{v} + a \mathbf{v} \,,\] where \(a\in C^\infty(M, \frak{conf}(k))\), i.e. \(a\) has the scalar diagonal part, since the Lie algebra \(\frak{conf}(k)\) is a direct product: \(\frak{conf}(k)=\frak{so}(k)\oplus I \mathbb{R}\), where \(I\) is the \(k\times k\) identity matrix.

All the above considerations, including the geodesic equations in Theorem 9, have the same form, where the matrix term \(a\), which before was skew-symmetric and hence zeros on the diagonal, may now have a scalar diagonal part: \[\dot{u} + \nabla_{{u}}{u} = 0 , \quad \dot{a} + \nabla_u a = 0, \quad \dot{\mathbf{v}} + L_{{u}} \mathbf{v} = a \mathbf{v} .\] Here, however, the scalar density \(\varrho = \lvert\mathbf{v}\rvert^2\) is not transported by \(u\) any longer, but changes as in the scalar unbalanced transport: \[\dot{\varrho} +L_u \varrho= -\frac{2}{k}\operatorname{tr}( a) \varrho\] due to the present diagonal part of \(a\).

5 Universal mass transport of matrix densities↩︎

The setting of matrix-valued densities requires a different acting group. Let \(M\) be a manifold of dimension \(n\), and fix a (normalized) reference density \(\mu\).

It turns out that, unlike the vector case, it is more natural to start with the unbalanced transport of matrix densities.

5.1 Unbalanced optimal transport of matrix-valued densities↩︎

Now we consider the semidirect product group \[\widehat G={\rm Diff}(M)\ltimes C^\infty(M, {\rm GL}(k, \mathbb{R})),\] where \(k\) and \(n\) could be different. This group acts on the spaces of matrix densities \({\rm MDens}=\Omega^n(M, {\rm Sym}_+(k))\) as follows. For \[(\varphi, A)\in {\rm Diff}(M)\ltimes C^\infty(M, {\rm GL}(k, \mathbb{R}))\] the action on a matrix density \[\mathbf{S} \in \Omega^n(M, {\rm Sym}_+(k))\subset \Omega^n(M) \otimes {\rm Sym}(k)\] is \[(\varphi, A)\cdot \mathbf{S} = A (\varphi_*\mathbf{S}) A^\top \,,\] where \(\varphi_*\) acts on each matrix element thought of as a top-form. The density associated with \(\mathbf{S}\) is thus \(\operatorname{tr}(\mathbf{S})\), so the total mass is \[{\rm Mass}(\mathbf{S}):=\int_M \operatorname{tr}\mathbf{S}.\]

Lift the action of the group \(\widehat G\) from the space of densities \(\mathrm{MDens}\) to the Hamiltonian action on its cotangent bundle \(T^*\mathrm{MDens}\). We identify \(T^*_{\mathbf{S}}\mathrm{MDens}\) with \(C^\infty(M, \mathfrak{gl}(k)/\mathfrak{so}(k))\) via the point-wise Frobenius inner product. For an element \(P\in C^\infty(M, \mathfrak{gl}(k))\) consider its coset \([P]\in C^\infty(M, \mathfrak{gl}(k)/\mathfrak{so}(k))\) defined by pointwise addition of an arbitrary skew-symmetric matrix.

As above, let now \(M\) be a Riemannian manifold with metric \((\cdot\,, \cdot)\). We fix an initial matrix density \(\mathbf{S}_0 \in \mathrm{MDens}\).

The consideration mimicking Proposition 6 shows that this metric is nondegenerate.

As before, these two metrics are defined as natural extensions of the classical Wasserstein metric: For any path between two matrix densities the first term penalizes if much mass is to be moved over \(M\), while the second evaluates how much two matrix valued masses differ by measuring how far one needs to rotate the coordinates to identify them.

For convenience, we now introduce the factorization \(\mathbf{S} = S\otimes\varrho\) where \(\operatorname{tr}S = 1\) (this corresponds to the factorization \(\mathbf{v} = v\otimes \sqrt{\varrho}\) of the vector half-density above). In this factorization, we have that \[\label{eq:rho95evolution95MDens} \dot{\varrho} = \frac{d}{dt}\operatorname{tr}(\mathbf{S}) = - L_u\varrho + 2\operatorname{tr}(a S)\varrho\tag{9}\] and \[\dot{S} = -L_u S + a S + S a^\top - 2 S \operatorname{tr}(a S).\]

We are now in a position to study geodesics. To this end, consider the quadratic Hamiltonian on \(\widehat{\mathfrak{g}}^*\times \mathrm{MDens}\) obtained via the Legendre transform 11 :

\[\label{eq:Hamiltonian95MDens} \widehat{H}(u^\flat\otimes\varrho, a\otimes\varrho, S\otimes\varrho) = \frac{1}{2}\int_M \Big( |u|^2 + \operatorname{tr}(aa^\top) \Big)\varrho\tag{12}\]

Note that if \(a\) is skew-symmetric then all the terms with \(\operatorname{tr}(aS)\) vanish (as well as the term \((|u|^2 + \operatorname{tr}(aa^\top))S\)), and thus the equations simplify noticeably. We will also observe similar simplifications for the balanced version of matrix transport in the next section. Again, the equations are on \(\hat{\mathfrak{g}}^*\times \mathrm{MDens}\), which is a reduced form of \(T^*\hat{G}=\hat{\mathfrak{g}}^*\times (\operatorname{Diff}(M)\ltimes C^\infty(M,\mathrm{Sym}_+(k)))\) taking into account the reduction of base isometries.

5.2 Balanced matrix transport↩︎

While the matrix version intrinsically favors the unbalanced version, one can develop the balanced matrix transport by normalizing the matrix-valued densities and changing the gauge group from \(\mathrm{GL}(k)\)-currents to \(\mathrm{PGL}(k)\)-currents.

The semidirect product group acting on this space is \[\widehat{PG}={\rm Diff}(M)\ltimes C^\infty(M, {\rm PGL}(k, \mathbb{R})).\] The group action is the composition of the previously defined (unbalanced) action and a pointwise rescaling as follows: For an element \[(\varphi, A)\in {\rm Diff}(M)\ltimes C^\infty(M, {\rm PGL}(k, \mathbb{R}))\] the action on a matrix density \(\mathbf{S}\) is \[(\varphi, A)\cdot \mathbf{S} = \pi\circ A (\varphi_*\mathbf{S}) A^\top \,,\] where \(\varphi_*\) acts on each matrix element thought of as a top-form, \(A\) acts by pointwise change of basis in the quadratic form, and the pointwise rescaling \(\pi\) takes positive-definite quadratic forms to those of unit trace. As before, the action of the group \(\widehat{PG}= {\rm Diff}(M) \ltimes C^\infty(M, {\rm PGL}(k, \mathbb{R}))\) on matrix densities \(\mathrm{MProb}\) of unit total masses is transitive.

The unbalanced results in Section 5.1 are readily adapted to the balanced case. In particular, the geodesic equations, given by Theorem 18 in the unbalanced case, now look as follows:

Note that for the balanced matrix transport we have the same alternative as with the scalar one, namely one can use only one parameter, the total mass, to normalize the density, as in simple unbalanced transport [13]. Above, for \(\mathrm{PGL}(k)\)-currents, we use the pointwise normalization.

6 Further ramifications of transport↩︎

The relations between different parts of this paper are summarized in Figure 2. We continue here with several further directions which, in our opinion, would be interesting to explore.

1. Do there exist interesting finite-dimensional models of the vector optimal transport, extending the action of linear transformations on Gaussians, cf. [15]?

2. By changing the discrete index (the density coordinates) to a continuous parameter, the velocity \(v\in C^\infty(M, \mathbb{R}^k)\) changes to \(v\in C^\infty(M\times \mathbb{R}, \mathbb{R})\), and this covers a new variety of the equations, with only cosmetic changes. It would be interesting to develop this setting and its applications, as it might be related to generalized flows as suggested by Brenier [16].

3. Adding a potential \(U\) to the Lagrangian we extend the domain of applicability of transports of vector densities. Namely, they are now governed not by geodesics, but by the Newton equation. The potential can encode various constraints and obstructions for the transport.

4. The groups considered above were related to densities and gauge transformations in a trivial bundle over an orientable manifold. A similar optimal transport theory should exist for a topologically non-trivial bundle, as well as for densities over a non-orientable manifold.

5. Is there a relation to the vector-, 1-form-, or matrix- densities considered in [17], [18]?

6. It would be intersting to see if the relativistic Burgers described by Brenier [19] could be recovered as a particular case of the matrix Burgers equation (or, rather, of a compressible fluid equation) described above.

7. Consider the Kähler setting, with vector half-densities over complex scalars, which gives a generalized Madelung transform via Fusca’s map [20]. It would be interesting to unify those two approaches in matrix transport and Kähler vector transport. They present essentially the same group action in two similar ways, in the real and complex directions. A natural question arises whether there is an analogue of Fusca’s momentum map for the unbalanced mass transport. And if so, what are its Riemannian submersion and Kähler properties?

More on motivation In the introduction we mentioned applications of optimal transport of vector and matrix densities to colored medical imaging and quantum spin liquids in physics. One more motivation comes from the spectral analysis of multi-variable time series (e.g., [9]). The related spectral content is assessed based on certain statistics related to moments of the corresponding power spectral density. It may drift across frequencies over time as well as across principle directions (e.g., echo locating a moving target using antenna arrays), as captured by vector-valued time series. Therefore, in the transport between matrix-valued densities, we consider both the cost of shifting power across frequencies and the cost of rotating the corresponding principle axes.

Another application is related to problems of optimal distribution in economics. Consider for instance the task of optimal distribution of vacation packages by wholesalers: assuming that one can change the length of stays at several resorts between a few standard ones, at a cost (i.e., the personnel’s salary) as compared to the set of various packages originally bought by the wholesaler in bulk. Minimization of the corresponding cost becomes a problem of optimal transport of vector densities, where the coordinates are packages of a given length and at a given place. The metrics proposed in this paper are straightforward analogs of the Wasserstein metric, being based on the differential geometry of the problem.

References↩︎