Linear optimal partial transport embedding

Let $\mathcal{P}(\Omega)$ be the set of Borel probability measures defined in a convex compact subset $\Omega$ of $\mathbb{R}^{d}$, and consider $\mu^{0},\mu^{j}\in\mathcal{P}(\Omega)$. The Optimal Transport (OT) problem between $\mu^{0}$ and $\mu^{j}$ is that of finding the cheapest way to transport all the mass distributed according to the reference measure $\mu^{0}$ onto a new distribution of mass determined by the target measure $\mu^{j}$. Mathematically, it was stated by Kantorovich as the minimization problem

	$\displaystyle OT(\mu^{0},\mu^{j}):=\inf_{\gamma\in\Gamma(\mu^{0},\mu^{j})}C(\gamma;\mu^{0},\mu^{j})$		(1)
	$\displaystyle\text{for }\quad C(\gamma;\mu^{0},\mu^{j}):=\int_{\Omega^{2}}\|x^{0}-x^{j}\|^{2}d\gamma(x^{0},x^{j}),$		(2)

where $\Gamma(\mu^{0},\mu^{j})$ is the set of all joint probability measures in $\Omega^{2}$ with marginals $\mu^{0}$ and $\mu^{j}$. A measure $\gamma\in\Gamma(\mu^{0},\mu^{j})$ is called a transportation plan, and given measurable sets $A,B\in\Omega$, the coupling $\gamma(A\times B)$ describes how much mass originally in the set $A$ is transported into the set $B$. The squared of the Euclidean distance¹¹1More general cost functions might be used, but they are beyond the scope of this article. $\|x^{0}-x^{j}\|^{2}$ is interpreted as the cost of transporting a unit mass located at $x^{0}$ to $x^{j}$. Therefore, $C(\gamma;\mu^{0},\mu^{j})$ represents the total cost of moving $\mu^{0}$ to $\mu^{j}$ according to $\gamma$. Finally, we will denote the set of all plans that achieve the infimum in (1), which is non-empty [40], as $\Gamma^{*}(\mu^{0},\mu^{j})$.

Under certain conditions (e.g. when $\mu^{0}$ has continuous density), an optimal plan $\gamma$ can be induced by a rule/map $T$ that takes all the mass at each position $x$ to a unique point $T(x)$. If that is the case, we say that $\gamma$ does not split mass and that it is induced by a map T. In fact, it is concentrated on the graph of $T$ in the sense that for all measurable sets $A,B\subset\Omega$, $\,\gamma(A\times B)=\mu^{0}(\{x\in A:\,T(x)\in B\})$, and we will write it as the pushforward $\gamma=(\mathrm{id}\times T)_{\#}\mu^{0}$. Hence, (1) reads as

$$OT(\mu^{0},\mu^{j})=\int_{\Omega}\|x-T(x)\|^{2}d\mu^{0}(x)$$

(3)

The function $T:\Omega\to\Omega$ is called a Monge map, and when $\mu^{0}$ is absolutely continuous it is unique [7].

Finally, the square root of the optimal value $OT(\cdot,\cdot)$ is exactly the so-called Wasserstein distance, $W_{2}$, in $\mathcal{P}(\Omega)$ [40, Th.7.3], and we will call it also as OT squared distance. In addition, with this distance, $\mathcal{P}(\Omega)$ is not only a metric space but also a Riemannian manifold [40]. In particular, the tangent space of any $\mu\in\mathcal{P}(\Omega)$ is $\mathcal{T}_{\mu}=L^{2}(\Omega;\mathbb{R}^{d},\mu)=\{u:\Omega\to\mathbb{R}^{d}:\|u\|_{\mu}^{2}<\infty\}$, where

$$\|u\|_{\mu}^{2}:=\int_{\Omega}\|u(x)\|^{2}d\mu(x).$$

(4)

To understand the framework of Linear Optimal Transport (LOT) we will use the dynamic formulation of the OT problem. Optimal plans and maps can be viewed as a static way of matching two distributions. They tell us where each mass in the initial distribution should end, but they do not tell the full story of how the system evolves from initial to final configurations.

In the dynamic formulation, we consider $\rho\in\mathcal{P}([0,1]\times\Omega)$ a curve of measures parametrized in time that describes the distribution of mass $\rho_{t}:=\rho(t,\cdot)\in\mathcal{P}(\Omega)$ at each instant $0\leq t\leq 1$. We will require the curve to be sufficiently smooth, to have boundary conditions $\rho_{0}=\mu^{0}$, $\rho_{1}=\mu^{j}$, and to satisfy the conservation of mass law. Then, it is well known that there exists a velocity vector field $v_{t}:=v(t,\cdot)$ such that $\rho_{t}$ satisfies the continuity equation²²2The continuity equation is satisfied weakly or in the sense of distributions. See [40, 38]. with boundary conditions

$$\partial_{t}\rho+\nabla\cdot\rho v=0,\qquad\rho_{0}=\mu^{0},\quad\rho_{1}=\mu^{j}.$$

(5)

The length³³3Precisely, the length of the curve $\rho$, with respect to the Wasserstein distance, should be $\int_{0}^{1}\|v_{t}\|_{\rho_{t}}dt$, but this will make no difference in the solutions of (6) since they are constant speed geodesics. of the curve can be stated as $\int_{[0,1]\times\Omega}\|v\|^{2}d\rho:=\int_{0}^{1}\|v_{t}\|_{\rho_{t}}^{2}dt$, for $\|\cdot\|_{\rho_{t}}$ as in (4), and $OT(\mu^{0},\mu^{j})$ coincides with the length of the shortest curve between $\mu^{0}$ and $\mu^{j}$ [6]. Hence, the dynamical formulation of the OT problem (1) reads as

$\displaystyle OT(\mu^{0},\mu^{j})=\inf_{(\rho,v)\in\mathcal{CE}(\mu^{0},\mu^{j})}\int_{[0,1]\times\Omega}\|v\|^{2}d\rho,$

(6)

where $\mathcal{CE}(\mu^{0},\mu^{j})$ is the set of pairs $(\rho,v)$, where $\rho\in\mathcal{P}([0,1]\times\Omega)$, and $v:[0,1]\times\Omega\to\mathbb{R}^{d}$, satisfying (5).

Under the assumption of existence of an optimal Monge map $T$, an optimal solution for (6) can be given explicitly and is pretty intuitive. If a particle starts at position $x$ and finishes at position $T(x)$, then for $0<t<1$ it will be at the point

$$T_{t}(x):=(1-t)x+tT(x).$$

(7)

Then, varying both the time $t$ and $x\in\Omega$, the mapping (7) can be interpreted as a flow whose time velocity⁴⁴4For each $(t,x)\in(0,1)\times\Omega$, the vector $v_{t}(x)$ is well defined as $T_{t}$ is invertible. See [38, Lemma 5.29]. is

$$v_{t}(x)=T(x_{0})-x_{0},\qquad\text{ for }x=T_{t}(x_{0}).$$

(8)

To obtain the curve of probability measures $\rho_{t}$, one can evolve $\mu^{0}$ through the flow $T_{t}$ using the formula $\rho_{t}(A)=\mu_{0}(T_{t}^{-1}(A))$ for any measurable set $A$. That is, $\rho_{t}$ is the push-forward of $\mu^{0}$ by $T_{t}$

$$\rho_{t}=(T_{t})_{\#}\mu^{0},\qquad 0\leq t\leq 1.$$

(9)

The pair $(\rho,v)$ defined by (9) and (8) satisfies the continuity equation (5) and solves (6). Moreover, the curve $\rho_{t}$ is a constant speed geodesic in $\mathcal{P}(\Omega)$ between $\mu^{0}$ and $\mu^{j}$ [22], i.e., it satisfies that for all $0\leq s\leq t\leq 1$

$$\sqrt{OT(\rho_{s},\rho_{t})}=(t-s)\sqrt{OT(\rho_{0},\rho_{1})}.$$

(10)

A confirmation of this comes from comparing the OT cost (3) with (8) obtaining

$\displaystyle OT(\mu^{0},\mu^{j})$

$\displaystyle=\int_{\Omega}\|v_{0}(x)\|^{2}d\mu^{0}(x)$

(11)

which tells us that we only need the speed at the initial time to compute the total length of the curve. Moreover, $OT(\mu^{0},\mu^{j})$ coincides with the squared norm of the tangent vector $v_{0}$ in the tangent space $\mathcal{T}_{\mu^{0}}$ of $\mathcal{P}(\Omega)$ at $\mu^{0}$.

Inspired by the induced Riemannian geometry of the $OT$ squared distance, [41] proposed the so-called Linear Optimal Transportation (LOT) framework. Given two target measures $\mu^{i},\mu^{j}$, the main idea relies on considering a reference measure $\mu^{0}$ and embed these target measures into the tangent space $\mathcal{T}_{\mu^{0}}$. This is done by identifying each measure $\mu^{j}$ with the curve (9) minimizing $OT(\mu^{0},\mu^{j})$ and computing its velocity (tangent vector) at $t=0$ using (8).

Formally, let us fix a continuous probability reference measure $\mu^{0}$. Then, the LOT embedding [34] is defined as

$\displaystyle\mu^{j}\mapsto u^{j}:=T^{j}-\mathrm{id}\qquad\forall\ \mu^{j}\in\mathcal{P}(\Omega)$

(12)

where $T^{j}$ is the optimal Monge map between $\mu^{0}$ and $\mu^{j}$. Notice that by (3), (4) and (11) we have

$$\|u^{j}\|^{2}_{\mu^{0}}=OT(\mu^{0},\mu^{j}).$$

(13)

After this embedding, one can use the distance in $\mathcal{T}_{\mu^{0}}$ between the projected measures to define a new distance in $\mathcal{P}(\Omega)$ that can be used to approximate $OT(\mu^{i},\mu^{j})$. The LOT squared distance is defined as

$\displaystyle LOT_{\mu^{0}}(\mu^{i},\mu^{j}):=\|u^{i}-u^{j}\|_{\mu^{0}}^{2}.$

(14)

For discrete probability measures $\mu^{0},\mu^{j}$ of the form

$$\mu^{0}=\sum_{n=1}^{N_{0}}p^{0}_{n}\delta_{x^{0}_{n}},\qquad\mu^{j}=\sum_{n=1}^{N_{j}}p^{j}_{n}\delta_{x^{j}_{n}},$$

(15)

a Monge map $T^{j}$ for $OT(\mu^{0},\mu^{j})$ may not exist. Following [41], in this setting, the target measure $\mu^{j}$ can be replaced by a new measure $\hat{\mu}^{j}$ for which an optimal transport Monge map exists. For that, given an optimal plan $\gamma^{j}\in\Gamma^{*}(\mu^{0},\mu^{j})$, it can be viewed as a $N_{0}\times N_{i}$ matrix whose value at position $(n,m)$ represents how much mass from $x^{0}_{n}$ should be taken to $x^{j}_{m}$. Then, we define the OT barycentric projection⁵⁵5We refer to [2] for the rigorous definition. of $\mu^{j}$ with respect to $\mu^{0}$ as

$$\hat{\mu}^{j}:=\sum_{n=1}^{N_{0}}p^{0}_{n}\delta_{\hat{x}_{n}^{j}},\,\text{where}\quad\hat{x}_{n}^{j}:=\frac{1}{p_{n}^{0}}\sum_{m=1}^{N_{j}}\gamma^{j}_{n,m}x_{m}^{j}.$$

(16)

The new measure $\hat{\mu}^{j}$ is regarded as an $N_{0}$-point representation of the target measure $\mu^{j}$. The following lemma guarantees the existence of a Monge map between $\mu^{0}$ and $\hat{\mu}^{j}$.

It is easy to show that if the optimal transport plan $\gamma^{j}$ is induced by a Monge map, then $\hat{\mu}^{j}=\mu^{j}$. As a consequence, the OT barycentric projection is an actual projection in the sense that it is idempotent.

Similar to the continuous case (12), given a discrete reference measure $\mu^{0}$, we can define the LOT embedding for a discrete measure $\mu^{j}$ as the rule

$$\mu^{j}\mapsto u^{j}:=[(\hat{x}_{1}^{j}-x^{0}_{1}),\ldots,(\hat{x}_{N_{0}}^{j}-x^{0}_{N_{0}})].$$

(17)

The range $\mathcal{T}_{\mu_{0}}$ of this application is identified with $\mathbb{R}^{d\times N_{0}}$ with the norm $\|u\|_{\mu^{0}}:=\sum_{n=1}^{N_{0}}\|u(n)\|^{2}p_{n}^{0}$, where $u(n)\in\mathbb{R}^{d}$ denotes the $n$th entry of $u$. We call $(\mathbb{R}^{d\times N_{0}},\|\cdot\|_{\mu^{0}})$ the embedding space.

By the discussion above, if the optimal plan $\gamma^{j}$ for problem $\text{OT}(\mu^{0},\mu^{j})$ is induced by a Monge map, then the discrete embedding is consistent with (13) in the sense that

$\displaystyle\|u^{j}\|_{\mu^{0}}^{2}=OT(\mu^{0},\hat{\mu}^{j})=OT(\mu^{0},\mu^{j}).$

(18)

Hence, as in section 2.3, we can use the distance between embedded measures in ($\mathbb{R}^{d\times{N_{0}}},\|\cdot\|_{\mu_{0}})$ to define a discrepancy in the space of discrete probabilities that can be used to approximate $OT(\mu^{i},\mu^{j})$. The LOT discrepancy⁷⁷7In [41], LOT is defined by the infimum over all possible optimal pairs $(\gamma^{i},\gamma^{j})$. We do not distinguish these two formulations for convenience in this paper. Additionally, (19) is determined by the choice of $(\gamma^{i},\gamma^{j})$. is defined as

$\displaystyle LOT_{\mu^{0}}(\mu^{i},\mu^{j}):=\|u^{i}-u^{j}\|_{\mu^{0}}^{2}.$

(19)

We call it a discrepancy because it is not a squared metric between discrete measures. It does not necessarily satisfy that $LOT(\mu^{i},\mu^{j})\not=0$ for every distinct $\mu^{i},\mu^{j}$. Nevertheless, $\|u^{i}-u^{j}\|_{\mu^{0}}$ is a metric in the embedding space.

Let $\mu^{i}$, $\mu^{j}$ be discrete probability measures as in (15) (with ‘$i$’ in place of $0$). If an optimal Monge map $T$ for $OT(\mu^{i},\mu^{j})$ exists, a constant speed geodesic $\rho_{t}$ between $\mu^{i}$ and $\mu^{j}$, for the OT squared distance, can be found by mimicking (9). Explicitly, with $T_{t}$ as in (7),

$$\rho_{t}=(T_{t})_{\#}\mu^{i}=\sum_{n=1}^{N_{i}}p_{n}^{i}\delta_{(1-t)x_{n}^{i}+tT(x_{n}^{i})}.$$

(20)

In practice, one replaces $\mu^{j}$ by its OT barycentric projection with respect to $\mu^{i}$ (and so, the existence of an optimal Monge map is guaranteed by Lemma 2.1).

Now, given a discrete reference $\mu^{0}$, the LOT discrepancy provides a new structure to the space of discrete probability densities. Therefore, we can provide a substitute for the OT geodesic (20) between $\mu^{i}$ and $\mu^{j}$. Assume we have the embeddings $\mu^{i}\mapsto u^{i}$, $\mu^{j}\mapsto u^{j}$ as in (17). The geodesic between $u^{i}$ and $u^{j}$ in the LOT embedding space $\mathbb{R}^{d\times N_{0}}$ has the simple form $u_{t}=(1-t)u^{i}+tu^{j}$. This correlates with the curve $\hat{\rho}_{t}$ in $\mathcal{P}(\Omega)$ induced by the map $\hat{T}:\hat{x}_{n}^{i}\mapsto\hat{x}_{n}^{j}$ ⁸⁸8This map can be understood as the one that transports $\hat{\mu}^{i}$ onto $\hat{\mu}^{j}$ pivoting on the reference: $\hat{\mu}^{i}\mapsto\mu^{0}\mapsto\hat{\mu}^{j}$. as

$$\hat{\rho}_{t}:=(\hat{T}_{t})_{\#}\hat{\mu}^{i}=\sum_{n=1}^{N_{0}}p_{n}^{0}\delta_{{x}_{n}^{0}+u_{t}(n)}.$$

(21)

By abuse of notation, we call this curve the LOT geodesic between $\mu^{i}$ and $\mu^{j}$. Nevertheless, it is a geodesic between their barycentric projections since it satisfies the following.

In addition to mass transportation, the OPT problem allows mass destruction at the source and mass creation at the target. Let $\mathcal{M}_{+}(\Omega)$ denote the set of all positive finite Borel measures defined on $\Omega$. For $\lambda\geq 0$ the OPT problem between $\mu^{0},\mu^{j}\in\mathcal{M}_{+}(\Omega)$ can be formulated as

	$\displaystyle OPT_{\lambda}(\mu^{0},\mu^{j}):=\inf_{\gamma\in\Gamma_{\leq}(\mu^{0},\mu^{j})}C(\gamma;\mu^{0},\mu^{j},\lambda)$		(23)
	$\displaystyle\text{for}\quad C(\gamma;\mu^{0},\mu^{j},\lambda):=\int_{\Omega^{2}}\|x^{0}-x^{j}\|^{2}d\gamma(x^{0},x^{j})+\lambda(|\mu^{0}-\gamma_{0}|+|\mu^{j}-\gamma_{1}|)$		(24)

where $|\mu^{0}-\gamma_{0}|$ is the total mass of $\mu^{0}-\gamma_{0}$ (resp. $|\mu^{j}-\gamma_{1}|$), and $\Gamma_{\leq}(\mu^{0},\mu^{j})$ denotes the set of all measures in $\Omega^{2}$ with marginals $\gamma_{0}$ and $\gamma_{1}$ satisfying $\gamma_{0}\leq\mu^{0}$ (i.e., $\gamma_{0}(E)\leq\mu^{0}(E)$ for all measurable set $E$), and $\gamma_{1}\leq\mu^{j}$. Here, the mass destruction and creation penalty is linear, parametrized by $\lambda$. The set of minimizers $\Gamma_{\leq}^{*}(\mu^{0},\mu^{j})$ of (23) is non-empty [20]. One can further restrict $\Gamma_{\leq}(\mu^{0},\mu^{j})$ to the set of partial transport plans $\gamma$ such that $\|x^{0}-x^{j}\|^{2}<2\lambda$ for all $(x^{0},x^{j})\in\operatorname{supp}(\gamma)$ [4, Lemma 3.2]. This means that if the usual transportation cost is greater than $2\lambda$, it is better to create/destroy mass.

Adding a forcing term $\zeta$ to the continuity equation (6), one can take into account curves that allow creation and destruction of mass. That is, those who break the conservation of mass law. Thus, it is natural that the minimization problem (23) can be rewritten [12, Th. 5.2] into a dynamic formulation as

$$OPT_{\lambda}(\mu^{0},\mu^{j})=\inf_{(\rho,v,\zeta)\in\mathcal{FCE}(\mu^{0},\mu^{j})}\int_{[0,1]\times\Omega}\|v\|^{2}d\rho+\lambda|\zeta|$$

(25)

where $\mathcal{FCE}(\mu^{0},\mu^{j})$ is the set of tuples $(\rho,v,\zeta)$ such that $\rho\in\mathcal{M}_{+}([0,1]\times\Omega)$, $\zeta\in\mathcal{M}([0,1]\times\Omega)$ (where $\mathcal{M}$ stands for signed measures) and $v:[0,1]\times\Omega\to\mathbb{R}^{d}$, satisfying

$$\partial_{t}\rho+\nabla\cdot\rho v=\zeta,\qquad\rho_{0}=\mu^{0},\quad\rho_{1}=\mu^{j}.$$

(26)

As in the case of OT, under certain conditions on the minimizers $\gamma$ of (23), one curve $\rho_{t}$ that minimizes the dynamic formulation (25) is quite intuitive. We show in the next proposition that it consists of three parts $\gamma_{t}$, $(1-t)\nu_{0}$ and $t\nu^{j}$ (see (27), (28), and (29) below). The first is a curve that only transports mass, and the second and third destroy and create mass at constant rates $|\nu_{0}|$, $|\nu^{j}|$, respectively.

In analogy to the OT squared distance, we also call the optimal partial cost (32) as the OPT squared distance.

Let us compare the LOPT (33) and LOT (12) embeddings. The first component $v_{0}$ represents the tangent of the curve that transports mass from the reference to the target. This is exactly the same as the LOT embedding. In contrast to LOT, the second component $\gamma_{0}$ is necessary since we need to specify what part of the reference is being transported. The third component $\nu^{j}$ can be thought of as the tangent vector of the part that creates mass. There is no need to save the destroyed mass because it can be inferred from the other quantities.

Now, let $\mu^{0}\wedge\mu^{j}$ be the minimum measure⁹⁹9Formally, ${\mu}^{0}\wedge{\mu}^{j}(B):=\inf\left\{{\mu}^{0}\left(B_{1}\right)+{\mu}^{j}\left(B_{2}\right)\right\}$ for every Borel set $B$, where the infimum is taken over all partitions of $B$., i.e. $B=B_{1}\cup B_{2}$, $B_{1}\cap B_{2}=\emptyset$, given by Borel sets $B_{1}$, $B_{2}$. between $\mu^{0}$ and $\mu^{j}$. By the above definition, $\mu^{0}\mapsto(u^{0},\bar{\mu}^{0},\nu^{0})=(0,\mu^{0},0)$. Therefore, (32) can be rewritten

$\displaystyle OPT_{\lambda}(\mu^{0},\mu^{j})$

$\displaystyle=\|u^{0}-u^{j}\|^{2}_{\bar{\mu}^{0}\wedge\bar{\mu}^{j},2\lambda}+\lambda(|\bar{\mu}^{0}-\bar{\mu}^{j}|+|\nu_{0}|+|\nu^{j}|)$

(34)

This motivates the definition of the LOPT discrepancy.¹⁰¹⁰10$LOPT_{\lambda}$ is not a rigorous metric.

Similar to the LOT framework, by equation (34), LOPT can recover OPT when $\mu^{i}=\mu^{0}$. That is,

$$LOPT_{\mu^{0},\lambda}(\mu^{0},\mu^{j})=OPT_{\lambda}(\mu^{0},\mu^{j}).$$

If $\mu^{0},\mu^{j}$ are $N_{0},N_{j}-$size discrete non-negative measures as in (15) (but not necessarily with total mass 1), the OPT problem (23) can be written as

$$\min_{\gamma\in\Gamma_{\leq}(\mu^{0},\mu^{j})}\sum_{n,m}\|x_{n}^{0}-x_{m}^{j}\|^{2}\gamma_{n,m}+\lambda(|p^{0}|+|p^{j}|-2|\gamma|)$$

where the set $\Gamma_{\leq}(\mu^{0},\mu^{j})$ can be viewed as the subset of $N_{0}\times N_{j}$ matrices with non-negative entries

$$\Gamma_{\leq}(\mu^{0},\mu^{j}):=\{\gamma\in\mathbb{R}_{+}^{N_{0}\times{N_{j}}}:\gamma 1_{N_{j}}\leq p^{0},\gamma^{T}1_{N_{0}}\leq p^{j}\},$$

where $1_{N_{0}}$ denotes the $N_{0}\times 1$ vector whose entries are $1$ (resp. $1_{N_{j}}$), $p^{0}=[p_{1}^{0},\ldots,p_{N_{0}}^{0}]$ is the vector of weights of $\mu^{0}$ (resp. $p^{j}$), $\gamma 1_{N_{j}}\leq p^{0}$ means that component-wise holds the ‘$\leq$’ (resp. $\gamma^{T}1_{N_{0}}\leq p^{j}$, where $\gamma^{T}$ is the transpose of $\gamma$), and $|p^{0}|=\sum_{n=1}^{N_{0}}|p^{0}_{n}|$ is the total mass of $\mu^{0}$ (resp. $|p^{j}|,|\gamma|$). The marginals are $\gamma_{0}:=\gamma 1_{N_{j}}$, and $\gamma_{1}:=\gamma^{T}1_{N_{0}}$.

Similar to OT, when an optimal plan $\gamma^{j}$ for $OPT_{\lambda}(\mu^{0},\hat{\mu}^{j})$ is not induced by a map, we can replace the target measure $\mu^{j}$ by an OPT barycentric projection $\hat{\mu}^{j}$ for which a map exists. Therefore, allowing us to apply the LOPT embedding (see (33) and (40) below).

It is worth noting that when we take a barycentric projection of a measure, some information is lost. Specifically, the information about the part of $\mu^{j}$ that is not transported from the reference $\mu^{0}$. This has some minor consequences.

First, unlike (18), the optimal partial transport cost $OPT_{\lambda}(\mu^{0},\mu^{j})$ changes when we replace $\mu^{j}$ by $\hat{\mu}^{j}$. Nevertheless, the following relation holds.

The second consequence¹²¹²12This is indeed an advantage since it allows the range of the embedding to always have the same dimension $N_{0}\times(d+1)$. is that the LOPT embedding of $\hat{\mu}^{j}$ will always have a null third component. That is,

$$\hat{\mu}^{j}\mapsto([\hat{x}_{1}^{j}-x_{1}^{0},\ldots,\hat{x}_{N_{0}}^{j}-x_{N_{0}}^{0}],\,\sum_{n=1}^{N_{0}}\hat{p}_{n}^{j}\delta_{x_{n}^{0}},\,0).$$

(40)

Therefore, we represent this embedding as $\hat{\mu}^{j}\mapsto(u^{j},\hat{p}^{j})$, for $u^{j}=[\hat{x}_{1}^{j}-x_{1}^{0},\ldots,\hat{x}_{N_{0}}^{j}-x_{N_{0}}^{0}]$ and $\hat{p}^{j}=[\hat{p}^{j}_{1},\ldots,\hat{p}^{j}_{N_{0}}]$. The last consequence is given in the next result.

As a byproduct we can define the LOPT discrepancy for any pair of discrete measures $\mu^{i},\mu^{j}$ as the right-hand side of (41). In practice, unless to approximate $OPT_{\lambda}(\mu^{i},\mu^{j})$, we set $C_{i,j}=0$ in (41). That is,

$$LOPT_{\mu^{0},\lambda}(\mu^{i},\mu^{j}):=LOPT_{\mu^{0},\lambda}(\hat{\mu}^{i},\hat{\mu}^{j}).$$

(42)

Inspired by OT and LOT geodesics as defined in section 2.5, but lacking the Riemannian structure provided by the OT squared norm, we propose an OPT interpolation curve and its LOPT approximation.

For the OPT interpolation between two measures $\mu^{i}$, $\mu^{j}$ for which exists $\gamma\in\Gamma_{\leq}^{*}(\mu^{i},\mu^{j})$ of the form $\gamma=(\mathrm{id}\times T)_{\#}\gamma_{0}\,$, a natural candidate is the solution $\rho_{t}$ of the dynamic formulation of $OPT_{\lambda}(\mu^{i},\mu^{j})$. The exact expression is given by Proposition 3.1. When working with general discrete measures $\mu^{i}$, $\mu^{j}$ (as in (15), with ‘$i$’ in place of $0$) such $\gamma$ is not guaranteed to exist. Then, we replace the latter with its OPT barycentric projection with respect to $\mu^{i}$. And by Theorem 3.5 the map $T:x_{n}^{i}\mapsto\hat{x}_{n}^{j}$ solves $OPT_{\lambda}(\mu^{i},\hat{\mu}^{j})$ and the OPT interpolating curve is¹³¹³13$\hat{p}_{n}^{j}$ are the coefficients of $\hat{\mu}^{j}$ with respect to $\mu^{i}$ analogous to (36).

$$t\mapsto\sum_{n=1}^{N_{i}}\hat{p}_{n}^{j}\delta_{(1-t)x_{n}^{i}+tT(x_{n}^{i})}+(1-t)\sum_{n=1}^{N_{i}}(p_{n}^{i}-\hat{p}_{n}^{j})\delta_{x_{n}^{i}}.$$

When working with a multitude of measures, it is convenient to consider a reference $\mu^{0}$ and embed the measures in $\mathbb{R}^{(d+1)\times N_{0}}$ using LOPT. Hence, doing computations in a simpler space. Below we provide the LOPT interpolation.