Disentangled Representation Learning with the Gromov-Monge Gap

## Abstract

Abstract Learning disentangled representations from unlabelled data is a fundamental challenge in machine learning. Solving it may unlock other problems, such as generalization, interpretability, or fairness. Although remarkably challenging to solve in theory, disentanglement is often achieved in practice through prior matching. Furthermore, recent works have shown that prior matching approaches can be enhanced by leveraging geometrical considerations, e.g., by learning representations that preserve geometric features of the data, such as distances or angles between points. However, matching the prior while preserving geometric features is challenging, as a mapping that fully preserves these features while aligning the data distribution with the prior does not exist in general. To address these challenges, we introduce a novel approach to disentangled representation learning based on quadratic optimal transport. We formulate the problem using Gromov-Monge maps that transport one distribution onto another with minimal distortion of predefined geometric features, preserving them as much as can be achieved . To compute such maps, we propose the Gromov-Monge-Gap (GMG), a regularizer quantifying whether a map moves a reference distribution with minimal geometry distortion. We demonstrate the effectiveness of our approach
for disentanglement across four standard benchmarks, outperforming other methods leveraging geometric considerations.

## 1 Introduction

Learning low-dimensional representations of high-dimensional data is a fundamental challenge in unsupervised deep learning (Bengio et al., ). Emphasis is put on learning representations that allow for efficient and robust adaptation across a wide range of tasks (Higgins et al., , Locatello et al., ). The fundamental property of disentanglement has shown significant promise to improve generalization (Locatello et al., , Roth et al., , Hsu et al., , Barin-Pacela et al., ), interpretability and fairness (Locatello et al., , Träuble et al., ). Most works regard disentanglement as a one-to-one map between learned representations and ground-truth latent factors, effectively seeking to recover these factors from data alone in an unsupervised fashion.
While unsupervised disentanglement is theoretically impossible (Locatello et al., ), the inductive biases of autoencoder architectures ensure effective disentanglement in practice (Rolinek et al., , Zietlow et al., ). Most approaches operate using variational autoencoder (VAE) frameworks (Kingma and Welling, ), using objectives that match latent VAE posteriors to factorized priors (Higgins et al., , Kim and Mnih, , Kumar et al., , Burgess et al., , Chen et al., ).

More recently, studies such as Gropp et al. (), Chen et al. (), Lee et al. (), Horan et al. (), Nakagawa et al. (), Huh et al. (), Hahm et al. () have provided a new perspective, showing that geometric constraints on representation spaces may also enable disentanglement.
Typically, latent representations are encouraged to preserve key geometric features of the data distribution, such as (scaled) distances or angles between samples. Horan et al. () even demonstrate that unsupervised disentanglement is *always* possible provided that the latent space is locally isometric to the data, further supporting the geometric desiderata. However, combining prior matching with these geometric aspects is challenging. In general, a mapping that perfectly aligns the data distribution with the prior while *fully* preserving the geometric features of interest may not exist. This leads to an inherent trade-off: Practitioners must carefully fine-tune regularization terms, either by altering prior matching to prioritize geometry preservation, or vice-versa.

In this work, we demonstrate how to effectively combine geometric desiderata with prior matching within the VAE framework, using optimal transport (OT) theory (Santambrogio, , Peyré and Cuturi, ). By treating mappings from the data manifold to the latent space (encoders) or vice versa (decoders) as transport maps $T:\mathcal{X}\to\mathcal{Y}$, we can leverage the Gromov-{Monge, Wasserstein} paradigm (Sturm, , Mémoli, ), which aligns two distributions by finding a mapping that minimizes the distortion between intra-domain cost functions defined on their supports. Specifically, we consider cost functions $c_{\mathcal{X}}(\mathbf{x},\mathbf{x}^{\prime})$ on $\mathcal{X}$ and $c_{\mathcal{Y}}(\mathbf{y},\mathbf{y}^{\prime})$ on $\mathcal{Y}$ that encode geometric features such as scaled distances or angles. Consequently, the resulting mapping transforms one distribution onto the other while preserving these geometric features as much as possible.

Our Contribution: A novel OT-based approach to disentanglement through geometric considerations.

- (i)
We address the challenge of learning disentangled representations using geometric constraints by leveraging Gromov-Monge mappings between the data and prior distributions. Since *fully* preserving geometric features—such as (scaled) distances or angles between points—during the alignment of these two distributions is generally impossible, we aim to find an alignment that, instead, minimizes the distortion of these features, thereby preserving them *as much as possible*.
- (ii)
Inspired by (Uscidda and Cuturi, ), we introduce the *Gromov-Monge Gap* (GMG), a regularizer that measures how closely a map $T$ approximates a Gromov-Monge map for costs $c_{\mathcal{X}},c_{\mathcal{Y}}$. GMG measures whether $T$ transports distributions with minimal distortion w.r.t. $c_{\mathcal{X}},c_{\mathcal{Y}}$. We propose an efficient procedure to compute GMG and describe how to integrate it within the VAE framework.
- (iii)
We show that when $c_{\mathcal{X}}$ and $c_{\mathcal{Y}}$ encode scaled distances or angles, the GMG and its finite-sample counterpart are weakly convex functions. In both cases, we precisely characterize the weak convexity constants and analyze their practical implications for practitioners.
- (iv)
Across four standard disentangled representation learning benchmarks, we show that incorporating geometry-preserving desiderata via the GMG significantly enhances disentanglement across various methods—from the standard $\beta$-VAE to the combination of $\beta$-TCVAE with HFS (Roth et al., )—notably outperforming other methods that leverage geometric constraints.

## 2 Background and Related Works

### 2.1 Disentangled Representation Learning

#### The Disentanglement Formalism.

Disentanglement has varying operational definitions. In this work, we follow the common understanding (Higgins et al., , Locatello et al., ; , Träuble et al., , Roth et al., ) where data $\mathbf{x}$ is generated by a process $p(\mathbf{x}|\mathbf{z})$ operating on ground-truth latent factors $\mathbf{z}\sim p(\mathbf{z})$, modeling underlying source of variations (s.a. object shape, color, background…). Given a dataset $\mathcal{D}=\{\mathbf{x}_{i}\}_{i=1}^{N}$, $\mathbf{x}_{i}\sim p_{\textrm{data}}$, unsupervised disentangled representation learning aims to find a mapping $e_{\phi}$ s.t. $e_{\phi}(\mathbf{x}_{i})\approx\mathbb{E}[\mathbf{z}|\mathbf{x}_{i}]$, up to element-wise transformations. Notably, this is to be achieved without prior information on $p(\mathbf{z})$ and $p(\mathbf{x}|\mathbf{z})$.

#### Unsupervised Disentanglement through Prior Matching.

Most unsupervised disentanglement methods operate on variational autoencoders (VAEs, Kingma and Welling ()), which define a generative model of the form $p_{\theta}(\mathbf{x},\mathbf{z})=p(\mathbf{z})p_{\theta}(\mathbf{x}|\mathbf{z})$. Here, $p_{\theta}(\mathbf{x}|\mathbf{z})$ is a product of exponential family distributions with parameters computed by a decoder $d_{\theta}(\mathbf{z})$. The latent prior $p(\mathbf{z})$ is usually chosen as a standard Gaussian $\mathcal{N}(\mathbf{z}|\mathbf{0}_{d},\mathbf{I}_{d})$, and the probabilistic encoder $q_{\phi}(\mathbf{z}|\mathbf{x})$ is implemented through neural networks $e_{\phi}(\mathbf{x}),\sigma_{\phi}(\mathbf{x})$ that predicts the latent parameters so that $q_{\phi}(\mathbf{z}|\mathbf{x})=\mathcal{N}(\mathbf{z}|e_{\phi}(\mathbf{x}),
\sigma^{2}_{\phi}(\mathbf{x}))$. The $\beta$-VAE (Higgins et al., ) achieves disentanglement by minimizing

$$ $\min_{\theta,\phi}\mathbb{E}_{\mathbf{x}\sim p_{\textrm{data}},\mathbf{z}\sim q _{\phi}(\mathbf{z}|\mathbf{x})}[\underbrace{-\log p_{\theta}(\mathbf{x}| \mathbf{z})}_{\textrm{(i) reconstruction}}+\underbrace{\beta D_{\text{KL}}(q_{ \phi}(\mathbf{z}|\mathbf{x})||p(\mathbf{z}))}_{\textrm{(ii) prior matching}}],$ (1) $$

which enforces $\beta$-weighted prior matching on top of the reconstruction loss, assuming statistical factor independence (Roth et al., ). Several follow-ups refine latent prior matching through different losses or prior choices (Kim and Mnih, , Chen et al., , Kumar et al., , Burgess et al., , Rolinek et al., , Moor et al., , Balabin et al., ).

#### Disentanglement through a Geometric Lens.

Recent research suggests that disentanglement can arise by imposing geometric constraints on representation spaces (Horan et al., ). Most works (Karras et al., , Huh et al., ) promote latent representations to preserve scaled distances between data points. Closest to our work, Lee et al. () use a Jacobian-based regularization within a VAE, while Nakagawa et al. () employ a distortion-based regularization, directly comparing pairwise squared Euclidean distances, within a Wasserstein AE (WAE, Tolstikhin et al. ()).

### 2.2 Quadratic Optimal Transport

#### Gromov-{Monge, Wasserstein} Formulations.

OT (Peyré and Cuturi, ) involves transferring a probability distribution to another with inductive biases. When these distributions lie on incomparable domains, one relies on the Gromov-Monge (GM) and GW problems, also known as quadratic formulations of OT. Consider two compact $\mathcal{X}\subset\mathbb{R}^{d_{\mathcal{X}}}$, $\mathcal{Y}\subset\mathbb{R}^{d_{\mathcal{Y}}}$, each of them equipped with an intra-domain cost $c_{\mathcal{X}}:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$ and $c_{\mathcal{Y}}:\mathcal{Y}\times\mathcal{Y}\rightarrow\mathbb{R}$. For technical reasons, we assume that $c_{\mathcal{X}}$ and $c_{\mathcal{Y}}$ (or $-c_{\mathcal{X}}$ and $-c_{\mathcal{Y}}$) are CPD kernels (see Def. (A.1)). For $p\in\mathcal{P}(\mathcal{X})$ and $q\in\mathcal{P}(\mathcal{Y})$—two distributions supported on each domain—the GM problem (Mémoli and Needham, ) seeks a map $T:\mathcal{X}\to\mathcal{Y}$ that push-forwards $p$ onto $q$, while minimizing the distortion of the costs:

$$ $\inf_{T:T\sharp p=q}\int_{\mathcal{X}\times\mathcal{X}}\tfrac{1}{2}|c_{ \mathcal{X}}(\mathbf{x},\mathbf{x}^{\prime})-c_{\mathcal{Y}}(T(\mathbf{x}),T( \mathbf{x}^{\prime}))|^{2}\mathop{}\!\mathrm{d}p(\mathbf{x})\mathop{}\!\mathrm {d}p(\mathbf{x}^{\prime})\,.$ (GMP) $$

When it exists, we call a solution $T^{\star}$ to (GMP) a Gromov-Monge map for costs $c_{\mathcal{X}},c_{\mathcal{Y}}$.
However, this formulation is ill-suited for discrete distributions $p,q$, as the constraint set might be empty in that case. Replacing maps by coupling $\pi\in\Pi(p,q)$, i.e. distributions on $\mathcal{X}\times\mathcal{Y}$ with marginals $p$ and $q$, we obtain the GW problem (Mémoli, , Sturm, )

$$ $\mathrm{GW}(p,q):=\min_{\pi\in\Pi(p,q)}\int_{(\mathcal{X}\times\mathcal{Y})^{2 }}\tfrac{1}{2}|c_{\mathcal{X}}(\mathbf{x},\mathbf{x}^{\prime})-c_{\mathcal{Y}} (\mathbf{y},\mathbf{y}^{\prime})|^{2}\mathop{}\!\mathrm{d}\pi(\mathbf{x}, \mathbf{y})\mathop{}\!\mathrm{d}\pi(\mathbf{x}^{\prime},\mathbf{y}^{\prime})\,.$ (GWP) $$

A solution $\pi^{\star}$ to (GWP) always exists, making $\mathrm{GW}(p,q)$ a well-defined quantity. It quantifies the minimal distortion of the geometries induced by $c_{\mathcal{X}}$ and $c_{\mathcal{Y}}$ achievable when coupling $p$ and $q$.

#### Discrete Solvers.

When both $p$ and $q$ are instantiated as samples, GW Prob. (GWP) translates to a quadratic assignment problem, whose objective can be regularized using entropy (Cuturi, , Peyré et al., ). For empirical measures ${p}_{n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{\mathbf{x}_{i}}$, ${q}_{n}=\frac{1}{n}\sum_{j=1}^{n}\delta_{\mathbf{y}_{j}}$ and $\varepsilon\geq 0$, we set:

$$ $\mathrm{GW}_{\varepsilon}({p}_{n},{q}_{n})\vcentcolon=\min_{\mathbf{P}\in U_{n }}\sum_{i,j,i^{\prime},j^{\prime}=1}^{n}(\mathbf{C}_{\mathcal{X}_{i,i^{\prime} }}-\mathbf{C}_{\mathcal{Y}_{j,j^{\prime}}})^{2}\,\mathbf{P}_{i,j}\mathbf{P}_{i ^{\prime},j^{\prime}}-\varepsilon H(\mathbf{P})\,,$ (EGWP) $$

with $\mathbf{C}_{\mathcal{X}}=[c_{\mathcal{X}}(\mathbf{x}_{i},\mathbf{x}_{i^{\prime
}})]_{i,i^{\prime}}$, $\mathbf{C}_{\mathcal{Y}}=[c_{\mathcal{Y}}(\mathbf{y}_{j},\mathbf{y}_{j^{\prime
}})]_{j,j^{\prime}}\in\mathbb{R}^{n\times n}$, $U_{n}=\{\mathbf{P}\in\mathbb{R}^{n\times n}_{+},\mathbf{P}\mathbf{1}_{n}=
\mathbf{P}^{T}\mathbf{1}_{n}=\tfrac{1}{n}\mathbf{1}_{n}\}$ and $H(\mathbf{P})=-\sum_{i,j=1}^{n}\mathbf{P}_{i,j}\log(\mathbf{P}_{i,j})$. As $\varepsilon\to 0$, we recover $\mathrm{GW}^{c_{\mathcal{X}},c_{\mathcal{Y}}}_{0}=\mathrm{GW}^{c_{\mathcal{X}}
,c_{\mathcal{Y}}}$. In addition to yielding greater convexity (Rioux et al., ) and better statistical (Zhang et al., ) and properties, entropic regularization improves computational performance. We can solve (EGWP) using a scheme that iterates the Sinkhorn algorithm (Peyré et al., ).
This solver has $\mathcal{O}(n^{2})$ memory complexity. Its time complexity depends on $c_{\mathcal{X}},c_{\mathcal{Y}}$. For general $c_{\mathcal{X}},c_{\mathcal{Y}}$, it runs in $\mathcal{O}(n^{3})$. However, for the most common practical choices of $c_{\mathcal{X}}=c_{\mathcal{Y}}=\langle\cdot,\cdot\rangle$ or $c_{\mathcal{X}}=c_{\mathcal{Y}}=\|\cdot-\cdot\|_{2}^{2}$, it can also be reduced to $\mathcal{O}(n^{2}(d_{\mathcal{X}}+d_{\mathcal{Y}}))$, as detailed by Scetbon et al. (, §3 & Alg. 2).

#### Neural GW.

While for classical OT, numerous neural methods have been proposed (Makkuva et al., , Korotin et al., , Tong et al., , Eyring et al., ), the GW setting has received less attention. Carrasco et al. () propose an approach for solving GM with inner product costs, that involves a min-max-min optimization procedure. Klein et al. () propose a framework to computeentropic GW couplings. Yet, these formulations are end-to-end, and not based on a regularizer that can be flexibly added to a loss function. Therefore, they cannot be combined with other models, as we do in this work with the $\beta$-VAE. In a concurrent work, Sotiropoulou and Alvarez-Melis () introduced the Gromov-Monge gap. However, (i) they do not apply it for disentangled representational learning, and (ii) they do not investigate its properties, as we do in this work.

## 3 Disentanglement with the Gromov-Monge gap

This section outlines our novel OT-based approach to disentanglement through geometric considerations within the VAE framework. First, we explore how an arbitrary map $T:\mathcal{X}\to\mathcal{Y}$ between two domains can be encouraged to preserve predefined geometric features. In a VAE, $T$ may represent either the encoder $e_{\phi}$, which generates latent codes from the data, or the decoder $d_{\theta}$, which reconstructs the data from these codes. In the case of the encoder, the source domain $\mathcal{X}$ corresponds to the data, and the target domain $\mathcal{Y}$ is the latent space, while for the decoder, these roles are swapped. Assuming that $d_{\theta}$ perfectly reconstructs the data from the latents produced by $e_{\phi}$, i.e., $e_{\phi}\circ d_{\theta}=\mathrm{I}d$, the preservation of geometric features by either $e_{\phi}$ or $d_{\theta}$ becomes equivalent. Therefore, in the following sections, we refer to $T$ as either the encoder or the decoder without loss of generality.

#### Outline.

Building on this perspective, this section begins by defining cost functions to encode geometric features and the notion of distortions in §3.1. We leverage this concept in §3.2 to introduce the Gromov-Monge Gap (GMG), a regularizer that measures whether a map moves distributions while preserving geometric features as much as possible, i.e., minimizing distortion.
§3.3 then shows how the GMG can be efficiently computed from samples to be practically applicable in the VAE framework. This transitions into §3.4 studying (weak) convexity properties of the GMG, as an operator. Finally, in §3.5, we describe how to integrate the GMG with disentangled representation learning objectives, effectively combining prior matching with geometric constraints.

### 3.1 From the distortion…

We encode the geometric features of interest using two cost functions, one for each domain: $c_{\mathcal{X}}:\mathcal{X}\times\mathcal{X}\to\mathbb{R}$ and $c_{\mathcal{Y}}:\mathcal{Y}\times\mathcal{Y}\to\mathbb{R}$. We aim for $T$ to preserve the geometric feature, which means that $T$ preserves these costs, that is, $c_{\mathcal{X}}(\mathbf{x},\mathbf{x^{\prime}})\approx c_{\mathcal{Y}}(T(
\mathbf{x}),T(\mathbf{x^{\prime}}))$ for $\mathbf{x},\mathbf{x^{\prime}}\in\mathcal{X}$. In practice, two types of cost function from this family are particularly relevant:

- [i]
(Scaled) squared L2 distance: $c_{\mathcal{X}}(\mathbf{x},\mathbf{x}^{\prime})=\|\mathbf{x}-\mathbf{x}^{
\prime}\|_{2}^{2}$ and $c_{\mathcal{Y}}(\mathbf{y},\mathbf{y}^{\prime})=\alpha^{2}\|\mathbf{y}-\mathbf
{y}\|_{2}^{2}$, with $\alpha>0$. A map $T$ preserving $c_{\mathcal{X}},c_{\mathcal{Y}}$ preserves the scaled distances between the points, i.e. it is a scaled isometry. When $\alpha=1$, we recover the standard definition of an isometry.
- [ii]
Cosine-Similarity: $c_{\mathcal{X}}(\mathbf{x},\mathbf{x}^{\prime})=\textrm{cos-sim}(\mathbf{x},
\mathbf{x}^{\prime})\vcentcolon=\langle\tfrac{\mathbf{x}}{\|\mathbf{x}\|_{2}},
\tfrac{\mathbf{x}^{\prime}}{\|\mathbf{x}^{\prime}\|_{2}}\rangle$ and $c_{\mathcal{Y}}(\mathbf{y},\mathbf{y}^{\prime})=\textrm{cos-sim}(\mathbf{y},
\mathbf{y}^{\prime})$ similarly. On has $\textrm{cos-sim}(\mathbf{x},\mathbf{x}^{\prime})=\cos(\theta_{\mathbf{x},
\mathbf{x}^{\prime}})$ where $\theta_{\mathbf{x},\mathbf{x}^{\prime}}$ is the angle between $\mathbf{x}$ and $\mathbf{x}^{\prime}$. A map $T$ preserving $c_{\mathcal{X}},c_{\mathcal{Y}}$ then preserves the angles between the points, i.e. it is a conformal map. Note that if $T$ is (scaled) isometry (see above), it is a conformal map.

We refer to these costs via $\mathbf{L2^{2}}$ for [i] with $\alpha=1$, $\mathbf{ScL2^{2}}$ for [i] with $\alpha\neq 1$ and $\mathbf{Cos}$ for [ii].
Introducing a reference distribution $r\in\mathcal{P}(\mathcal{X})$, weighting the areas of $\mathcal{X}$ where we penalize deviations of $c_{\mathcal{X}}(\mathbf{x},\mathbf{x}^{\prime})$ from $c_{\mathcal{Y}}(T(\mathbf{x}),T(\mathbf{x}^{\prime}))$, we can quantify this property using the following criterion:

###### Definition 3.1 (Distortion) .

The distortion (DST) of a map $T$, for cost functions $c_{\mathcal{X}},c_{\mathcal{Y}}$ and reference distribution $r$, is defined as:

$$ $\displaystyle\mathrm{DST}_{r}(T):=\int_{\mathcal{X}\times\mathcal{X}}\tfrac{1} {2}(c_{\mathcal{X}}(\mathbf{x},\mathbf{x}^{\prime})-c_{\mathcal{Y}}(T(\mathbf{ x}),T(\mathbf{x}^{\prime})))^{2}\mathop{}\!\mathrm{d}r(\mathbf{x})\mathop{}\! \mathrm{d}r(\mathbf{x}^{\prime})\,.$ ( DST ) $$

$\mathrm{DST}_{r}(T)$ quantifies how much $T$ distorts geometric features encoded by $c_{\mathcal{X}},c_{\mathcal{Y}}$ on the support of $r$, that is, when $\mathrm{DST}_{r}(T)=0$, one has $c_{\mathcal{X}}(\mathbf{x},\mathbf{x}^{\prime})=c_{\mathcal{Y}}(T(\mathbf{x}),
T(\mathbf{x}^{\prime}))$ for $\mathbf{x},\mathbf{x}^{\prime}\in\mathrm{Spt}(r)$. Nakagawa et al. () suggest promoting geometry preservation by regularizing the encoder $d_{\theta}$ using the DST, with $\mathbf{ScL2^{2}}$ as costs, and the latent representation as reference distribution, namely $r=e_{\phi}\sharp p_{\textrm{data}}$. While they use it within a WAE, translated in our VAE setting, this corresponds to minimizing

$$ $\min_{\theta,\phi}\mathbb{E}_{\mathbf{x}\sim p_{\textrm{data}},\mathbf{z}\sim q _{\phi}(\mathbf{z}|\mathbf{x})}[\underbrace{-\log p_{\theta}(\mathbf{x}| \mathbf{z})}_{\textrm{(i) reconstruction}}+\underbrace{\beta D_{\text{KL}}(q_{ \phi}(\mathbf{z}|\mathbf{x})||p(\mathbf{z}))}_{\textrm{(ii) prior matching}}]+ \underbrace{\lambda\,\mathrm{DST}_{r}(d_{\theta})}_{\textrm{(iii) geom. preservation}},$ (2) $$

where $\lambda>0$. Given the choice of the costs, $d_{\theta}$ is distortion-free (i.e., $\mathrm{DST}_{r}(d_{\theta})=0$), if it is a scaled isometry. Since a scaled isometry that maps the prior onto the data distribution may not exist, there is an inherent trade-off between minimizing terms (ii) and (iii), which are responsible for achieving practical disentanglement.
As these terms cannot be simultaneously minimized to 0 0, the DST loss will move away from accurately matching the prior, which will negatively impact the quality of the learned latent representations. This naturally raises the question of how to avoid this over-penalization. Instead of seeking a distortion-free decoder, we should seek a decoder that transports the prior to the data distribution with minimal distortion of the costs $c_{\mathcal{X}},c_{\mathcal{Y}}$. In other words, the decoder should be a Gromov-Monge map between the prior and the data distributions for costs $c_{\mathcal{X}},c_{\mathcal{Y}}$ (see 2.2). Conversely, if we choose to regularize the encoder, the same reasoning applies by swapping the roles of the prior and the data distribution. In the next section, we introduce a regularizer to fit Gromov-Monge maps, which we will use as a replacement to the DST.

###### Definition 3.1 (Distortion) .

### 3.2 …to The Gromov-Monge Gap

Figure: Figure 1: Sketch of $\mathrm{GMG}_{\boldsymbol{{\color[rgb]{0.1328125,0.546875,0.1328125} \definecolor[named]{pgfstrokecolor}{rgb}{0.1328125,0.546875,0.1328125}r_{n}}}} (\boldsymbol{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1 }T}})$ for two different maps $\boldsymbol{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1} T}}$. We use a discrete reference distribution $\boldsymbol{{\color[rgb]{0.1328125,0.546875,0.1328125}\definecolor[named]{ pgfstrokecolor}{rgb}{0.1328125,0.546875,0.1328125}r_{n}}}$ on 3 points, and $c_{\mathcal{X}}=c_{\mathcal{Y}}=\|\cdot-\cdot\|_{2}$, hence measuring if $\boldsymbol{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1} T}}$ minimally distorts the distances. On the left, $\boldsymbol{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1} T}}$ is the optimal map, $\boldsymbol{{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{ 1,.5,0}T^{\star}}}$, and maps the three points with minimal (yet non-zero) distortion, which is measured as the sum of the squared lengths of the orange segments. This results in $\mathrm{GMG}_{\boldsymbol{{\color[rgb]{0.1328125,0.546875,0.1328125} \definecolor[named]{pgfstrokecolor}{rgb}{0.1328125,0.546875,0.1328125}r_{n}}}} (\boldsymbol{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1 }T}})=0$. On the right, $\boldsymbol{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1} T}}$ swaps two points compared to $\boldsymbol{{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{ 1,.5,0}T^{\star}}}$—specifically, $\boldsymbol{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0} \mathbf{x}_{2}}}$ and $\boldsymbol{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0} \mathbf{x}_{3}}}$—causing a higher distortion than the minimal one, and measured as the sum of the squared lengths of the red segments. This results in $\mathrm{GMG}_{\boldsymbol{{\color[rgb]{0.1328125,0.546875,0.1328125} \definecolor[named]{pgfstrokecolor}{rgb}{0.1328125,0.546875,0.1328125}r_{n}}}} (\boldsymbol{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1 }T}})>0$, equal to the gap between these distortions.
Refer to caption: x1.png

Recently, Uscidda and Cuturi () introduced the Monge gap, a regularizer that measures whether a map $T$ transports a reference distribution at the minimal displacement cost. Building on this concept, we replace "displacement" with "distortion" to introduce the Gromov-Monge gap, a regularizer that assesses whether a map $T$ transports a reference distribution at the minimal distortion cost.

###### Definition 3.2 (Gromov-Monge gap) .

The Gromov-Monge gap (GMG) of a map $T$, for cost functions $c_{\mathcal{X}},c_{\mathcal{Y}}$ and reference distribution $r$, is defined as:

$$ $\displaystyle\mathrm{GMG}_{r}(T)\vcentcolon=\mathrm{DST}_{r}(T)-\mathrm{GW}(r, T\sharp r)$ ( GMG ) $$

We recall from Eq. (GWP) that $\mathrm{GW}(r,T\sharp r)$ represents the minimal distortion of $c_{\mathcal{X}},c_{\mathcal{Y}}$ achievable when transporting $r$ to $T\sharp r$. Thus, the GMG quantifies the difference between the distortion incurred when transporting $r$ to $T\sharp r$ via $T$, and this minimal distortion. Formally, when Prob. (GMP) and Prob. (GWP) between $r$ and $T\sharp r$ are equivalent, the the GMG is the suboptimality gap of $T$ in Prob. (GMP). This is the case, for example, when $r$ is a density and $c_{\mathcal{X}}=c_{\mathcal{Y}}=\langle\cdot,\cdot\rangle$ (Dumont et al., ). Otherwise, the GMG is the suboptimality gap of $\pi=(\mathrm{I}_{d},T)\sharp r$ in Prob. (GWP) between $r$ and $T\sharp r$. In light of this, it is a well-defined quantity and:

- •
The GMG measures how close $T$ is to be a Gromov-Monge map for costs $c_{\mathcal{X}},c_{\mathcal{Y}}$. Indeed, $\mathrm{GMG}_{r}(T)\geq 0$ with equality if $T$ is a Gromov-Monge map solution of Prob. (GMP) between $r$ and $T\sharp r$, i.e., $T$ moves $r$ with minimal (but eventually non zero) distortion; see App. B.1.
- •
When transport without distortion is possible, the GMG coincides with the distortion. When there exists another map $U:\mathcal{X}\to\mathcal{Y}$ transporting $r$ to $T\sharp r$ with zero distortion, i.e., $U\sharp r=T\sharp r$ and $\mathrm{DST}_{r}(U)=0$, then $\mathrm{GMG}_{r}(T)=\mathrm{DST}_{r}(T)$. Indeed, $\mathrm{GW}(r,T\sharp r)=0$ in that case, as the coupling $\pi=(\mathrm{Id},U)\sharp r$ sets the GW objective to zero, thereby minimizing it.

The last point illustrates how the GMG functions as a debiased distortion. It compares the distortion induced by $T$ to a baseline distortion, defined as the minimal achievable distortion when transforming $r$ into $T\sharp r$. Thus, when transformation without distortion is achievable, this baseline becomes zero, and the GMG equals the distortion. Consequently, the GMG offers the optimal compromise: it avoids the over-penalization induced by the distortion when fully preserving $c_{\mathcal{X}},c_{\mathcal{Y}}$ is not feasible, yet it coincides with it when such full preservation is feasible. See Fig. 1 for a simple illustration.

###### Definition 3.2 (Gromov-Monge gap) .

#### The Influence of the Reference Distribution.

A crucial property of $\mathrm{DST}_{r}$ is that if $T$ transforms $r$ without distortion, it will also apply distortion-free to any distribution $s$ whose support is contained within that of $r$. Formally, if $\mathrm{DST}_{r}(T)=0$ and $s\in\mathcal{P}(\mathcal{X})$ with $\mathrm{Spt}(s)\subseteq\mathrm{Spt}(r)$, then $\mathrm{DST}_{s}(T)=0$. This raises a question for the GMG: If $T$ maps $r$ with minimal distortion, does it similarly map $s$ with minimal distortion? We answer this question positively with Prop. (3.3).

###### Proposition 3.3 .

If $\mathrm{GMG}_{r}(T)=0$, $\forall s\in\mathcal{P}(\mathcal{X})$ s.t. $\mathrm{Spt}(s)\subseteq\mathrm{Spt}(r)$, one has $\mathrm{GMG}_{s}(T)=0$.

###### Proposition 3.3 .

### 3.3 Estimation and Computation from Samples

Plug-In Estimation. In practice, we estimate Eq. (DST) and Eq. (GMG) from samples $\mathbf{x}_{1},...,\mathbf{x}_{n}\sim r$. We consider the empirical version ${r}_{n}\vcentcolon=\tfrac{1}{n}\sum_{i=1}^{n}\delta_{\mathbf{x}_{i}}$ of $r$ and use plug-in estimators, i.e.

$$ $\begin{split}\mathrm{DST}_{{r}_{n}}(T)=\tfrac{1}{n^{2}}\sum_{i,j=1}^{n}(c_{ \mathcal{X}}(\mathbf{x}_{i},\mathbf{x}_{j})-c_{\mathcal{Y}}(T(\mathbf{x}_{i}), T(\mathbf{x}_{j})))^{2}\,,\end{split}$ (3) $$

and $\mathrm{GMG}_{{r}_{n}}(T)=\mathrm{DST}_{{r}_{n}}(T)-\mathrm{GW}({r}_{n},T
\sharp{r}_{n})$, where $T\sharp{r}_{n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{T(\mathbf{x}_{i})}$.

Figure: Algorithm 1 GMG$(\mathbf{x}_{1},\dots\mathbf{x}_{n},T,\varepsilon)$.

#### Efficient and Stable Computation.

Computing the GMG requires solving a discrete GW problem between $r_{n}$ and $T\sharp r_{n}$ to get $\mathrm{GW}({r}_{n},T\sharp{r}_{n})$. We compute this term using an entropic regularization $\varepsilon\geq 0$, as in Eq. (EGWP):

$$ $\displaystyle\mathrm{GMG}_{{r}_{n},\varepsilon}(T)\vcentcolon=\mathrm{DST}_{{r }_{n}}(T)$ (4) $\displaystyle\,\,-\mathrm{GW}_{\varepsilon}({r}_{n},T\sharp{r}_{n})\,.$ $$

Choosing $\varepsilon=0$, we recover $\mathrm{GMG}_{{r}_{n},0}(T)=\mathrm{GMG}_{{r}_{n}}(T)$. Moreover, the entropic estimator preserves positivity, as for $\varepsilon\geq 0$, $\mathrm{GMG}_{r_{n},\varepsilon}(T)\geq 0$ (see B.3). We compute $\mathrm{GW}_{\varepsilon}({r}_{n},T\sharp{r}_{n})$ using Peyré et al. ()’s solver introduced in 2.2. We use the implementation provided by ott-jax (Cuturi et al., ).
In practice, we select $\varepsilon$ based on (positive) statistics from the cost matrices $\mathbf{C}_{\mathcal{X}}$,$\mathbf{C}_{\mathcal{Y}}$. We define a scale $\varepsilon_{0}$ and set $\varepsilon=\varepsilon_{0}\cdot\texttt{stat}(\mathbf{C}_{\mathcal{X}})\cdot
\texttt{stat}(\mathbf{C}_{\mathcal{Y}})$. Standard options for the statistic include $\texttt{stat}\in\{\texttt{mean},\texttt{max},\texttt{std}\}$. This procedure is equivalent to running the entropic GW solver on the re-scaled cost matrices $\mathbf{C}_{\mathcal{X}}/\texttt{stat}(\mathbf{C}_{\mathcal{X}})$ and $\mathbf{C}_{\mathcal{Y}}/\texttt{stat}(\mathbf{C}_{\mathcal{Y}})$ with $\varepsilon=\varepsilon_{0}$; see App. B.2. It is a common practical trick, initially suggested for stabilizing the Sinkhorn algorithm Cuturi (), then extended to the entropic GW solver (Cuturi et al., ). Additionally, as suggested by Scetbon et al. (), to promote the solver’s stability, we always initialize it with $\mathbf{P}_{\textrm{init}}=\tfrac{1}{n^{2}}\mathbf{1}_{n}\mathbf{1}_{n}^{\top}$.

#### Computational Complexity.

For usual costs, including inner products, $\ell_{p}^{q}$ distances, and standard CPD kernels, the computation of $c_{\mathcal{X}}(\mathbf{x},\mathbf{x}^{\prime})$ (resp. $c_{\mathcal{Y}}(\mathbf{y},\mathbf{y}^{\prime})$) can be done in $\mathcal{O}(d_{\mathcal{X}})$ time (resp. $\mathcal{O}(d_{\mathcal{Y}})$). Consequently, the DST can be computed in $\mathcal{O}(n^{2}(d_{\mathcal{X}}+d_{\mathcal{Y}}))$ time. Furthermore, as discussed in § 2.2, the time complexity of the entropic GW solver is $\mathcal{O}(n^{3})$ in general, but can be reduced to $\mathcal{O}(n^{2}(d_{\mathcal{X}}+d_{\mathcal{Y}}))$ when $c_{\mathcal{X}}=c_{\mathcal{Y}}=\langle\cdot,\cdot\rangle$, or $c_{\mathcal{X}}=\|\cdot-\cdot\|_{2}^{2},c_{\mathcal{Y}}=\alpha\|\cdot-\cdot\|_
{2}^{2}$. Therefore, since the cosine similarity is equivalent to the inner product, up to pre-normalization of $\mathbf{x}_{i}$
and $T(\mathbf{x}_{i})$, this solver runs in $\mathcal{O}(n^{2}(d_{\mathcal{X}}+d_{\mathcal{Y}}))$ for the costs of interest $\mathbf{(Sc)L2^{2}}$ and $\mathbf{Cos}$. The complete algorithm, along with a time complexity analysis of each step, is described in Alg. 1.

### 3.4 (Weak) Convexity of the Gromov-Monge gap

As laid out, the GMG can be used as a regularization loss to push any model $T$ to move distributions with minimal distortion. A natural question arises: is this regularizer convex? In the following, we study the convexity of $T\mapsto\mathrm{GMG}_{r}(T)$ and its finite-sample counterpart $T\mapsto\mathrm{GMG}_{{r}_{n}}(T)$. We focus on the costs $\mathbf{L2}$ and $\mathbf{Cos}$. For simplicity, we replace $\mathbf{Cos}$ with $\langle\cdot,\cdot\rangle$, as these costs are equivalent, up to normalization of $r$ and $T$. We respectively denote by $\mathrm{GMG}_{r}^{2}$ and $\mathrm{GMG}_{r}^{\langle\cdot,\cdot\rangle}$ the GMG for these costs.
We start by introducing a weaker notion of convexity, previously defined on $\mathbb{R}^{d}$ (Davis et al., ), which we extend here to $L_{2}(r)\vcentcolon=\{T\,|\,\|T\|_{L_{2}(r)}^{2}\vcentcolon=\int_{\mathcal{X}}
\|T(\mathbf{x})\|_{2}^{2}\mathop{}\!\mathrm{d}r(\mathbf{x})<+\infty\}$.

###### Definition 3.4 (Weak convexity.) .

With $\gamma>0$, a functional $\mathcal{F}:L_{2}(r)\to\mathbb{R}$ is $\gamma$-weakly convex if $\mathcal{F}_{\gamma}:T\mapsto\mathcal{F}(T)+\tfrac{\gamma}{2}\|T\|_{L_{2}(r)}^
{2}$ is convex.

A weakly convex functional is convex up to an additive quadratic perturbation. The weak convexity constant $\gamma$ quantifies the magnitude of this perturbation and indicates a degree of non-convexity of $\mathcal{F}$. A lower $\gamma$ suggests that $\mathcal{F}$ is closer to being convex, while a higher $\gamma$ indicates greater non-convexity.

###### Theorem 3.5 .

Both $\mathrm{GMG}_{r}^{2}$ and $\mathrm{GMG}_{r}^{\langle\cdot,\cdot\rangle}$, as well as their finite sample versions, are weakly convex.

- •
Finite sample. We note $\mathbf{X}\in\mathbb{R}^{n\times d}$ the matrix that stores the $\mathbf{x}_{i}$, i.e. the support of $r_{n}$, as rows. Then, (i) $\mathrm{GMG}_{r_{n}}^{2}$ and (ii) $\mathrm{GMG}_{r_{n}}^{\langle\cdot,\cdot\rangle}$ are respectively (i) $\gamma_{2,n}$ and (ii) $\gamma_{\textrm{inner},n}$-weakly convex, where: $\gamma_{\textrm{inner},n}=\lambda_{\max}(\tfrac{1}{n}\mathbf{X}\mathbf{X}^{
\top})-\lambda_{\min}(\tfrac{1}{n}\mathbf{X}\mathbf{X}^{\top})$ and $\gamma_{2,n}=\gamma_{\textrm{inner},n}+\max_{i=1\dots n}\|\mathbf{x}_{i}\|_{2}
^{2}$.
- •
Asymptotic. (i) $\mathrm{GMG}_{r}^{2}$ and (ii) $\mathrm{GMG}_{r}^{\langle\cdot,\cdot\rangle}$ are respectively (i) $\gamma_{2}$ and (ii) $\gamma_{\textrm{inner}}$-weakly convex, where: $\gamma_{\textrm{inner}}=\lambda_{\max}(\mathbb{E}_{\mathbf{x}\sim r}[\mathbf{x
}\mathbf{x}^{\top}])$ and $\gamma_{2,n}=\gamma_{\textrm{inner}}+\max_{\mathbf{x}\in\mathrm{Spt}(r)}\|
\mathbf{x}\|_{2}^{2}$.

From a practitioner’s perspective, we analyze the insights provided by Thm. (3.5) in three parts.

- •
First, we have $\gamma_{2}\geq\gamma_{\textrm{inner}}$. Therefore, $\mathrm{GMG}_{r}^{2}$ is less convex than $\mathrm{GMG}_{r}^{\langle\cdot,\cdot\rangle}$, making it harder to optimize, and the same argument holds for their estimator. In other words, we provably recover that, in practice, preserving the (scaled) distances is harder than simply preserving the angles.
- •
Second, as $\gamma_{\textrm{inner}}=\lambda_{\max}(\mathbb{E}_{\mathbf{x}\sim r}[\mathbf{x
}\mathbf{x}^{\top}])\geq\lambda_{\max}(\mathrm{Cov}_{\mathbf{x}\sim r}[\mathbf
{x}])$, this exhibits a tradeoff w.r.t. Prop. (3.3): by choosing a bigger reference distribution $r$, we trade the convexity of the GMG. For $\gamma_{2}$, the dependency in $r$ is even worse. In practice, we then choose $r$ with support as small as possible, precisely where we want $T$ to move points with minimal distortion.
- •
Third, and most surprising, the finite sample GMG is more convex in high dimension. Indeed, $\gamma_{\textrm{inner},n}$ is the spectral width of $\tfrac{1}{n}\mathbf{X}\mathbf{X}^{\top}$, which contains the (rescaled) inner products between $\mathbf{x}_{i}\sim r$.
When $n>d$, $\lambda_{\min}(\mathbf{X}\mathbf{X}^{\top})=0$ as $\mathrm{rank}(\mathbf{X}\mathbf{X}^{\top})=d$. Then, $\gamma_{\textrm{inner},n}$ increases, which in turn decreases the GMG’s convexity. However, when $d>n$, $\lambda_{\min}(\mathbf{X}\mathbf{X}^{\top})>0$ if $\mathbf{X}$ is full rank. Intuitively, $\mathrm{GMG}_{r_{n}}^{\langle\cdot,\cdot\rangle}$ is nearly convex when $\mathbf{X}\mathbf{X}^{\top}$ is well conditioned. Assuming that the $\mathbf{x}_{i}$ are normalized, this might happen in high dimension, as they will be orthogonal with high probability. This suggests that, contrary to the insights provided by the statistical OT literature (Weed and Bach, , Zhang et al., ), the GMG might not benefit a large sample size.

###### Definition 3.4 (Weak convexity.) .

###### Theorem 3.5 .

### 3.5 Learning with the Gromov-Monge gap

#### General Learning Procedure.

Given a source $p$ and a target distribution $q$, we can use the GMG to guide a parameterized map $T_{\theta}$ towards approximating a Gromov-Monge map between $p$ and $q$. We handle the marginal constraint $T_{\theta}\sharp p=q$ separately through a fitting loss $\Delta(T_{\theta},p,q)$. Provided any reference $r$ s.t. $\mathrm{Spt}(p)\subset\mathrm{Spt}(r)$, we minimize

$$ $\min_{\theta}\Delta(T_{\theta},p,q)+\lambda\mathrm{GMG}_{r}(T_{\theta})$ (5) $$

$\Delta$ can operate on paired (e.g., in VAE, the reconstruction loss), or unpaired (e.g., in VAE, the KL loss in VAE), samples of $p$ and $q$.
Note that, in theory and as stated in Prop. (3.3), we can select any reference $r$ such that $\mathrm{Spt}(p)\subset\mathrm{Spt}(r)$. However, based on the insights from Thm. (3.4), we typically choose $r$ with minimal support size and, in practice, set $r=p$. This learning procedure is illustrated in Fig. 2, where we also explore the effect of replacing $\mathrm{GMG}_{r}(T_{\theta})$ by $\mathrm{DST}_{r}(T_{\theta})$ in Eq. (5).

#### VAE Learning Procedure.

In the VAE setting, we can use the GMG promote the (i) encoder $e_{\phi}$ or the (ii) decoder $d_{\theta}$ to mimic a Gromov-Monge map. In (i) we use $r_{e}=p_{\textrm{data}}$ the data distribution as reference $r$, while in (ii) we use the latent distribution $r_{d}=e_{\phi}\sharp p_{\textrm{data}}$. Introducing weightings $\lambda_{e},\lambda_{d}\geq 0$, determining which mapping we regularize, this remains to minimize

$$ $\min_{\theta,\phi}\mathbb{E}_{\mathbf{x}\sim p_{\textrm{data}},\mathbf{z}\sim q _{\phi}(\mathbf{z}|\mathbf{x})}[\underbrace{-\log p_{\theta}(\mathbf{x}| \mathbf{z})}_{\textrm{(i) reconstruction}}+\underbrace{\beta D_{\text{KL}}(q_{ \phi}(\mathbf{z}|\mathbf{x})||p(\mathbf{z}))}_{\textrm{(ii) prior matching}}]+ \underbrace{\lambda_{e}\mathrm{GMG}_{r_{e}}(e_{\phi})+\lambda_{d}\mathrm{GMG}_ {r_{d}}(d_{\theta})}_{\textrm{(iii) geom. preservation}},$ (6) $$

With this loss, prior matching and geometric desiderata can be efficiently combined, as terms (ii) and (iii) can simultaneously be 0. Note that this loss can be easily extended to more advanced prior matching objectives, such as the $\beta$-TCVAE loss (Chen et al., ), and can be combined with other regularizers, such as the HFS (Roth et al., ). We explore this strategy in experiments § 4.2.

Figure: Figure 2: Learning of geometry-preserving maps with the DST and the GMG. Provided a mixture of Gaussian source distribution $p$ and a uniform or circular target distribution $q$, we minimize a fitting loss together with a geometry-preserving regularization. Each line correspond to a different data setup and cost. As costs on the top line, we use $\mathbf{L2^{2}}=\|\cdot-\cdot\|_{2}^{2}$, to preserve distances between points while on the bottom line, we use $\mathbf{Cos}=\textrm{cos-sim}(\cdot,\cdot)$, to preserve angles.
Refer to caption: x2.png

## 4 Experiments

### 4.1 GMG vs. DST: Illustrative Example

In Fig. 2, we start by illustrating the difference between the DST and GMG on synthetic data. We train $T_{\theta}$ to map from a $3$D source $p$ to a $2$D target distribution $q$ (first and second column) by minimizing $\mathcal{L}(\theta)\vcentcolon=\mathrm{S}_{\varepsilon}(T_{\theta}\sharp p,q)+
\mathrm{R}(T_{\theta})$, where $\mathrm{S}_{\varepsilon}$ is the Sinkhorn divergence (Feydy et al., ) and $\mathrm{R}$ a geometry-preserving regularizer. We compare three settings, [i] no regularization $\mathrm{R}=0$, [ii] $\mathrm{R}=\mathrm{DST}_{p}$ as regularizer , and [iii] $\mathrm{R}=\mathrm{GMG}_{p}$. For all cases, we plot the transported distribution $T_{\theta}\sharp p$ after training. Without regularization (third column), we fit the marginal constraint $T_{\theta}\sharp p=q$ but do not preserve the geometric features. With the DST (fourth column), we preserve geometric features but do not fit the marginal constraint, i.e., $T_{\theta}\sharp p\neq q$. On the other hand, with the GMG (fifth column), we get the best compromise by approximating a Gromov-Monge map: we fit $T_{\theta}\sharp p=q$, while preserving the geometric features as fully as possible.

### 4.2 Leveraging the Gromov-Monge Gap for Disentanglement

Experimental Setup.
Having demonstrated how the DST enables (i) fitting a marginal constraint while (ii) preserving geometric features as much as possible, we now apply it to disentangled representation learning.
Our primary goal is to investigate whether the GMG results in enhanced disentanglement compared to the DST by efficiently combining (i) prior matching with (ii) geometric constraints on the representation space. Moreover, we aim to determine which costs $c_{\mathcal{X}},c_{\mathcal{Y}}$ to choose in the geometric regularization and what part of the pipeline should be regularized, the encoder $e_{\phi}$, or the decoder $d_{\theta}$. The experimental setup is described below.

- •
Baselines. We use the standard $\beta$-VAE and $\beta$-TCVAE as our starting models, with the option to apply the recent HFS regularization  (Roth et al., ) to each, resulting in a total of four base configurations. Note that the latter does not leverage geometric constraints; it is only used to enhance prior matching. We then investigate the effect of various geometry-preserving regularizations on disentanglement on top of these four base configurations. We consider GMG, DST and the Jacobian-based (Jac) regularization (Lee et al., ) discussed in 2.1. Given the inclusion of the DST, we naturally consider Nakagawa et al. () as a baseline.
- •
Metrics. We evaluate the learned representations using the DCI-D (Eastwood and Williams, ) as it was found that it is the metric most suitable for measuring the disentanglement (Locatello et al., , Dittadi et al., ). We report mean and standard deviation over 5 seeds.
- •
Datasets. We benchmark over four standard datasets: Shapes3D (Kim and Mnih, ), DSprites Higgins et al. (), SmallNORB LeCun et al. (), and Cars3D Reed et al. ().
- •
Hyperparameters. To ensure a fair experimental comparison, we follow recent works (Locatello et al., ; , Roth et al., ) by using the same architecture and hyperparameters. We perform a similar small grid search for the weighting terms: $\beta$ for the KL loss and $\gamma$ for HFS. Additionally, we include the weighting $\lambda$ for the geometry-preserving regularizer in the grid search. Note that we search over the same loss weightings $\lambda$ for DST, GMG, and Jac. For all experiments with the GMG, we compute it with Alg. 1, and systematically use $\varepsilon_{0}=0.1$ and $\texttt{stat}=\texttt{mean}$. As a result, $\varepsilon$ is not included in the grid search. See Appendix C for full details.

**Table 1: Effect of different regularization on disentanglement (DCI-D on Shapes3D). We highlight the best method per regularization type ($\mathbf{L2^{2}}$, $\mathbf{ScL2^{2}}$, or $\mathbf{Cos}$), and the best/second best per column.**
|  | $\beta$-VAE | $\beta$-TCVAE | $\beta$-VAE + HFS | $\beta$-TCVAE + HFS |
| --- | --- | --- | --- | --- |
| Base | $65.8$ $\pm 15.6$ | $75.0$ $\pm 3.4$ | $88.1$ $\pm 7.4$ | $90.2$ $\pm 7.5$ |
| $\mathbf{L2^{2}}:c_{\mathcal{X}}=c_{\mathcal{Y}}=\|\cdot-\cdot\|_{2}^{2}$ |  |  |  |  |
| + Enc-DST | $59.6$ $\pm 6.9$ | $75.7$ $\pm 3.0$ | $88.7$ $\pm 7.1$ | $90.3$ $\pm 7.9$ |
| + Enc-GMG | $62.3$ $\pm 8.4$ | $75.4$ $\pm 5.3$ | $88.4$ $\pm 7.7$ | $90.1$ $\pm 4.3$ |
| + Dec-DST | $71.5$ $\pm 3.6$ | $75.8$ $\pm 6.6$ | $92.1$ $\pm 9.7$ | $90.9$ $\pm 7.6$ |
| + Dec-GMG | $72.0$ $\pm 8.5$ | $78.9$ $\pm 5.0$ | $92.5$ $\pm 4.4$ | $91.7$ $\pm 6.0$ |
| $\mathbf{ScL2^{2}}:c_{\mathcal{X}}=\|\cdot-\cdot\|_{2}^{2},c_{\mathcal{Y}}=<br>\alpha\|\cdot-\cdot\|_{2}^{2}$, $\alpha>0$ learnable |  |  |  |  |
| + Jac | $61.4$ $\pm 12.8$ | $76.7$ $\pm 4.5$ | $90.5$ $\pm 3.8$ | $91.5$ $\pm 5.6$ |
| + Enc-DST | $65.8$ $\pm 11.9$ | $73.0$ $\pm 7.9$ | $92.4$ $\pm 3.7$ | $89.2$ $\pm 3.8$ |
| + Enc-GMG | $65.1$ $\pm 5.5$ | $76.1$ $\pm 7.7$ | $90.8$ $\pm 9.2$ | $92.0$ $\pm 5.3$ |
| + Dec-DST | $67.4$ $\pm 7.1$ | $77.9$ $\pm 4.5$ | $93.2$ $\pm 9.7$ | $94.5$ $\pm 6.9$ |
| + Dec-GMG | 70.0 $\pm 5.9$ | 81.0 $\pm 3.2$ | 93.3 $\pm 8.6$ | 96.1 $\pm 3.8$ |
| $\mathbf{Cos}:c_{\mathcal{X}}=c_{\mathcal{Y}}=\textrm{cos-sim}(\cdot,\cdot)$ |  |  |  |  |
| + Enc-DST | $69.2$ $\pm 9.1$ | $77.2$ $\pm 7.5$ | $87.7$ $\pm 7.7$ | $90.5$ $\pm 5.9$ |
| + Enc-GMG | $70.9$ $\pm 9.5$ | $79.6$ $\pm 6.6$ | $92.5$ $\pm 5.9$ | $93.5$ $\pm 6.9$ |
| + Dec-DST | 76.8 $\pm 4.1$ | 81.3 $\pm 4.7$ | $87.5$ $\pm 3.3$ | $91.9$ $\pm 9.4$ |
| + Dec-GMG | 82.1 $\pm 4.5$ | 83.7 $\pm 8.8$ | 95.7 $\pm 5.8$ | 96.9 $\pm 4.9$ |

Which costs $c_{\mathcal{X}},c_{\mathcal{Y}}$ should we choose?  The first question that naturally arises when using a geometry-preserving regularizer is: Which geometric features should be preserved? Previous works (Nakagawa et al., , Lee et al., , Huh et al., ) focused on preserving scaled distances between points, with the scale being learnable. We follow and extend this approach by also investigating plain distances and angles. This leads to three choices for $c_{\mathcal{X}},c_{\mathcal{Y}}$ ($\mathbf{L2^{2}},\mathbf{ScL2^{2}},\mathbf{Cos}$), as introduced in §3.1, following the hierarchy of geometry-preserving mappings proposed in Lee et al. (). We benchmark these on Shapes3D across various settings including DST, GMG, and Jac. Table 1 shows that angle preservation ($\mathbf{Cos}$), previously unconsidered for disentangled representation learning, consistently outperforms (scaled) distance preservation. This result is intuitive, as preserving angles imposes a weaker constraint, allowing for greater latent space expressiveness. In practice, preserving scaled distances seems to overly restrict the expressiveness of the latent space.

Should we regularize the encoder $e_{\phi}$, or the decoder $d_{\theta}$?  The next question we aim to answer is whether the decoder or encoder should be regularized. Therefore, we follow the previous setup on Shapes3D and benchmark all geometry-preserving regularizers on $e_{\phi}$ and $d_{\theta}$ as reported in Table 1. We find that regularizing the decoder is beneficial over regularizing the encoder. We hypothesize this is due to the regularization of $d_{\theta}$ offering a stronger signal as its gradients impact both $\phi$ and $\theta$, as in this case, the reference $r=e_{\phi}\sharp p_{\textrm{data}}$ is the distribution of encoded images. Our findings align with prior works (Lee et al., , Nakagawa et al., ), which focus on regularizing $d_{\theta}$ yet do not offer this type of analysis. Additionally, we find that the GMG consistently outperforms the DST over all costs. Overall, the GMG on $d_{\theta}$ with $\mathbf{Cos}$ achieves best DCI-D results over all baselines. Consequently, moving forward we regularize the decoder for angle preservation.

**Table 2: Effect of GMG and DST with $\mathbf{Cos}$ as costs on disentanglement, as measured by DCI-D, over three datasets. We highlight the best, and second best result for each dataset and method.**
| With $\mathbf{Cos}$ costs | $\beta$-VAE | $\beta$-TCVAE | $\beta$-VAE + HFS | $\beta$-TCVAE + HFS |
| --- | --- | --- | --- | --- |
| DSprites Higgins et al. () |  |  |  |  |
| Base | $26.2$ $\pm 18.5$ | ${32.3}$ $\pm 19.3$ | $33.6$ $\pm 17.9$ | $48.7$ $\pm 10.2$ |
| + Dec-DST | $\underline{28.6}$ $\pm 19.3$ | $32.4$ $\pm 8.5$ | $39.3$ $\pm 18.1$ | $49.0$ $\pm 11.2$ |
| + Dec-GMG | 39.5 $\pm 15.2$ | 42.2 $\pm 3.6$ | 46.7 $\pm 2.0$ | 50.1 $\pm 8.5$ |
| SmallNORB LeCun et al. () |  |  |  |  |
| Base | $26.8$ $\pm 0.2$ | $29.8$ $\pm 0.4$ | $26.8$ $\pm 0.2$ | $29.8$ $\pm 0.4$ |
| + Dec-DST | $28.2$ $\pm 0.3$ | 29.9 $\pm 0.4$ | $28.2$ $\pm 0.3$ | 29.9 $\pm 0.4$ |
| + Dec-GMG | 28.3 $\pm 0.6$ | 29.9 $\pm 0.5$ | 28.3 $\pm 0.6$ | 29.9 $\pm 0.5$ |
| Cars3D Reed et al. () |  |  |  |  |
| Base | $29.6$ $\pm 5.7$ | $32.3$ $\pm 4.6$ | $29.6$ $\pm 5.7$ | $32.3$ $\pm 4.6$ |
| + Dec-DST | $26.8$ $\pm 3.6$ | $33.7$ $\pm 4.2$ | $26.8$ $\pm 3.6$ | $33.7$ $\pm 4.2$ |
| + Dec-GMG | 30.1 $\pm 5.6$ | 36.4 $\pm 5.7$ | 30.1 $\pm 5.6$ | 36.4 $\pm 5.7$ |

GMG Consistently Enhances Disentanglement.  To further validate our findings, we benchmark the GMG for decoder regularization with angle preservation ($\mathbf{Cos}$) against its distortion counterpart across three more datasets. We report full results in Table 2. Again we observe that the GMG outperforms or performs equally well to its distortion equivalent, confirming the benefits of accounting for the optimal possible mapping in the regularization. Note that for SmallNORB and Cars3D, we found no benefits with respect to DCI-D in adding an HFS regularization and obtained the best results without it. We emphasize that using the GMG with $\mathbf{Cos}$ regularization significantly improves results for all datasets versus not using any isometric regularization. This establishes the GMG as an effective tool for enhanced disentanglement.

Figure: Figure 3: Stability analysis for the DST and the GMG applied to the decoder $d_{\theta}$ with cost $\mathbf{Cos}$. For each regularizer, we assess stability by measuring the alignment of its gradients across 5 batches. We compute the cosine similarity between all pairs of gradients. Values closer to 1 indicates better alignment, hence better stability.
Refer to caption: x3.png

Stability of the GMG. Furthermore, we investigate the stability of the GMG compared to the DST. Computing the GMG using Algorithm 1 involves solving an optimization problem through a GW solver. This experiment aims to demonstrate that our proposed method for solving the GW problem leads to a stable loss in the GMG . We assess the stability by measuring the alignment of GMG gradients $\nabla_{\theta}\mathrm{GMG}_{r_{n}}(T_{\theta})$ across 5 randomly sampled batches from each dataset $r_{n}\sim\mathcal{D}$, for a fixed neural map $T_{\theta}$. We repeat this procedure for each of the four datasets $\mathcal{D}$ considered and apply the same methodology to the DST. Figure 4 presents results for our best-performing configuration, which regularizes the decoder (i.e., $T_{\theta}=d_{\theta}$) and employs the $\mathbf{Cos}$ cost. Results for encoder regularization (i.e., $T_{\theta}=e_{\phi}$) are reported in Appendix D.1. We observe that the GMG’s gradients exhibit significantly higher alignment compared to those of the DST, demonstrating greater stability of our proposed regularizer. This suggests that the GMG ’s formulation, which accounts for the the minimal distortion, effectively mitigates the inherent variability inherent of the distortion, leading to more stable gradient computations.

## 5 Conclusion

In this work, we introduce an OT perspective on unsupervised disentangled representation learning to incorporate general latent geometrical constraints. We derive the GMG, a provably weakly convex regularizer that measures whether a map $T$ transports a fixed reference distribution with minimal distortion of some predefined geometric features. By formulating disentangled representation learning as a transport problem, we integrate the GMG into standard training objectives, allowing for incorporating and studying various geometric constraints on the learned representation spaces. We show significant performance benefits of our approach on four standard disentanglement benchmarks.

## 6 Reproduciblity

In this work, we introduce the GMG, computed as detailed in Algorithm 1. To facilitate reproducibility, we provide the implementation code for computing the GMG on source and target batches in Appendix E. Comprehensive proofs, including all underlying assumptions, are presented in Appendix B. For our experiments on disentanglement benchmarks, we adhere to standard practices, employing streamlined preprocessing across all datasets. Detailed descriptions of these procedures are available in Appendix C. All experiments described in this paper can be conducted using a single RTX 2080TI GPU, ensuring accessibility and replicability of our results.

## Acknowledgements

Co-funded by the European Union (ERC, DeepCell - 101054957). Views and opinions expressed are, however, those of the author(s) only and do not necessarily reflect those of the European Union
or the European Research Council. Neither the European Union nor the granting authority can be
held responsible for them. Luca Eyring and Karsten Roth thank the European Laboratory for Learning and Intelligent Systems (ELLIS) PhD program for support. Karsten Roth also thanks the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for support.
Zeynep Akata was supported by BMBF FKZ: 01IS18039A, by the ERC (853489 - DEXIM), by EXC number 2064/1 – project number 390727645. Fabian J. Theis consults for Immunai Inc., Singularity Bio B.V., CytoReason Ltd,
Cellarity, and has ownership interest in Dermagnostix GmbH and Cellarity.

## Appendix

The Appendix is organized as follows:

- •
Section A provides additional background information to supplement the main text.
- •
Section B presents all theoretical proofs, including detailed assumptions.
- •
Section C outlines comprehensive experimental details, ensuring reproducibility.
- •
Section D offers supplementary empirical results that further support our findings.
- •
Section E includes the implementation code for computing the GMG.

## Appendix A Additional Background

### A.1 Reminders on Monge and Kantorovich OT

In this section, we recall the Monge and Kantorovich formulations of OT, which we will use to prove various results. These are the classical formulations of OT. Although we introduce them here after discussing the Gromov-Monge and Gromov-Wasserstein formulations, it should be noted that they are generally introduced beforehand. Indeed, the Gromov-Monge and Gromov-Wasserstein formulations were historically developed to derive OT formulations for comparing measures supported on incomparable spaces.

#### Monge Formulation.

Instead of intra-domain cost functions, we consider here an inter-domain continuous cost function $c:\mathcal{X}\times\mathcal{Y}\to\mathbb{R}$. This assumes that we have a meaningful way to compare elements $\mathbf{x},\mathbf{y}$ from the source and target domains. The Monge () problem (MP) between $p\in\mathcal{P}(\mathcal{X})$ and $p\in\mathcal{P}(\mathcal{Y})$ consists of finding a map $T:\mathcal{X}\to\mathcal{Y}$ that push-forwards $p$ onto $p$, while minimizing the average displacement cost quantified by $c$

$$ $\inf_{T:T\sharp p=p}\int_{\mathcal{X}}c(\mathbf{x},T(\mathbf{x}))\mathop{}\! \mathrm{d}p(\mathbf{x})\,.$ (MP) $$

We call any solution $T^{\star}$ to this problem a Monge map between $p$ and $q$ for cost $c$.
Similarly to the Gromov-Monge Problem (GMP), solving the Monge Problem (MP) is difficult, as the constraint set is not convex and might be empty, especially when $p,q$ are discrete.

#### Kantorovich Formulation.

Instead of transport maps, the problem (KP) seeks a couplings $\pi\in\Pi(p,q)$:

$$ $\mathrm{W}(p,q):=\min_{\pi\in\Pi(p,q)}\int_{\mathcal{X}\times\mathcal{Y}}c( \mathbf{x},\mathbf{y})\mathop{}\!\mathrm{d}\pi(\mathbf{x},\mathbf{y})\,.$ (KP) $$

An optimal coupling $\pi^{\star}$ solution of (KP), always exists. Studying the equivalence between (MP) and (KP) is easier than in the Gromov-Monge and Gromov-Wasserstein cases. Indeed,
when (MP) is feasible, the Monge and Kantorovich formulations coincide and $\pi^{\star}=(\mathrm{Id},T^{\star})\sharp p$.

### A.2 Conditionally Positive Kernels

In this section, we recall the definition of a conditionally positive kernel, which is involved in multiple proofs relying on the linearization of the Gromov-Wasserstein problem as a Kantorovich problem.

###### Definition A.1 .

A kernel $k:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ is CPD, i.e., conditionally positive, if it is symmetric and for any $\mathbf{x}_{1},...,\mathbf{x}_{n}\in\mathbb{R}^{d}$ and $\mathbf{a}\in\mathbb{R}^{n}$ s.t. $\mathbf{a}^{\top}\mathbf{1}_{n}=0$, one has

$$ $\sum_{i,j=1}^{n}\mathbf{a}_{i}\mathbf{a}_{j}\,k(\mathbf{x}_{i},\mathbf{x}_{j})\geq 0$ $$

CPD include all positive kernels, such as the inner-product $k(\mathbf{x},\mathbf{x}^{\prime})=\langle\mathbf{x},\mathbf{x}^{\prime}\rangle$, or the cosine similarity $k(\mathbf{x},\mathbf{x}^{\prime})=\textrm{cos-sim}(\mathbf{x},\mathbf{x}^{
\prime})=\langle\tfrac{\mathbf{x}}{\|\mathbf{x}\|_{2}},\tfrac{\mathbf{x}^{
\prime}}{\|\mathbf{x}^{\prime}\|_{2}}\rangle$, but also the (scaled) negative squared Euclidean distance $k(\mathbf{x},\mathbf{x}^{\prime})=-\alpha\|\mathbf{x}-\mathbf{x}^{\prime}\|_{2
}^{2}$, $\alpha>0$. Therefore, each of the costs of interest is either a conditionally positive kernel - for the inner product and the cosine distance - or its opposite is - for the squared Euclidean distance. Additionally, CPD kernels also include more exotic cost functions, such as:

- •
The RBF kernel $k(\mathbf{x},\mathbf{x}^{\prime})=\exp(-\|\mathbf{x}-\mathbf{y}\|_{2}^{2}/
\gamma),\gamma>0$.
- •
The power kernel $k(\mathbf{x},\mathbf{x}^{\prime})=-\|\mathbf{x}-\mathbf{x}^{\prime}\|_{2}^{p},
0<p<2$.
- •
The thin plate spline kernel $k(\mathbf{x},\mathbf{x}^{\prime})=\|\mathbf{x}-\mathbf{x}^{\prime}\|_{2}^{2}
\log(\|\mathbf{x}-\mathbf{x}^{\prime})$
- •
The inverse multi-quadratic kernel $k(\mathbf{x},\mathbf{x}^{\prime})=1/\sqrt{\|\mathbf{x}-\mathbf{x}^{\prime}\|_{
2}^{2}+c^{2}},c\in\mathbb{R}$.

As a result, the family of CPD kernels includes a large variety of cost functions, which can be used to define various GMGs.

###### Definition A.1 .

## Appendix B Proofs

### B.1 The GMG characterizes Gromov-Monge Optimality

We show here that if $\mathrm{GMG}_{r}(T)=0$, then $T$ is a Gromov-Monge map between $r$ and $T\sharp r$ for costs $c_{\mathcal{X}},c_{\mathcal{Y}}$. As the set of deterministic couplings $\{\pi_{F}\vcentcolon=(\mathrm{I}_{d},F)\sharp r|F:\mathcal{X}\to\mathcal{Y}\,,
F\sharp r=T\sharp r\}\subset\Pi(r,T\sharp r\}$, we immediately get that

$$ $\inf_{F\sharp r=T\sharp r}\int_{\mathcal{X}\times\mathcal{X}}(c_{\mathcal{X}}( \mathbf{x},\mathbf{x}^{\prime})-c_{\mathcal{Y}}(F(\mathbf{x}),F(\mathbf{x}^{ \prime})))^{2}\mathop{}\!\mathrm{d}r(\mathbf{x})\mathop{}\!\mathrm{d}r(\mathbf {x}^{\prime})\geq\mathrm{GW}(r,T\sharp r)$ (7) $$

On the other hand, if $\mathrm{GMG}_{r}(T)=0$, one has

$$ $\mathrm{GW}(r,T\sharp r)=\int_{\mathcal{X}\times\mathcal{X}}(c_{\mathcal{X}}( \mathbf{x},\mathbf{x}^{\prime})-c_{\mathcal{Y}}(T(\mathbf{x}),T(\mathbf{x}^{ \prime})))^{2}\mathop{}\!\mathrm{d}r(\mathbf{x})\mathop{}\!\mathrm{d}r(\mathbf {x}^{\prime})$ (8) $$

Therefore, combining Eq. (7) and Eq. (8), we get

$$ $\begin{split}&\inf_{F\sharp r=T\sharp r}\int_{\mathcal{X}\times\mathcal{X}}(c_ {\mathcal{X}}(\mathbf{x},\mathbf{x}^{\prime})-c_{\mathcal{Y}}(F(\mathbf{x}),F( \mathbf{x}^{\prime})))^{2}\mathop{}\!\mathrm{d}r(\mathbf{x})\mathop{}\!\mathrm {d}r(\mathbf{x}^{\prime})\\ =&\int_{\mathcal{X}\times\mathcal{X}}(c_{\mathcal{X}}(\mathbf{x},\mathbf{x}^{ \prime})-c_{\mathcal{Y}}(T(\mathbf{x}),T(\mathbf{x}^{\prime})))^{2}\mathop{}\! \mathrm{d}r(\mathbf{x})\mathop{}\!\mathrm{d}r(\mathbf{x}^{\prime})\end{split}$ (9) $$

Finally, as $T$ naturally satisfies the marginal constraint, we conclude that $T$ is a Gromov-Monge map between $r$ and $T\sharp r$ for costs $c_{\mathcal{X}},c_{\mathcal{Y}}$.

### B.2 On rescaling the costs matrices in the entropic GW solver

We remind, from Eq. (EGWP), that

$$ $\mathrm{GW}_{\varepsilon}({p}_{n},{q}_{n})=\min_{\mathbf{P}\in U_{n}}\sum_{i,j ,i^{\prime},j^{\prime}=1}^{n}(\mathbf{C}_{\mathcal{X}_{i,i^{\prime}}}-\mathbf{ C}_{\mathcal{Y}_{j,j^{\prime}}})^{2}\,\mathbf{P}_{i,j}\mathbf{P}_{i^{\prime},j ^{\prime}}-\varepsilon H(\mathbf{P})\,.$ (10) $$

By developping each terms, and using the fact that $\mathbf{P}\in U_{n}$, we get

$$ $\begin{split}\mathrm{GW}_{\varepsilon}({p}_{n},{q}_{n})&=\min_{\mathbf{P}\in U _{n}}\tfrac{1}{n^{2}}\langle\mathbf{C}_{\mathcal{X}}^{\odot 2}\mathbf{1}_{n}, \mathbf{1}_{n}\rangle+\tfrac{1}{n^{2}}\langle\mathbf{C}_{\mathcal{Y}}^{\odot 2 }\mathbf{1}_{n},\mathbf{1}_{n}\rangle-2\langle\mathbf{C}_{\mathcal{X}}\mathbf{ P}\mathbf{C}_{\mathcal{Y}},\mathbf{P}\rangle-\varepsilon H(\mathbf{P})\\ &=\tfrac{1}{n^{2}}\langle\mathbf{C}_{\mathcal{X}}^{\odot 2}\mathbf{1}_{n}, \mathbf{1}_{n}\rangle+\tfrac{1}{n^{2}}\langle\mathbf{C}_{\mathcal{Y}}^{\odot 2 }\mathbf{1}_{n},\mathbf{1}_{n}\rangle+\min_{\mathbf{P}\in U_{n}}-2\langle \mathbf{C}_{\mathcal{X}}\mathbf{P}\mathbf{C}_{\mathcal{Y}},\mathbf{P}\rangle- \varepsilon H(\mathbf{P})\,\end{split}$ (11) $$

where $\mathbf{C}_{\mathcal{X}}^{\odot 2}=\mathbf{C}_{\mathcal{X}}\odot\mathbf{C}_{
\mathcal{X}}$, with $\odot$ the Hadamard (i.e., elementwise) product, and similarly for $\mathbf{C}_{\mathcal{Y}}^{\odot 2}$.
As we can see that the two terms on the left do not depend on $\mathbf{P}$, they do not impact the minimization, an OT coupling $\mathbf{P}^{\star}$ solving the problem satisfies:

$$ $\begin{split}\mathbf{P}^{\star}\in\operatorname*{arg\,min}_{\mathbf{P}\in U_{n }}\,-2\langle\mathbf{C}_{\mathcal{X}}\mathbf{P}\mathbf{C}_{\mathcal{Y}}, \mathbf{P}\rangle-\varepsilon H(\mathbf{P})\,.\end{split}$ (12) $$

As a result, if we now replace $\mathbf{C}_{\mathcal{X}}$ and $\mathbf{C}_{\mathcal{Y}}$ by $\mathbf{C}_{\mathcal{X}}/\texttt{stat}(\mathbf{C}_{\mathcal{X}})$ and $\mathbf{C}_{\mathcal{Y}}/\texttt{stat}(\mathbf{C}_{\mathcal{Y}})$, respectively, the new OT coupling $\tilde{\mathbf{P}}^{\star}$ solving the problem satisfies

$$ $\begin{split}&\tilde{\mathbf{P}}^{\star}\in\operatorname*{arg\,min}_{\mathbf{P }\in U_{n}}\,-2\langle\frac{\mathbf{C}_{\mathcal{X}}}{\texttt{stat}(\mathbf{C} _{\mathcal{X}})}\mathbf{P}\frac{\mathbf{C}_{\mathcal{Y}}}{\texttt{stat}( \mathbf{C}_{\mathcal{Y}})},\mathbf{P}\rangle-\varepsilon H(\mathbf{P})\,\\ \Leftrightarrow\quad&\tilde{\mathbf{P}}^{\star}\in\operatorname*{arg\,min}_{ \mathbf{P}\in U_{n}}\,-\frac{2}{{\texttt{stat}(\mathbf{C}_{\mathcal{X}})}\cdot \texttt{stat}(\mathbf{C}_{\mathcal{Y}})}\langle\mathbf{C}_{\mathcal{X}}\mathbf {P}\mathbf{C}_{\mathcal{Y}},\mathbf{P}\rangle-\varepsilon H(\mathbf{P})\,\\ \Leftrightarrow\quad&\tilde{\mathbf{P}}^{\star}\in\operatorname*{arg\,min}_{ \mathbf{P}\in U_{n}}\,-2\langle\mathbf{C}_{\mathcal{X}}\mathbf{P}\mathbf{C}_{ \mathcal{Y}},\mathbf{P}\rangle-\texttt{stat}(\mathbf{C}_{\mathcal{X}})\cdot \texttt{stat}(\mathbf{C}_{\mathcal{Y}})\cdot\varepsilon H(\mathbf{P})\,,\end{split}$ (13) $$

where in the last line, we use the fact that $\texttt{stat}(\mathbf{C}_{\mathcal{X}}),\texttt{stat}(\mathbf{C}_{\mathcal{Y}}
)>0$. This yields the desired equivalence on scaling the cost matrix and adapting the entropic regularization strength.

### B.3 Positivity of the Entropic GMG estimator

Recall that

$$ $\displaystyle\begin{split}\mathrm{GMG}_{{r}_{n},\varepsilon}(T)&=\mathrm{DST}_ {r_{n}}(T)-\mathrm{GW}_{\varepsilon}({r}_{n},T\sharp{r}_{n})\,\\ &=\mathrm{DST}_{r_{n}}(T)-\min_{\mathbf{P}\in U_{n}}\sum_{i,j,i^{\prime},j^{ \prime}=1}^{n}(c_{\mathcal{X}}(\mathbf{x}_{i},\mathbf{x}_{j})-c_{\mathcal{Y}}( \mathbf{y}_{i},\mathbf{y}_{j}))^{2}\mathbf{P}_{ij}\mathbf{P}_{i^{\prime}j^{ \prime}}-\varepsilon H(\mathbf{P})\,,\end{split}$ $$

For any coupling $\mathbf{P}\in U_{n}$, since $-\varepsilon H(\mathbf{P})=-\varepsilon\sum_{i,j=1}^{n}\mathbf{P}_{ij}\log(
\mathbf{P}_{ij})<0$, one has:

$$ $\sum_{i,j,i^{\prime},j^{\prime}=1}^{n}(c_{\mathcal{X}}(\mathbf{x}_{i},\mathbf{ x}_{j})-c_{\mathcal{Y}}(\mathbf{y}_{i},\mathbf{y}_{j}))^{2}\mathbf{P}_{ij} \mathbf{P}_{i^{\prime}j^{\prime}}-\varepsilon H(\mathbf{P})<\sum_{i,j,i^{ \prime},j^{\prime}=1}^{n}(c_{\mathcal{X}}(\mathbf{x}_{i},\mathbf{x}_{j})-c_{ \mathcal{Y}}(\mathbf{y}_{i},\mathbf{y}_{j}))^{2}\mathbf{P}_{ij}\mathbf{P}_{i^{ \prime}j^{\prime}}$ $$

As a result, applying minimization on both sides yields that $\mathrm{GW}_{\varepsilon}({r}_{n},T\sharp{r}_{n})<\mathrm{GW}_{0}({r}_{n},T
\sharp{r}_{n})=\mathrm{GW}({r}_{n},T\sharp{r}_{n})$, and therefore:

$$ $\mathrm{GMG}_{r_{n},\varepsilon}(T)>\mathrm{GMG}_{r_{n},0}(T)=\mathrm{GMG}_{r_ {n}}(T)\geq 0.$ $$

### B.4 Proofs of Prop. 3.3

See 3.3

###### Proof.

Let $T,r,s$ as described and suppose that $\mathcal{GM}_{r}^{c}(T)=0$. Then, $\pi^{r}:=(\mathrm{Id},T)\sharp r$ is an optimal Gromov-Wasserstein coupling, solution of Problem (GWP) between $r$ and $T\sharp r$ for costs $c_{\mathcal{X}}$ and $c_{\mathcal{Y}}$. Therefore, from (Séjourné et al., , Theorem. 3), $\pi^{r}$ is an optimal Kantorvich coupling, solution of Problem (KP) between $r$ and $T\sharp r$ for the linearized cost:

$$ $\tilde{c}:(\mathbf{x},\mathbf{y})\in\mathcal{X}\times\mathcal{Y}\mapsto\int_{ \mathcal{X}\times\mathcal{Y}}\tfrac{1}{2}|c_{\mathcal{X}}(\mathbf{x},\mathbf{x }^{\prime})-c_{\mathcal{Y}}(\mathbf{y},\mathbf{y}^{\prime})|^{2}\mathop{}\! \mathrm{d}\pi^{r}(\mathbf{x}^{\prime},\mathbf{y}^{\prime})$ (14) $$

Additionally, $\mathcal{X}\times\mathcal{Y}$ is a compact set as a product of compact sets, so since $(\mathbf{x},\mathbf{y})\mapsto|c_{\mathcal{X}}(\mathbf{x},\mathbf{x}^{\prime})
-c_{\mathcal{Y}}(\mathbf{y},\mathbf{y}^{\prime})|^{2}$ is continuous as $c_{\mathcal{X}}$ and $c_{\mathcal{Y}}$ are continuous, it is bounded on $\mathcal{X}\times\mathcal{Y}$. Afterward, since $\pi^{r}$ has finite mass, by Lebesgue’s dominated convergence Theorem, it follows that $\tilde{c}$ is continuous, and hence uniformly continuous, again since $\mathcal{X}\times\mathcal{Y}$ is compact.

Afterwards, by virtue of (Santambrogio, , Theorem 1.38), $\mathrm{Spt}\left(\pi^{r}\right)$ is a $\tilde{c}$-cyclically monotone (CM) set (see (Santambrogio, , Definition. 1.36)). From the definition of cyclical monotonicity, this property translates to subsets. Then, by defining $\pi^{s}=(\mathrm{Id},T)\sharp s$, as $\mathrm{Spt}(p)\subset\mathrm{Spt}(r)$, one has $\mathrm{Spt}(\pi^{s})=\mathrm{Spt}((\mathrm{Id},T)\sharp s)\subset\mathrm{Spt}
((\mathrm{Id},T)\sharp r)=\mathrm{Spt}(\pi^{r})$, so $\mathrm{Spt}(\pi^{s})$ is $\tilde{c}$-CM. Finally, since $\mathcal{X}$ and $\mathcal{Y}$ are compact, and $\tilde{c}$ is uniformly continuous, the $\tilde{c}$-cyclical monotonicity of its support implies that the coupling $\pi^{p}$ is a Kantorovich optimal coupling between its marginals for cost $\tilde{c}$, thanks to (Santambrogio, , Theorem 1.49). By re-applying (Séjourné et al., , Theorem. 3), we get that $\pi^{s}$ solves the Gromov-Wasserstein problem between its marginals for costs $c_{\mathcal{X}}$ and $c_{\mathcal{Y}}$. In other words, $\pi^{s}=(\mathrm{Id},T)\sharp s$ is Gromov-Wasserstein optimal coupling between $s$ and $T\sharp s$ so $T$ is a Gromov-Monge map between $s$ and $T\sharp s$ and $\mathrm{GMG}_{s}(T)=0$.
∎

###### Proof.

### B.5 Proofs of Thm. 3.5

See 3.5

Before proving Thm. 3.5, we first demonstrate some technical results that will be useful later.

#### Reformulation of the empirical GMG using permutations.

We start by showing that $\mathrm{GMG}_{{r}_{n}}(T)$ is always the sub-optimality gap of $T$ in Prob. (GMP) between $r_{n}$ and $T\sharp r_{n}$. This occurs because Prob. (GMP) and Prob. (GWP) coincide when applied between empirical measures on the same number of points. In other words, we can reformulate Prob. (GWP) between $r_{n}$ and $T\sharp r_{n}$ using permutation matrices, instead of (plain) couplings.

###### Proposition B.1 .

The empirical GMG reads

$$ $\displaystyle\begin{split}\mathrm{GMG}_{{r}_{n}}(T)=\mathrm{DST}_{{r}_{n}}(T)- \min_{\sigma\in\mathcal{S}_{n}}\tfrac{1}{n^{2}}\sum_{i,j=1}^{n}\left(c_{ \mathcal{X}}(\mathbf{x}_{i},\mathbf{x}_{j})-c_{\mathcal{Y}}(T(\mathbf{x}_{ \sigma(i)}),T(\mathbf{x}_{\sigma(j)}))\right)^{2}\end{split}$ (15) $$

###### Proof.

We first show a more general results, stating that when $c_{\mathcal{X}},c_{\mathcal{Y}}$ are conditionally positive kernels (see A.1), the discrete GW couplings between uniform, empirical distributions supported on the same number of points, ae permutation matrices.

###### Proposition B.2 (Equivalence between Gromov-Monge and Gromov-Wasserstein problems in the discrete case.) .

Let ${p}_{n}=\tfrac{1}{n}\sum_{i=1}^{n}\delta_{\mathbf{x}_{i}}$ and ${q}_{n}=\tfrac{1}{n}\sum_{i=1}^{n}\delta_{\mathbf{y}_{i}}$ two uniform, empirical measures, supported on the same number of points. We denote by $P_{n}=\{\mathbf{P}\in\mathbb{R}^{n\times n},\exists\sigma\in\mathcal{S}_{n},
\mathbf{P}_{ij}:=\delta_{j,\sigma(i)}\}$ the set set of permutation matrices.
Assume that $c_{\mathcal{X}}$ and $c_{\mathcal{Y}}$ (or $-c_{\mathcal{X}}$ and $-c_{\mathcal{Y}}$) are conditionally positive kernels (see A.1).
Then, the GM and GW formulations coincide, in the sense that we can restrict the GW problem to permutations, namely

$$ $\begin{split}\mathrm{GW}({p}_{n},{p}_{n})&=\min_{\mathbf{P}\in U_{n}}\sum_{i,j ,i^{\prime},j^{\prime}=1}^{n}(c_{\mathcal{X}}(\mathbf{x}_{i},\mathbf{x}_{i^{ \prime}})-c_{\mathcal{Y}}(\mathbf{y}_{j},\mathbf{y}_{j^{\prime}}))^{2}\mathbf{ P}_{ij}\mathbf{P}_{i^{\prime}j^{\prime}}\\ &=\tfrac{1}{n^{2}}\min_{\mathbf{P}\in P_{n}}\sum_{i,j,i^{\prime},j^{\prime}=1} ^{n}(c_{\mathcal{X}}(\mathbf{x}_{i},\mathbf{x}_{i^{\prime}})-c_{\mathcal{Y}}( \mathbf{y}_{j},\mathbf{y}_{j^{\prime}}))^{2}\mathbf{P}_{ij}\mathbf{P}_{i^{ \prime}j^{\prime}}\\ &=\tfrac{1}{n^{2}}\min_{\sigma\in\mathcal{S}_{n}}\sum_{i,j=1}^{n}(c_{\mathcal{ X}}(\mathbf{x}_{i},\mathbf{x}_{j})-c_{\mathcal{Y}}(\mathbf{y}_{\sigma(i)}, \mathbf{y}_{\sigma(j)}))^{2}\end{split}$ (16) $$

###### Proof.

Let $\mathbf{P}^{\star}\in U_{n}$ solution of the Gromov-Wasserstein between ${p}_{n}$ and ${p}_{n}$, i.e.

$$ $\mathbf{P}^{\star}\in\operatorname*{arg\,min}_{\mathbf{P}\in U_{n}}\sum_{i,j,i ^{\prime},j^{\prime}=1}^{n}(c_{\mathcal{X}}(\mathbf{x}_{i},\mathbf{x}_{i^{ \prime}})-c_{\mathcal{Y}}(\mathbf{y}_{j},\mathbf{y}_{j^{\prime}}))^{2}\mathbf{ P}_{ij}\mathbf{P}_{i^{\prime}j^{\prime}}$ $$

that always exists by continuity of the GW objective function on the compact $U_{n}$. We show that $\mathbf{P}^{\star}$ can be chosen as a (rescaled) permutation matrix without loss of generality.

As we assume that $c_{\mathcal{X}}$ and $c_{\mathcal{Y}}$ (or $-c_{\mathcal{X}}$ and $-c_{\mathcal{Y}}$) are conditionally positive kernels, from (Séjourné et al., , Theorem. 3), $\mathbf{P}^{\star}$ also solves:

$$ $\mathbf{P}^{\star}\in\operatorname*{arg\,min}_{\mathbf{Q}\in U_{n}}\sum_{i,j,i ^{\prime},j^{\prime}=1}^{n}(c_{\mathcal{X}}(\mathbf{x}_{i},\mathbf{x}_{i^{ \prime}})-c_{\mathcal{Y}}(\mathbf{y}_{j},\mathbf{y}_{j^{\prime}}))^{2}\mathbf{ P}^{\star}_{ij}\mathbf{Q}_{i^{\prime}j^{\prime}}$ (17) $$

We then define the linearized cost matrix $\tilde{C}\in\mathbb{R}^{n\times n}$, s.t.

$$ $\tilde{\mathbf{C}}_{ij}=\sum_{i^{\prime},j^{\prime}=1}^{n}(c_{\mathcal{X}}( \mathbf{x}_{i},\mathbf{x}_{i^{\prime}})-c_{\mathcal{Y}}(\mathbf{y}_{j},\mathbf {y}_{j^{\prime}}))^{2}\mathbf{P}^{\star}_{ij}$ $$

which allows us to reformulate Eq. (17) as

$$ $\mathbf{P}^{\star}\in\operatorname*{arg\,min}_{\mathbf{Q}\in U_{n}}\langle \tilde{\mathbf{C}},\mathbf{Q}\rangle$ (18) $$

’s theorem states that the extremal points of $U_{n}$
are the permutation matrices $P_{n}$. Moreover, a seminal theorem of linear programming (Bertsimas and Tsitsiklis, , Theorem 2.7) states that the minimum of a linear objective on a bounded polytope, if finite, is reached at an extremal point of the
polyhedron. Therefore, as $\mathbf{P}^{\star}$ solves Eq. (18), it is an extremal point of $U_{n}$, so it can always be chosen as a permutation matrix. Therefore, the equivalence between GW and GM follows.

∎

To conclude the proof of Prop. B.1, we simply remark that $r_{n}=\tfrac{1}{n}\sum_{i=1}^{n}\delta_{\mathbf{x}_{i}}$ and $T\sharp r_{n}=\tfrac{1}{n}\sum_{i=1}^{n}\delta_{T(\mathbf{x}_{i})}$ are uniform, empirical distribution, and supported on the same number of points.

∎

###### Proposition B.1 .

###### Proof.

###### Proposition B.2 (Equivalence between Gromov-Monge and Gromov-Wasserstein problems in the discrete case.) .

###### Proof.

#### Consistency of the empirical GMG.

We continue by proving a consistency result for the empirical GMG, which we will later use to deduce the asymptotic weak convexity constant from the finite-sample case.

###### Proposition B.3 .

For both $c_{\mathcal{X}}=c_{\mathcal{Y}}=\|\cdot-\cdot\|_{2}^{2}$ and $c_{\mathcal{X}}=c_{\mathcal{Y}}=\langle\cdot,\cdot\rangle$, one has
$\mathrm{GMG}_{{r}_{n}}(T)\rightarrow\mathrm{GMG}_{r}(T)$ almost surely.

###### Proof.

We first note that the empirical estimator of the distortion is consistent, as both costs are continuous, and $\mathcal{X}$ is compact. We then need to study, in both cases, the convergence of $\mathrm{GW}(r_{n},T\sharp r_{n})$ to $\mathrm{GW}(r_{n},T\sharp r)$.

To that end, we first remark that as, almost surely, $r_{n}\to r$ in distribution, one also has that, almost surely, $T\sharp r_{n}\to T\sharp r$ in distribution. Indeed, since $\mathcal{Y}$ is compact, $T$ is bounded so for any bounded and continuous $f:\mathcal{Y}\rightarrow\mathbb{R}$ and $X\sim r$, $f\circ T(X)$ is well defined and bounded so integrable. Afterwards, one can simply adapt the proof of the almost sure weak convergence of empirical measure based on the strong law of large numbers to show that, almost surely, $T\sharp{r}_{n}\rightarrow T\sharp r$ in distribution. See for instance (, Theorem 10.4.1).

We start with the squared Euclidean distance. As, almost surely, both $r_{n}\to r$ and $T\sharp{r}_{n}\to T\sharp r$ in distribution, the results follows from (Mémoli, , Thm 5.1, (e)).

We continue with the inner product. As noticed by Rioux et al. (, Lemma 2)–in the first version of the paper – the GW for inner product costs can be reformulated as:

$$ $\begin{split}\mathrm{GW}^{\langle\cdot,\cdot\rangle}(p,q)=&\int_{\mathcal{X} \times\mathcal{X}}\langle\mathbf{x},\mathbf{x}^{\prime}\rangle\mathop{}\! \mathrm{d}p(\mathbf{x})\mathop{}\!\mathrm{d}p(\mathbf{x}^{\prime})+\int_{ \mathcal{Y}\times\mathcal{Y}}\langle\mathbf{y},\mathbf{y}^{\prime}\rangle \mathop{}\!\mathrm{d}q(\mathbf{y})\mathop{}\!\mathrm{d}q(\mathbf{y}^{\prime}) \\ +&\min_{\mathbf{M}\in\mathcal{M}}\min_{\pi\in\Pi(p,q)}\int_{\mathcal{X}\times \mathcal{Y}}-4\langle\mathbf{M}\mathbf{x},\mathbf{y}\rangle\mathop{}\!\mathrm{ d}\pi(\mathbf{x},\mathbf{y})+4\|\mathbf{M}\|_{2}^{2}\,,\end{split}$ (19) $$

where we define $\mathcal{M}=[-M/2,M/2]^{d_{\mathcal{X}}\times d_{\mathcal{Y}}}$ with $M=\sqrt{\int_{\mathcal{X}}\|\mathbf{x}\|_{2}^{2}\mathop{}\!\mathrm{d}p(\mathbf
{x})\int_{\mathcal{Y}}\|\mathbf{y}\|_{2}^{2}\mathop{}\!\mathrm{d}q(\mathbf{y})}$.
In particular, they show this result for the entropic GW problem with $\varepsilon>0$, but their proof is also valid for $\varepsilon=0$. The above terms only involving the marginal, i.e., not involved in the minimization, are naturally stable under convergence in distribution, as $\mathcal{X}$ and $\mathcal{Y}$ are compact, so as $\mathcal{X}\times\mathcal{X}$ and $\mathcal{Y}\times\mathcal{Y}$.
As a result, we only need to study the stability of this quantity under the convergence in distribution of the following functional:

$$ $\mathcal{F}(p,q)=\min_{\mathbf{M}\in\mathcal{M}}\min_{\pi\in\Pi(p,q)}\int_{ \mathcal{X}\times\mathcal{Y}}-4\langle\mathbf{M}\mathbf{x},\mathbf{y}\rangle \mathop{}\!\mathrm{d}\pi(\mathbf{x},\mathbf{y})+4\|\mathbf{M}\|_{2}^{2}\,,$ (20) $$

We first remark that:

$$ $\begin{split}&|\mathcal{F}(p,q)-\mathcal{F}(p_{n},q_{n})|\\ \leq&\sup_{M\in\mathcal{M}}|\min_{\pi\in\Pi(p,q)}\int_{\mathcal{X}\times \mathcal{Y}}-4\langle\mathbf{M}\mathbf{x},\mathbf{y}\rangle\mathop{}\!\mathrm{ d}\pi(\mathbf{x},\mathbf{y})-\min_{\pi\in\Pi(p,q)}\int_{\mathcal{X}\times \mathcal{Y}}-4\langle\mathbf{M}\mathbf{x},\mathbf{y}\rangle\mathop{}\!\mathrm{ d}\pi(\mathbf{x},\mathbf{y})|\\ \leq&\sup_{M\in\mathcal{M}}|\min_{\pi\in\Pi(p,q)}\int_{\mathcal{X}\times \mathcal{Y}}2\|\mathbf{M}\mathbf{x}-\mathbf{y}\|_{2}^{2}\mathop{}\!\mathrm{d} \pi(\mathbf{x},\mathbf{y})-\min_{\pi\in\Pi(p_{n},q_{n})}\int_{\mathcal{X} \times\mathcal{Y}}2\|\mathbf{M}\mathbf{x}-\mathbf{y}\|_{2}^{2}\mathop{}\! \mathrm{d}\pi(\mathbf{x},\mathbf{y})2|\\ +&2\cdot\,\sup_{M\in\mathcal{M}}|\int_{\mathcal{X}}\|\mathbf{M}\mathbf{x}\|_{2 }^{2}\mathop{}\!\mathrm{d}p(\mathbf{x})-\int_{\mathcal{X}}\|\mathbf{M}\mathbf{ x}\|_{2}^{2}\mathop{}\!\mathrm{d}p_{n}(\mathbf{x})|\\ +&2\cdot\,|\int_{\mathcal{Y}}\|\mathbf{y}\|_{2}^{2}\mathop{}\!\mathrm{d}q( \mathbf{y})-\int_{\mathcal{Y}}\|\mathbf{y}\|_{2}^{2}\mathop{}\!\mathrm{d}q_{n} (\mathbf{y})|\end{split}$ (21) $$

Then, we show the convergence of each term separately.

- •
For the first term, we remark that (up to a constant factor) it can be reformulated:
$\sup_{M\in\mathcal{M}}|\mathrm{W}_{2}^{2}(\mathbf{M}\sharp p,q)-\mathrm{W}_{2}
^{2}(\mathbf{M}\sharp p_{n},q_{n})|$
where we remind that that $\mathrm{W}_{2}^{2}$ is the (squared) Wasserstein distance, solution of Eq. (KP) induced by $c(\mathbf{x},\mathbf{y})=\|\mathbf{x}-\mathbf{y}\|_{2}^{2}$. By virtue of (Manole and Niles-Weed, , Theorem 2), there exists a constant $C>0$, s.t. we can uniformly bound
$\sup_{M\in\mathcal{M}}|\mathrm{W}_{2}^{2}(\mathbf{M}\sharp p,q)-\mathrm{W}_{2}
^{2}(\mathbf{M}\sharp p_{n},q_{n})|\leq Cn^{-1/d}$
and the convergence follows.
- •
For the second one, this follows from from the convergence in distribution of $p_{n}$ to $p$ along with the Ascoli-Arzela theorem, since both $\mathcal{M}$ and $\mathcal{X}$ are compact sets, so the $\{f_{\mathbf{M}}\,|\,f_{\mathbf{M}}:\mathbf{x}\mapsto\|\mathbf{M}\mathbf{x}\|_
{2}^{2}\}$ are uniformly bounded and equi-continuous.
- •
For the third one, this follows from the convergence in distribution of $q_{n}$ to $q$.

As a result, we finally get $\mathrm{GW}^{\langle\cdot,\cdot\rangle}(p_{n},q_{n})\to\mathrm{GW}^{\langle
\cdot,\cdot\rangle}(p,q)$.

∎

###### Proposition B.3 .

###### Proof.

#### Weak convexity.

Finally, we demonstrate some useful results on weakly convex functions on $\mathbb{R}^{d}$.

###### Definition B.4 .

A function $f:\mathbb{R}^{d}\to\mathbb{R}$ is $\gamma$-weakly convex if $f+\gamma\|\cdot\|_{2}^{2}$ is convex.

From the definition, we see that if $f$ is $\gamma$-weakly convex, than $f$ is also $\gamma^{\prime}$ weakly convex for any $\gamma^{\prime}\geq\gamma$. This naturally extends to weakly convex functionals $\mathcal{F}$ on $L_{2}(r)$.

###### Lemma B.5 .

Let $\mathbf{A}\in S_{d}(\mathbb{R})$ a symmetric matrix and define the quadratic form $f_{\mathbf{A}}:\mathbf{x}\in\mathbb{R}^{d}\mapsto\mathbf{x}^{\top}\mathbf{A}
\mathbf{x}$. Then, $f_{\mathbf{A}}$ is $\max(0,-\lambda_{\min}(\mathbf{A}))$-weakly convex.

###### Proof.

We use the fact that a twice continuously differentiable function is convex i.f.f. its hessian is positive semi-definite (Boyd and Vandenberghe, , §(3.1.4)). Therefore, $f_{\mathbf{A}}$ is convex i.f.f. $\nabla^{2}f_{\mathbf{A}}=\mathbf{A}\geq 0$. If $\lambda_{\min}(\mathbf{A})\geq 0$, then $\mathbf{A}\geq 0$ so $f_{\mathbf{A}}$ is convex, i.e. 0 0-weakly convex. Otherwise, $f_{\mathbf{A}}-\tfrac{1}{2}\lambda_{\min}(\mathbf{A})\|\cdot\|_{2}^{2}$ has hessian $A-\lambda_{\min}(\mathbf{A})\geq 0$, so it is convex, which yields that $f_{\mathbf{A}}$ is $-\lambda_{\min}(\mathbf{A})$-weakly convex.
∎

###### Lemma B.6 .

Let $(f_{i})_{i\in I}$ a family of $\gamma$-weakly convex functions, with potentially infinite $I$. Then, $f:\mathbf{x}\in\mathbb{R}^{d}\mapsto\sup_{i\in I}f_{i}(\mathbf{x})$ is $\gamma$-weakly convex.

###### Proof.

As the $f_{i}$ are $\gamma$-weakly convex, $f_{i}+\tfrac{1}{2}\gamma$ is convex, so $\mathbf{x}\mapsto\sup_{i\in I}f_{i}(\mathbf{x})+\tfrac{1}{2}\gamma\|x\|_{2}^{2
}=(\sup_{i\in I}f_{i}(\mathbf{x}))+\tfrac{1}{2}\gamma\|x\|_{2}^{2}$ is convex (Boyd and Vandenberghe, , Eq. (3.7)). Therefore, the $\gamma$-weak convexity of $f$ follows
∎

Let’s now proceed to prove the main Thm. (3.5).

###### Proof of Thm. ( 3.5 )..

Finite sample. We first study the weak convexity of $\mathcal{GM}^{\langle\cdot,\cdot\rangle}_{{r}_{n}}$, i.e. the Gromov-Monge gap for the inner product. For a map $T\in L_{2}(r)$, it reads

$$ $\begin{split}\mathrm{GMG}^{\langle\cdot,\cdot\rangle}_{{r}_{n}}(T)&=\tfrac{1}{ n^{2}}\sum_{i,j=1}^{n}\tfrac{1}{2}|\langle\mathbf{x}_{i},\mathbf{x}_{j}\rangle -\langle T(\mathbf{x}_{i}),T(\mathbf{x}_{j})\rangle|^{2}\\ &-\min_{\mathbf{P}\in U_{n}}\sum_{i,j,i^{\prime},j^{\prime}=1}^{n}\tfrac{1}{2} |\langle\mathbf{x}_{i},\mathbf{x}_{i^{\prime}}\rangle-\langle T(\mathbf{x}_{j} ),T(\mathbf{x}_{j^{\prime}})\rangle|^{2}\mathbf{P}_{ij}\mathbf{P}_{i^{\prime}j ^{\prime}}\end{split}$ $$

As ${r}_{n}$ and $T\sharp{r}_{n}$ are uniform empirical supported on the same number of points, using Prop. B.2, we can reformulate the RHS with permutation matrices, which yields

$$ $\begin{split}\mathrm{GMG}^{\langle\cdot,\cdot\rangle}_{{r}_{n}}(T)&=\tfrac{1}{ n^{2}}\sum_{i,j=1}^{n}\tfrac{1}{2}|\langle\mathbf{x}_{i},\mathbf{x}_{j}\rangle -\langle T(\mathbf{x}_{i}),T(\mathbf{x}_{j})\rangle|^{2}\\ &-\tfrac{1}{n^{2}}\min_{\mathbf{P}\in P_{n}}\sum_{i,j,i^{\prime},j^{\prime}=1} ^{n}\tfrac{1}{2}|\langle\mathbf{x}_{i},\mathbf{x}_{i^{\prime}}\rangle-\langle T (\mathbf{x}_{j}),T(\mathbf{x}_{j^{\prime}})\rangle|^{2}\mathbf{P}_{ij}\mathbf{ P}_{i^{\prime}j^{\prime}}\end{split}$ $$

From this expression, $\mathrm{GMG}^{\langle\cdot,\cdot\rangle}_{{r}_{n}}$ can be reformulated as a matrix input function. Indeed, it only depends on the map $T$ via its values on the support of ${r}_{n}$, namely $\mathbf{x}_{1},...,\mathbf{x}_{n}$. Therefore, we write $\mathbf{t}_{i}\vcentcolon=T(\mathbf{x}_{i})$, and define $\mathbf{X},\mathbf{T}\in\mathbb{R}^{n\times d}$ which contain observations $\mathbf{x}_{i}$ and $\mathbf{t}_{i}$ respectively, stored as rows. Then, studying $\mathrm{GMG}^{\langle\cdot,\cdot\rangle}_{{r}_{n}}$ remains to study

$$ $f(\mathbf{T})\vcentcolon=\tfrac{1}{n^{2}}\sum_{i,j=1}^{n}\tfrac{1}{2}|\langle \mathbf{x}_{i},\mathbf{x}_{j}\rangle-\langle\mathbf{t}_{i},\mathbf{t}_{j} \rangle|^{2}-\tfrac{1}{n^{2}}\min_{\mathbf{P}\in P_{n}}\sum_{i,j,i^{\prime},j^ {\prime}=1}^{n}\tfrac{1}{2}|\langle\mathbf{x}_{i},\mathbf{x}_{i^{\prime}} \rangle-\langle\mathbf{t}_{j},\mathbf{t}_{j^{\prime}}\rangle|^{2}\mathbf{P}_{ ij}\mathbf{P}_{i^{\prime}j^{\prime}}$ $$

By developing each term and exploiting that for any $\mathbf{P}\in P_{n}$, $\mathbf{P}\mathrm{1}_{n}=\mathbf{P}^{\top}\mathrm{1}_{n}=\tfrac{1}{n}\mathrm{1
}_{n}$, we derive

$$ $\begin{split}f(\mathbf{T})&=\tfrac{1}{n^{2}}\sum_{i,j=1}^{n}-\langle\mathbf{x} _{i},\mathbf{x}_{j}\rangle\cdot\langle\mathbf{t}_{i},\mathbf{t}_{j}\rangle- \min_{\mathbf{P}\in P_{n}}\tfrac{1}{n^{2}}\sum_{i,j,i^{\prime},j^{\prime}=1}^{ n}-\langle\mathbf{x}_{i},\mathbf{x}_{i^{\prime}}\rangle\cdot\langle\mathbf{t}_ {j},\mathbf{t}_{j^{\prime}}\rangle\mathbf{P}_{ij}\mathbf{P}_{i^{\prime}j^{ \prime}}\\ &=\max_{\mathbf{P}\in P_{n}}\tfrac{1}{n^{2}}\sum_{i,j,i^{\prime},j^{\prime}=1} ^{n}\langle\mathbf{x}_{i},\mathbf{x}_{i^{\prime}}\rangle\cdot\langle\mathbf{t} _{j},\mathbf{t}_{j^{\prime}}\rangle\mathbf{P}_{ij}\mathbf{P}_{i^{\prime}j^{ \prime}}-\tfrac{1}{n^{2}}\sum_{i,j=1}^{n}\langle\mathbf{x}_{i},\mathbf{x}_{j} \rangle\cdot\langle\mathbf{t}_{i},\mathbf{t}_{j}\rangle\\ &=\max_{\mathbf{P}\in P_{n}}\langle\tfrac{1}{n^{2}}\mathbf{P}^{\top}\mathbf{X} \mathbf{X}^{\top}\mathbf{P},\mathbf{T}\mathbf{T}^{\top}\rangle-\langle\tfrac{1 }{n^{2}}\mathbf{X}\mathbf{X}^{\top},\mathbf{T}\mathbf{T}^{\top}\rangle\\ &=\max_{\mathbf{P}\in P_{n}}\langle\tfrac{1}{n^{2}}(\mathbf{P}^{\top}\mathbf{X }\mathbf{X}^{\top}\mathbf{P}-\mathbf{X}\mathbf{X}^{\top}),\mathbf{T}\mathbf{T} ^{\top}\rangle\\ &=\max_{\mathbf{P}\in P_{n}}\langle\tfrac{1}{n^{2}}(\mathbf{P}^{\top}\mathbf{X }\mathbf{X}^{\top}\mathbf{P}-\mathbf{X}\mathbf{X}^{\top})\mathbf{T},\mathbf{T} \rangle\\ &=\max_{\mathbf{P}\in P_{n}}\langle\mathbf{A}_{\mathbf{X},\mathbf{P}}\mathbf{T },\mathbf{T}\rangle\end{split}$ $$

where we define $\mathbf{A}_{\mathbf{X},\mathbf{P}}\vcentcolon=\tfrac{1}{n^{2}}(\mathbf{P}^{
\top}\mathbf{X}\mathbf{X}^{\top}\mathbf{P}-\mathbf{X}\mathbf{X}^{\top})\in
\mathbb{R}^{n\times n}$. To study the convexity of this matrix input function, we vectorize it. From (Petersen and Pedersen, , Eq. (520)), we note that, for any $\mathbf{M}\in\mathbb{R}^{n\times n}$

$$ $\begin{split}\langle\mathbf{M}\mathbf{T},\mathbf{T}\rangle=\mathbf{vec}( \mathbf{T})^{\top}\mathbf{vec}(\mathbf{M}\mathbf{T})=\mathbf{vec}(\mathbf{T})^ {\top}(\mathbf{M}\otimes I_{n})\mathbf{vec}(\mathbf{T})\end{split}$ $$

where $\mathbf{vec}$ is the vectorization operator, raveling a matrix along its rows, and $\otimes$ is the Kronecker product. Applying this identity, we reformulate:

$$ $f(\mathbf{T})=\max_{\mathbf{P}\in U_{n}}\mathbf{vec}(\mathbf{T})^{\top}( \mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n})\mathbf{vec}(\mathbf{T})$ (22) $$

To study the convexity of $r$, we study the convexity of each $r_{\mathbf{A}_{\mathbf{X},\mathbf{P}}}(\mathbf{T})\vcentcolon=\mathbf{vec}(
\mathbf{T})^{\top}(\mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n})\mathbf{vec
}(\mathbf{T})$, which are quadratic forms induced by the $\mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n}$. This remains to study the (semi-) positive definiteness of the matrices $\mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n}$. As each $\mathbf{A}_{\mathbf{X},\mathbf{P}}\in\mathbb{R}^{n\times n}$ is symmetric and square, $\mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n}$ is also symmetric and from (Petersen and Pedersen, , Eq. (519)) its eigenvalues are the outer products of the eigenvalues of $\mathbf{A}_{\mathbf{X},\mathbf{P}}$ and $I_{n}$, namely

$$ $\begin{split}\mathbf{eig}(\mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n})&=\{ \lambda_{i}(\mathbf{A}_{\mathbf{X},\mathbf{P}})\cdot\lambda_{j}(I_{n})\}_{1 \leq i,j\leq n}\\ &=\{\underbrace{\lambda_{1}(\mathbf{A}_{\mathbf{X},\mathbf{P}}),\dots,\lambda_ {1}(\mathbf{A}_{\mathbf{X},\mathbf{P}})}_{\textrm{n times}},\dots,\underbrace{ \lambda_{n}(\mathbf{A}_{\mathbf{X},\mathbf{P}}),\dots,\lambda_{n}(\mathbf{A}_{ \mathbf{X},\mathbf{P}})}_{\textrm{n times}}\}\end{split}$ (23) $$

It follows that the minimal eigenvalue of $\mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n}$ is $\lambda_{\min}(\mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n})=\lambda_{\min}
(\mathbf{A}_{\mathbf{X},\mathbf{P}})$. Utilizing the expression of $\mathbf{A}_{\mathbf{X},\mathbf{P}}$

$$ $\begin{split}\lambda_{\min}(\mathbf{A}_{\mathbf{X},\mathbf{P}})&=\tfrac{1}{n^{ 2}}\lambda_{\min}(\mathbf{P}^{\top}\mathbf{X}\mathbf{X}^{\top}\mathbf{P}- \mathbf{X}\mathbf{X}^{\top})\\ &\geq\tfrac{1}{n^{2}}(\lambda_{\min}(\mathbf{P}^{\top}\mathbf{X}\mathbf{X}^{ \top}\mathbf{P})+\lambda_{\min}(-\mathbf{X}\mathbf{X}^{\top}))\\ &=\tfrac{1}{n^{2}}(\lambda_{\min}(\mathbf{P}^{\top}\mathbf{X}\mathbf{X}^{\top} \mathbf{P})-\lambda_{\max}(\mathbf{X}\mathbf{X}^{\top}))\end{split}$ (24) $$

Reminding that $\mathbf{P}\in U_{n}$, one has $\mathbf{P}^{\top}=\mathbf{P}^{-1}$, so $\mathbf{P}^{\top}\mathbf{X}\mathbf{X}^{\top}$ and $\mathbf{X}\mathbf{X}^{\top}$ are similar, and they have the same eigenvalues. In particular $\lambda_{\min}(\mathbf{P}^{\top}\mathbf{X}\mathbf{X}^{\top}\mathbf{P})=\lambda
_{\min}(\mathbf{X}\mathbf{X}^{\top})$. Combining these results, it follows that

$$ $\lambda_{\min}(\mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n})=\lambda_{\min} (\mathbf{A}_{\mathbf{X},\mathbf{P}})\geq\tfrac{1}{n^{2}}(\lambda_{\min}( \mathbf{X}\mathbf{X}^{\top})-\lambda_{\max}(\mathbf{X}\mathbf{X}^{\top}))$ (25) $$

We then remind that each $r_{\mathbf{A}_{\mathbf{X},\mathbf{P}}}$ is the quadratic form defined by $\mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n}$, so by applying Prop. B.5, it is $\mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n}$-weakly convex, and hence $\tfrac{1}{n^{2}}(\lambda_{\max}(\mathbf{X}\mathbf{X}^{\top})-\lambda_{\min}(
\mathbf{X}\mathbf{X}^{\top}))$-weakly convex. Therefore, applying Prop. (B.6), $r$ is $\tfrac{1}{n^{2}}(\lambda_{\max}(\mathbf{X}\mathbf{X}^{\top})-\lambda_{\min}(
\mathbf{X}\mathbf{X}^{\top}))$-weakly convex, in $\mathbb{R}^{d}$. Reminding that $\gamma_{\mathrm{inner}}=\tfrac{1}{n}(\lambda_{\max}(\mathbf{X}\mathbf{X}^{\top
})-\lambda_{\min}(\mathbf{X}\mathbf{X}^{\top}))$, $r$ is $\tfrac{1}{n}\gamma_{\mathrm{inner}}$ weakly convex.
This implies that $\mathbf{T}\mapsto f(\mathbf{T})+\tfrac{1}{n}\gamma_{\mathrm{inner}}\|\mathbf{T
}\|_{2}^{2}$ is convex. By reminding that $\mathbf{T}$ stores the $T(\mathbf{x}_{i})$ as rows, $\tfrac{1}{n}\|\mathbf{T}\|_{2}^{2}=\|T\|_{L_{2}(r_{n})}$. Consequently, $\mathrm{GMG}_{r_{n}}^{\langle\cdot,\cdot\rangle}$ is $\gamma_{\mathrm{inner}}$ in $L_{2}(r_{n})$.

We then study the convexity of $\mathrm{GMG}_{r_{n}}^{2}$. We follow exactly the same approach. One has:

$$ $\begin{split}\mathrm{GMG}^{2}_{{r}_{n}}(T)&=\tfrac{1}{n^{2}}\sum_{i,j=1}^{n} \tfrac{1}{2}|\|\mathbf{x}_{i}-\mathbf{x}_{j}\|_{2}^{2}-\|T(\mathbf{x}_{i})-T( \mathbf{x}_{j})\|_{2}^{2}|^{2}\\ &-\tfrac{1}{n^{2}}\min_{\mathbf{P}\in P_{n}}\sum_{i,j,i^{\prime},j^{\prime}=1} ^{n}\tfrac{1}{2}|\|\mathbf{x}_{i}-\mathbf{x}_{j}\|_{2}^{2}-\|T(\mathbf{x}_{i}) -T(\mathbf{x}_{j})\|_{2}^{2}|^{2}|^{2}\mathbf{P}_{ij}\mathbf{P}_{i^{\prime}j^{ \prime}}\end{split}$ $$

Similarly, studying the convexity of $\mathrm{GMG}^{2}_{{r}_{n}}(T)$ remains to study the convexity of the matrix input function:

$$ $\begin{split}g(\mathbf{T})&\vcentcolon=\tfrac{1}{n^{2}}\sum_{i,j=1}^{n}\tfrac{ 1}{2}|\|\mathbf{x}_{i}-\mathbf{x}_{j}\|_{2}^{2}-\|\mathbf{t}_{i}-\mathbf{t}_{j }\|_{2}^{2}|^{2}\\ &-\tfrac{1}{n^{2}}\min_{\mathbf{P}\in P_{n}}\sum_{i,j,i^{\prime},j^{\prime}=1} ^{n}\tfrac{1}{2}|\|\mathbf{x}_{i}-\mathbf{x}_{j}\|_{2}^{2}-\|\mathbf{t}_{i}- \mathbf{t}_{j}\|_{2}^{2}|^{2}\mathbf{P}_{ij}\mathbf{P}_{i^{\prime}j^{\prime}} \end{split}$ $$

As before, by developing each term, one has:

$$ $\begin{split}g(\mathbf{T})=\max_{\mathbf{P}\in P_{n}}&\tfrac{1}{n^{2}}\sum_{i, j,i^{\prime},j^{\prime}=1}^{n}\langle\mathbf{x}_{i},\mathbf{x}_{i^{\prime}} \rangle\cdot\langle\mathbf{t}_{j},\mathbf{t}_{j^{\prime}}\rangle\mathbf{P}_{ij }\mathbf{P}_{i^{\prime}j^{\prime}}+\tfrac{1}{2n}\sum_{i,j=1}^{n}\mathbf{P}_{ij }\|\mathbf{x}_{i}\|_{2}^{2}\|\mathbf{t}_{i}\|_{2}^{2}\\ -&\left(\tfrac{1}{n^{2}}\sum_{i,j=1}^{n}\langle\mathbf{x}_{i},\mathbf{x}_{j} \rangle\cdot\langle\mathbf{t}_{i},\mathbf{t}_{j}\rangle+\tfrac{1}{2n}\sum_{i,j =1}^{n}\|\mathbf{x}_{i}\|_{2}^{2}\|\mathbf{t}_{i}\|_{2}^{2}\right)\end{split}$ $$

The quadratic terms in $\mathbf{P}$ can be factorized as before using $\mathbf{A}_{\mathbf{X},\mathbf{P}}$. For the new terms w.r.t. the inner product case, we introduce $\mathbf{D}_{\mathbf{X}}\vcentcolon=\mathrm{diag}(\|\mathbf{x}_{1}\|_{2}^{2},
\dots,\|\mathbf{x}_{n}\|_{2}^{2})$, and remark that we can rewrite:

$$ $\begin{split}\tfrac{1}{2n}\sum_{i,j=1}^{n}\mathbf{P}_{ij}\|\mathbf{x}_{i}\|_{2 }^{2}\|\mathbf{t}_{i}\|_{2}^{2}-\tfrac{1}{2n}\sum_{i,j=1}^{n}\|\mathbf{x}_{i} \|_{2}^{2}\|\mathbf{t}_{i}\|_{2}^{2}=\mathbf{vec}(T)^{\top}\left(\tfrac{1}{2n} (\mathbf{P}^{\top}-I_{n})\otimes\mathbf{D}_{\mathbf{X}}\right)\mathbf{vec}(T) \end{split}$ $$

As we can always symetrize the matrix when considering its associated quadratic form, we have:

$$ $\begin{split}\tfrac{1}{2n}\sum_{i,j=1}^{n}\mathbf{P}_{ij}\|\mathbf{x}_{i}\|_{2 }^{2}\|\mathbf{t}_{i}\|_{2}^{2}-\tfrac{1}{2n}\sum_{i,j=1}^{n}\|\mathbf{x}_{i} \|_{2}^{2}\|\mathbf{t}_{i}\|_{2}^{2}=\mathbf{vec}(T)^{\top}\left(\tfrac{1}{2}( \tfrac{1}{2n}(\mathbf{P}^{\top}+\mathbf{P})-I_{n})\otimes\mathbf{D}_{\mathbf{X }}\right)\mathbf{vec}(T)\end{split}$ $$

As a result, we denote $\mathbf{B}_{\mathbf{X},\mathbf{P}}=\tfrac{1}{n}(\tfrac{1}{2}(\mathbf{P}^{\top}
+\mathbf{P})-I_{n})\otimes\mathbf{D}_{\mathbf{X}}$ and finally get:

$$ $\begin{split}g(\mathbf{T})&=\max_{\mathbf{P}\in P_{n}}\mathbf{vec}(T)^{\top} \left(\mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n}+\mathbf{B}_{\mathbf{X}, \mathbf{P}}\right)\mathbf{vec}(T)\end{split}$ $$

As we did for $f$, studying the weak convexity of $f$ remains to lower bound the minimal eigenvalue of $\mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n}+\mathbf{B}_{\mathbf{X},\mathbf
{P}}$. First, one remark that:

$$ $\lambda_{\min}(\mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n}+\mathbf{B}_{ \mathbf{X},\mathbf{P}})\geq\lambda_{\min}(\mathbf{A}_{\mathbf{X},\mathbf{P}} \otimes I_{n})+\lambda_{\min}(\mathbf{B}_{\mathbf{X},\mathbf{P}})$ $$

As we we have already lower bounded $\lambda_{\min}(\mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n})\geq\tfrac{1}{n
^{2}}(\lambda_{\min}(\mathbf{X}\mathbf{X}^{\top})-\lambda_{\max}(\mathbf{X}
\mathbf{X}^{\top}))$, we focus on the RHS. Similarly, one has:

$$ $\begin{split}\lambda_{\min}(\mathbf{B}_{\mathbf{X},\mathbf{P}})&=\lambda_{\min }\left(\tfrac{1}{2n}(\tfrac{1}{2}(\mathbf{P}^{\top}+\mathbf{P})-I_{n})\otimes \mathbf{D}_{\mathbf{X}}\right)\\ &\geq\lambda_{\min}\left(\tfrac{1}{4n}(\mathbf{P}^{\top}+\mathbf{P})\otimes \mathbf{D}_{\mathbf{X}}\right)+\lambda_{\min}\left(-\tfrac{1}{2n}I_{n}\otimes \mathbf{D}_{\mathbf{X}}\right)\\ &\geq\lambda_{\min}\left(\tfrac{1}{4n}(\mathbf{P}^{\top}+\mathbf{P})\otimes \mathbf{D}_{\mathbf{X}}\right)-\lambda_{\max}\left(\tfrac{1}{2n}I_{n}\otimes \mathbf{D}_{\mathbf{X}}\right)\end{split}$ (26) $$

For both terms, we apply again (Petersen and Pedersen, , Eq. (519)). For the LHS, one has:

$$ $\mathbf{eig}\left(\tfrac{1}{4n}(\mathbf{P}^{\top}+\mathbf{P})\otimes\mathbf{D} _{\mathbf{X}}\right)=\{\lambda_{i}(\tfrac{1}{4n}(\mathbf{P}^{\top}+\mathbf{P}) )\lambda_{j}(\mathbf{D}_{\mathbf{X}})\}_{1\leq i,j\leq n}$ (27) $$

We remark that $\tfrac{1}{2}(\mathbf{P}^{\top}+\mathbf{P})$ is a symetric bi-stochastic matrix, so $\lambda_{\min}(\tfrac{1}{2}(\mathbf{P}^{\top}+\mathbf{P}))\geq-1$. Therefore, $\lambda_{\min}(\tfrac{1}{4n}(\mathbf{P}^{\top}+\mathbf{P}))\geq-\tfrac{1}{2n}$. As a result, since the eigenvalues of $\mathbf{D}_{\mathbf{X}}$ are the $\|\mathbf{x}_{i}\|_{2}^{2}$, this yields:

$$ $\lambda_{\min}\left(\tfrac{1}{4n}(\mathbf{P}^{\top}+\mathbf{P})\otimes\mathbf{ D}_{\mathbf{X}}\right)\geq-\tfrac{1}{2n}\max_{i=1,\dots,n}\|\mathbf{x}_{i}\|_{ 2}^{2}$ $$

Similarly, we have:

$$ $-\lambda_{\max}\left(\tfrac{1}{2n}I_{n}\otimes\mathbf{D}_{\mathbf{X}}\right) \geq-\tfrac{1}{2n}\max_{i=1,\dots,n}\|\mathbf{x}_{i}\|_{2}^{2}$ $$

from which we deduce that:

$$ $\lambda_{\min}(\mathbf{B}_{\mathbf{X},\mathbf{P}})\geq-\tfrac{1}{n}\max_{i=1, \dots,n}\|\mathbf{x}_{i}\|_{2}^{2}$ $$

We can then lower bound:

$$ $\begin{split}\lambda_{\min}(\mathbf{A}_{\mathbf{X},\mathbf{P}}\otimes I_{n}+ \mathbf{B}_{\mathbf{X},\mathbf{P}})&\geq\tfrac{1}{n^{2}}(\lambda_{\min}( \mathbf{X}\mathbf{X}^{\top})-\lambda_{\max}(\mathbf{X}\mathbf{X}^{\top}))- \tfrac{1}{n}\max_{i=1,\dots,n}\|\mathbf{x}_{i}\|_{2}^{2}\\ &=-\tfrac{1}{n}\gamma_{2,n}\end{split}$ (28) $$

which yields the $\tfrac{1}{n}\gamma_{2,n}$-weak convexity of $g$, and finally the $\gamma_{2,n}$-weak convexity of $\mathcal{GM}^{2}_{{r}_{n}}$.

Asymptotic. For any $T$, we note that, almost surely, $\|T\|^{2}_{L_{2}(r_{n})}\to\|T\|^{2}_{L_{2}(r)}$. As a result, since convexity is preserved under pointwise convergence and by virtue of Prop. (B.3), we study the (almost sure) convergence of $\gamma_{\textrm{inner},n}$ and $\gamma_{2,n}$.

We start by $\gamma_{\textrm{inner},n}$. We first remark that $\lambda_{\max}(\tfrac{1}{n}\mathbf{X}\mathbf{X}^{\top})=\lambda_{\max}(\tfrac{
1}{n}\mathbf{X}^{\top}\mathbf{X})$. Moreover, as $\mathbf{A}\in S_{d}^{+}(\mathbb{R})\mapsto\lambda_{\max}(\mathbf{A})$ is continuous and $\tfrac{1}{n}\mathbf{X}^{\top}\mathbf{X}\to\mathbb{E}_{\mathbf{x}\sim r}[
\mathbf{x}\mathbf{x}^{\top}]$ almost surely, one has $\lambda_{\max}(\tfrac{1}{n}\mathbf{X}\mathbf{X}^{\top})\to\lambda_{\max}(
\mathbb{E}_{\mathbf{x}\sim r}[\mathbf{x}\mathbf{x}^{\top}])$ almost surely. Moreover, for any $n>d$, $\lambda_{\min}(\tfrac{1}{n}\mathbf{X}\mathbf{X}^{\top})=0$. As a result, $\gamma_{\textrm{inner},n}\to\lambda_{\max}(\mathbb{E}_{\mathbf{x}\sim r}[
\mathbf{x}\mathbf{x}^{\top}])$ almost surely, which provides the desired asymptotic result.

We continue with $\gamma_{2,n}$. We first remark that $\max_{i=1,\dots,n}\|\mathbf{x}_{i}\|_{2}^{2}\leq\sup_{\mathbf{x}\in\mathrm{Spt
}(r)}\|\mathbf{x}\|_{2}^{2}$. As a result, by defining $\tilde{\gamma}_{2,n}=\gamma_{\textrm{inner},n}+\max_{\mathbf{x}\in\mathrm{Spt}
(r)}\|\mathbf{x}\|_{2}^{2}$, $\mathrm{GMG}^{2}_{{r}_{n}}$ is also $\tilde{\gamma}_{2,n}$-weakly convex. Moreover, $\max_{\mathbf{x}\in\mathrm{Spt}(r)}\|\mathbf{x}\|_{2}^{2}$ does not depends on $n$, $\tilde{\gamma}_{2,n}\to\lambda_{\max}(\mathbb{E}_{\mathbf{x}\sim r}[\mathbf{x}
\mathbf{x}^{\top}])+\max_{\mathbf{x}\in\mathrm{Spt}(r)}\|\mathbf{x}\|_{2}^{2}$ almost surely,
which also provides the desired asymptotic result.

∎

###### Definition B.4 .

###### Lemma B.5 .

###### Proof.

###### Lemma B.6 .

###### Proof.

###### Proof of Thm. ( 3.5 )..

## Appendix C Experimental Details

All our experiments build on python and the jax-framework (Babuschkin et al., ), alongside ott-jax for optimal transport utilities.

### C.1 Details on Disentanglement Benchmark

To effectively conduct comprehensive and representative research on disentangled representation learning, we convert the public PyTorch framework proposed in  Roth et al. () to an equivalent jax variant. We verify our implementation through replications of baseline and HFS results in Roth et al. (), mainting relative performance orderings and close absolute disentanglement scores (as measured using DCI-D, whose implementation directly follows from Locatello et al. () and leverages gradient boosted tree implementations from scikit-learn).

For exact and fair comparison, we utilize standard hyperparamater choices from Roth et al. () (which leverages hyerparameters directly from Locatello et al. (), Locatello et al. () and [https://github.com/google-research/disentanglement_lib](https://github.com/google-research/disentanglement_lib)).
Consequently, the base VAE architecture utilized across all experiment is the same as the one utilized in Roth et al. () and Locatello et al. (): With image input sizes of $64\times 64\times N_{c}$ (with $N_{c}$ the number of input image channels, usually 3). The latent dimensionality, if not otherwise specified, is set to 10. The exact VAE model architecture is as follows:

- •
Encoder: [conv($32,4\times 4$, stride 2) + ReLU] $\times$ 2, [conv($64,4\times 4$, stride 2) + ReLU] $\times$ 2, MLP(256), MLP(2 $\times$ 10)
- •
Decoder: MLP(256), [upconv($64,4\times 4$, stride 2) + ReLU] $\times$ 2, [upconv($32,4\times 4$, stride 2) + ReLU], [upconv($n_{c},4\times 4$, stride 2) + ReLU]

Similar, we retain all training hyperparameters from (Roth et al., ) and (Locatello et al., ): Using an Adam optimizer ((Kingma and Ba, ), $\beta_{1}=0.9,\beta_{2}=0.999,\epsilon=10^{-8}$) and a learning rate of $10^{-4}$. Following Locatello et al. (), Roth et al. () we utilize a batch-size of $64$, for which we also ablate all baseline methods. The total number of training steps is set to $300000$.

**Table 3: Hyperparameter grid searches for different baseline and proposed methods.**
| Method | Parameter | Values |
| --- | --- | --- |
| $\beta$-VAE | $\beta$ | [2, 4, 6, 8, 10, 16] |
| $\beta$-TCVAE | $\beta$ | [2, 4, 6, 8, 10, 16] |
| + HFS | $\gamma$ | [1, 10] |
| + DST | $\lambda$ | [0.1, 1, 5, 10, 20] |
| + GMG | $\lambda$ | [0.1, 1, 5, 10, 20] |
| + Jac | $\lambda$ | [0.1, 1, 5, 10, 20] |

As commonly done for this setting (Locatello et al., ; , Roth et al., ), we also perform a small grid search over all the hyperparameters. We report the full details in Tab. 3.

For $\lambda_{e}$ and $\lambda_{d}$, we set $\lambda_{e}=0$ for the Decoder setting and $\lambda_{d}=0$ for the Encoder setting while altering the weighting for the other $\lambda$. All experiments run on a single RTX 2080TI GPU.

### C.2 Stability Analysis

For the gradient stability analysis experiment, we repeat the following experiment for each of the four image datasets $\mathcal{D}$ that we consider. We first considered a fixed neural map $T_{\theta}$, which we choose to be a randomly initialized neural network, consisting of encoder and decoder, before any training. For $k=1,\dots,5$, we sample a batch $\mathbf{x}_{1}^{k},\dots,\mathbf{x}_{n}^{k}\sim\mathcal{D}$ and let $r_{n}^{k}=\tfrac{1}{n}\sum_{i=1}^{n}\delta_{\mathbf{x}_{i}^{k}}$. We report the pariwise cosine similarity between the gradients of the DST and the GMG. Formally, we compute $\textrm{cos-sim}(\nabla_{\theta}\mathrm{DST}_{r_{n}^{k}}(T_{\theta}),\nabla_{
\theta}\mathrm{DST}_{r_{n}^{l}}(T_{\theta}))$, and $\textrm{cos-sim}(\nabla_{\theta}\mathrm{GMG}_{r_{n}^{k}}(T_{\theta}),\nabla_{
\theta}\mathrm{GMG}_{r_{n}^{l}}(T_{\theta}))$, for $k,l=1,\dots,5$.

## Appendix D Additional Empirical Results

In this section, we report additional empirical results revolving around the regularization of encoder. First in  D.1, we conduct the stability analysis when regularizing the encoder. Then, we report further results of encoder and decoder regularization on DSprites. Lastly, we take a first exploratory step towards decoder-free disentangled representation learning in D.3.

### D.1 Encoder Stability Analysis

We repeat the following experiment for each of the four image datasets $\mathcal{D}$ that we consider. We first considered a fixed neural map $T_{\theta}$. For $i=1,\dots,5$, we sample a batch $\mathbf{x}_{1}^{i},\dots,\mathbf{x}_{n}^{i}\sim\mathcal{D}$ and let $r_{n}^{i}=\tfrac{1}{n}\sum_{i=1}^{n}\delta_{\mathbf{x}_{n}^{i}}$. We report the pariwise cosine similarity between the gradients of the DST. Namely, for $i,j=1,\dots,5$, we compute $\textrm{cos-sim}(\nabla_{\theta}\mathrm{DST}_{r_{n}^{i}}(T_{\theta}),\nabla_{
\theta}\mathrm{DST}_{r_{n}^{j}}(T_{\theta}))$, and the GMG, $\textrm{cos-sim}(\nabla_{\theta}\mathrm{GMG}_{r_{n}^{i}}(T_{\theta}),\nabla_{
\theta}\mathrm{GMG}_{r_{n}^{j}}(T_{\theta}))$.

Figure: Figure 4: Gradient stability analysis on the DST and GMG as Encoder regularizations. The cosine similarity is computed between all pairs of gradients $\nabla_{\theta}$ obtained through 5 randomly sampled batches and a fixed network $T_{\theta}$ for each dataset.
Refer to caption: x4.png

### D.2 Encoder Analysis on DSprites

**Table 4: Disentanglement of regularizing the Encoder and the Encoder and Decoder as measured by DCI-D on DSprites. We highlight best, second best, and third best results for each method and dataset.**
| DCI-D | $\beta$-VAE | $\beta$-TCVAE | $\beta$-VAE + HFS | $\beta$-TCVAE + HFS |
| --- | --- | --- | --- | --- |
| DSprites Higgins et al. () |  |  |  |  |
| Base | $27.6$ $\pm 13.4$ | $36.0$ $\pm 5.3$ | 38.7 $\pm 15.7$ | $48.1$ $\pm 10.8$ |
| + Enc-DST | 32.8 $\pm 15.0$ | $36.5$ $\pm 5.9$ | $33.9$ $\pm 15.9$ | $48.9$ $\pm 11.1$ |
| + Enc-GMG | $27.5$ $\pm 14.3$ | 37.4 $\pm 5.8$ | $31.0$ $\pm 14.3$ | $45.9$ $\pm 10.9$ |
| + Dec-DST | $28.6$ $\pm 19.3$ | $32.4$ $\pm 8.5$ | $39.3$ $\pm 18.1$ | $49.0$ $\pm 11.2$ |
| + Dec-GMG | 39.5 $\pm 15.2$ | 42.2 $\pm 3.6$ | 46.7 $\pm 2.0$ | 50.1 $\pm 8.5$ |

### D.3 Towards Decoder-free Disentanglement

Table: Table 5: Disentanglement (DCI-D) without a decoder trained with various regularizations on Shapes3D (Burgess and Kim, ).

Recently, works such as Burns et al. (), von Kügelgen et al. (), Eastwood et al. (), Matthes et al. (), Aitchison and Ganev () have shown the possibility of disentanglement through self-supervised, contrastive learning objectives in an effort to align with the scalability of encoder-only representation learning (Chen et al., , Zbontar et al., , Bardes et al., , Garrido et al., ). However, these encoder-only approaches still require weak supervision or access to multiple views of an image to learn meaningful representations of the data samples.

As the goal of geometry preservation connects the data manifold and the latent domain through a minimal distortion objective and is applicable to both the encoder and decoder of a VAE (§3, Table 4), we posit that its application may provide sufficient training signal to learn meaningful representations and encourage disentanglement, eliminating the need for a reconstruction loss and decoder. Table 5 shows preliminary results on unsupervised decoder-free disentangled representation learning on the Shapes3D benchmark, where the decoder and associated reconstruction objective have been removed.

Standard approaches such as $\beta$-VAE or $\beta$-TCVAE collapse and do not achieve measurable disentanglement (DCI-D of $0.0$). However, the inclusion of either DST or GMG significantly raises achievable disentanglement and, combined with the $\beta$-TCVAE matching objective, can achieve DCI-D scores of up to $53.5$ without needing any decoder or reconstruction loss. While these are preliminary insights, we believe they offer promise for more scalable approaches to unsupervised disentangled representation learning and potential bridges to popular and scalable self-supervised representation learning approaches. Note, that here the distortion loss significantly outperforms the GMG. This is expected due to the nature of the GMG, as the distortion loss offers a more restrictive and, thus, stronger signal for learning representations, which is necessary in the absence of a reconstruction objective. This highlights that while in most scenarios (§ 2.1, Figure 2), the GMG is preferable over the distortion loss, there also exist settings where a more restrictive optimization signal is desirable.

## Appendix E Python Code for the Computation of the Gromov-Monge Gap

import jax
import jax.numpy as jnp

from ott.geometry import costs, geometry
from ott.solvers.quadratic import gromov_wasserstein
from ott.problems.quadratic import quadratic_problem

def gromov_monge_gap_from_samples(
source: jax.Array,
target: jax.Array,
cost_fn: costs.CostFn = costs.Cosine(),
epsilon: float = 0.1,
stat_fn: Callable, # usually computes the mean of the cost matrix
**kwargs,
) -> float:
"""Gromov Monge gap regularizer on samples."""

# define source and target geometries
cost_matrix_x = cost_fn.all_pairs(x=source, y=source)
scale_cost_x = stat_fn(scale_cost_x)
cost_matrix_x = cost_matrix_x / jax.lax.stop_gradient(scale_cost_x)
geom_xx = geometry.Geometry(cost_matrix=cost_matrix_x)

cost_matrix_y = cost_fn.all_pairs(x=target, y=target)
scale_cost_y = stat_fn(cost_matrix_y)
cost_matrix_y = cost_matrix_y / jax.lax.stop_gradient(scale_cost_y)
geom_yy = geometry.Geometry(cost_matrix=cost_matrix_y)

# define and solve entropic GW problem
prob = quadratic_problem.QuadraticProblem(geom_xx, geom_yy)

solver = gromov_wasserstein.GromovWasserstein(
epsilon=epsilon, **kwargs
)
out = solver(prob)

# compute the distortion induced by the map
distortion_cost = jnp.nanmean(
(geom_xx.cost_matrix - geom_yy.cost_matrix)**2
)

# compute optimal (entropic) gromov-monge displacement
reg_gw_cost = out.reg_gw_cost
ent_reg_gw_cost = reg_gw_cost - 2 * epsilon * jnp.log(len(source))

# compute gromov-monge gap
loss = distortion_cost - ent_reg_gw_cost

return loss * jax.lax.stop_gradient(scale_cost_x * scale_cost_y)

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: - Considering the entropic regularization parameter plays a role in the optimization, how does adjusting the hyperparameter affect the computational efficiency and convergence of the GMG computation? An ablation study of varying hyperparameters would be nice.
- How scalable is GMG to high-dimensional real world datasets for the VAE configuration?
soundness: 4
strengths: - The paper is well-organized and clearly written.
- Introducing the Gromov-Monge Gap as a regularizer provides a fresh approach to minimizing geometric feature distortion, presenting a promising advancement for unsupervised disentangled representation learning.
- The authors offer valuable insights, particularly in showing that preserving angles through cosine similarity can enhance disentanglement more effectively than preserving distances.
- The proof demonstrating the GMG as a weakly convex regularizer is a useful contribution for future work in this area.
summary: This paper introduces a novel approach for unsupervised disentangled representation learning, aiming to balance prior alignment with the preservation of geometric features within latent spaces. Achieving disentangled representations without supervision is difficult; recent studies suggest that maintaining the geometric structure of the data distribution can promote disentanglement. However, aligning data with a prior distribution while fully preserving geometric properties often requires trade-offs. Therefore, the authors introduce the Gromov-Monge Gap (GMG), a regularizer based on optimal transport theory that learns a mapping with minimal distortion.
weaknesses: - The paper provides limited discussion on the scalability of GMG for high-dimensional real world datasets.
- There is an absence of ablation studies on key hyperparameters, such as entropic regularization.

---

confidence: 2
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. Why does angle preservation work better than distance preservation? Is this finding specific to the datasets tested?

2. How sensitive is the method to the entropic regularization parameter $\epsilon_0$?
soundness: 3
strengths: 1. This paper provides a fresh perspective on combining geometric constraints with prior matching in disentanglement learning. Different from previous works that use direct penalties for geometric preservation, the authors propose using Gromov-Monge maps to find mappings that optimally preserve geometric features while aligning distributions. The theoretical analysis proves weak convexity of the GMG regularizer and provides precise characterization of convexity constants.

2. The authors have conducted comprehensive experiments across multiple benchmarks and architectural choices. Through thorough ablation studies, they demonstrate that angle preservation outperforms distance preservation, decoder regularization is more effective than encoder regularization, and the GMG shows better gradient stability compared to distortion-based approaches. Additionally, they show the method can be integrated with existing approaches like β-VAE and β-TCVAE to enhance their performance.
summary: This paper introduces a new method for disentangled representation learning based on quadratic optimal transport theory using Gromov-Monge maps. The authors leverage Gromov-Monge Gap (GMG) to measure how well a mapping preserves geometric features while transporting between distributions with minimal distortion. The approach aims to effectively combine geometric constraints with prior matching by finding mappings that optimally balance geometric preservation and distribution alignment. The method is validated on four standard disentanglement benchmarks (Shapes3D, DSprites, SmallNORB, Cars3D) showing consistent improvements over existing disentanglement approaches.
weaknesses: 1. It remains unclear to me why the Gromov-Monge map framework is the optimal choice for combining geometric preservation with distribution alignment. While the authors show empirically that GMG outperforms direct distortion penalties, they don't fully explain why measuring suboptimality in the Gromov-Monge problem provides a better learning signal than other potential optimal transport formulations or geometric metrics.

2. My another concern is about the practicality of solving Gromov-Wasserstein problems during training. Although the authors provide computational complexity analysis and propose using entropic regularization with scaled costs, they don't thoroughly investigate how different solver parameters affect the trade-off between computational efficiency and accuracy of the approximation. The sensitivity analysis of the entropic regularization parameter $\epsilon_0$ and its impact on training dynamics needs more exploration.

---

confidence: 3
contribution: 2
initial_rating: 5
presentation: 3
questions: 1. Have you applied the method on datasets other than 3D dataset? For example, MNIST. 

2. Have you tested the method on noisy point clouds? Or any other noisy dataset? 

3. The whole framework is very computational expensive, do you think how it can be applied to large-scale setting? Have you thought about acceleration?
soundness: 3
strengths: 1. The intuition of this idea is good, it tries to minimize the difference between the previous Distortion and Gromov-Wasserstein distance so that the mapping between source and target should preserve the full features with minimal distortion. 
2. They show that the GMG and its counterpart are weakly convex functions. 
3. The experiments in 3D Shapes dataset and the clustering illustration sound persuasive.
summary: This paper studies learning disentangled representations in an unsupervised setting by introducing Gromov-Wasserstein formulation into the distortion framework. They propose GMG quantifying the difference between the distortion and the Gromov-Wasserstein distance, meanwhile, they show that GMG is weakly convex. They validate the effectiveness of GMG in four 3D datasets.
weaknesses: 1. The computation of Gromov-Wasserstein distance is very expensive even when entropy regularizer is employed, the time complexity is still around O(N^5). This makes the GMG not practical when involving the method in a large-scale data setting. 
2. Limited experiments: the datasets in the experiments are all about 3D shapes. How about 2D shapes? I would like to see how it can apply to the MNIST, another simple 2D digit. 
3. I believe that the robustness should also be important part but this work seems to ignore it. What if the data has out-of-distritbution points? In fact, DST is not robust in the noisy data setting.

---

confidence: 5
contribution: 1
initial_rating: 5
presentation: 2
questions: Please refer to the weaknesses.

My main confusion comes from the logical connection of this paper from isometry to disentanglement to OT. Please try to address this issue in the rebuttal.
soundness: 3
strengths: 1. The empirical performance is good, indicating that adding this neural GM gap can improve the disentanglement score for many architectures.
summary: This paper proposes to learn disentangled representation from the perspective of preserving geometric features. They propose to learn a pair of isometric encoder and decoder, hoping these inductive biases can benefit disentanglement. Based on this motivation, they propose to leverage Gromov-Monge mappings to enforce such an optimal transport (OT) mapping.
weaknesses: 1. The motivation and logical connection is not clear. There are three concerned concepts in the paper: isometry, OT, and disentanglement. The authors claim that adding geometrical constraints (isometry) can help disentanglement, but this connection is not explained well in the paper, which makes it hard to understand how disentanglement connects isometry. Even if there is such a connection, it is not clear why OT is required for isometry. You can just do an isometry mapping. I would expect the authors to explain these two connections more clearly in the revision.

2. The experiments are not set up well to support the main contribution of the paper. If the authors would like to show the benefit of isometry, the authors can follow homeomorphic VAE [1] to show the isometry can hep data on different groups/manifolds. If the authors would like to show the benefit of OT, there is Wasserstein auto-encoder. If the authors want to show the benefit of OT in disentanglement, there are [2] and [3] which learn an OT mapping to model disentangled sequential transformations. I do not see a direct connection between isometric OT mapping and disentanglement.

3. How does the method compare with the vanilla differentiable Sinkhorn algorithm [4]? Will there be any theoretical guarantees? Beyond the vanilla OT problem, there exist other OT solver like entropic OT and unbalanced OT. Do these techniques apply to disentanglement?

4. In a disentanglement paper, it is rare to not see figures about latent traversal which show the exact results of disentanglement. 

5. How does the method work in more scaled settings, i.e., high-resolution disentanglement datasets like Falcol3D and Issac3D [5]?

>[1] Explorations in Homeomorphic Variational Auto-Encoding. ICML 2019.
>
>[2] Flow Factorized Representation Learning. NeurIPS 2023.
>
>[3] Unsupervised Representation Learning from Sparse Transformation Analysis. Arxiv 2024.
>
>[4] Sinkhorn distances: Lightspeed computation of optimal transport. NeurIPS 2013.
>
>[5] Semi-Supervised StyleGAN for Disentanglement Learning. ICML 2020.

Meta-review: This paper proposes a quadratic optimal transport-based disentanglement approach to match data distributions with the priors while preserving geometry features. The problem is formulated  using Gromov-Monge maps that transport one distribution onto another with minimal distortion of predefined geometric features. The effectiveness of the method is demonstrated on standard benchmarks, outperforming similar approaches that leveraging geometric constraints. The main concerns from the reviewers are 1) the unclear motivation of using Gromov-Monge, 2) the computational complexity is relatively high, 3) the datasets used in the experiments are small scale. The authors did a great job in the rebuttal and clarified the concerns. The remaining concern on MNIST, and Facol3D, Issac3D datasets does not seem to be a strong reason to reject this paper. I recommend the paper to be accepted, though the overall rating is below 6.

Decision: Accept (Poster)