Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groups

## Abstract

Abstract The generative modeling of data on manifold is an important task, for which diffusion models in flat spaces typically need nontrivial adaptations. This article demonstrates how a technique called ‘trivialization’ can transfer the effectiveness of diffusion models in Euclidean spaces to Lie groups. In particular, an auxiliary momentum variable was algorithmically introduced to help transport the position variable between data distribution and a fixed, easy-to-sample distribution. Normally, this would incur further difficulty for manifold data because momentum lives in a space that changes with the position. However, our trivialization technique creates to a new momentum variable that stays in a simple fixed vector space . This design, together with a manifold preserving integrator, simplifies implementation and avoids inaccuracies created by approximations such as projections to tangent space and manifold, which were typically used in prior work, hence facilitating generation with high-fidelity and efficiency. The resulting method achieves state-of-the-art performance on protein and RNA torsion angle generation and sophisticated torus datasets. We also, arguably for the first time, tackle the generation of data on high-dimensional Special Orthogonal and Unitary groups, the latter essential for quantum problems.

## 1 Introduction

Diffusion-based [e.g., , , ] and flow-based [e.g., , , ] generative models have significantly impacted the landscape of various fields such as computer vision, largely due to their remarkable ability in modeling data that follow complicated and/or high-dimensional probability distributions. However, in many application domains, data explicitly reside on manifolds. Note this is different from the popular data manifold assumption which is implicit; here the manifold is a priori fixed due to, e.g., physics. Such cases occur, for example, in protein modeling [, ], cell development [], geographical sciences [], robotics [], and high-energy physics []. The naive application of standard generative models to these cases via embedding data in ambient Euclidean spaces often results in suboptimal performance []. This is partly due to the lack of appropriate geometric inductive biases
and potential encounters with singularities [].

Pioneering works suggest generalizing (continuous) neural ODE [] to manifold [, , ] with maximum-likelihood training. [, ] develop simulation-free algorithm but their objective is unscalable or biased []. Recent milestones, such as Riemannian Score-based Model (RSGM) [] and Riemannian Diffusion Model (RDM) [], have successfully demonstrated the potential to extend diffusion models onto Riemannian manifolds. RSGM explores the effectiveness and complexity of various variants of score matching loss on a general manifold and their applicable scenarios, and RDM discusses techniques such as variance reduction for the training objective via importance sampling and likelihood estimation. These models learn to reverse the diffusion process on a manifold, either by utilizing the heat equation defined on the manifold or by extending the Stochastic Differential Equation to the manifold. This is achieved through the employment of Riemannian score matching methods, which serve as simulation-based objectives for the optimization of the model. However, due to the inherent geometric complexity of the data, the training and sampling processes of such models necessitate multiple approximations. In particular, they require the projection of vector field (i.e score) to the tangent space which subsequently serves as the training label for the neural network during training phase. Furthermore, to mitigate numerical integration errors during the sampling process, there is a requirement for the projection of samples to the original data manifold. Moreover, among most scenarios, as these models are simulation-based algorithms, an additional approximation is introduced during the collection of training data from the simulation during training. This process also necessitates the projection of data to the manifold, which is analogous to the sampling phase. The combination of all these three approximations can compromise the quality of generation.

Even more recent advancements, such as Riemannian Flow Matching (RFM) [] and Scaled-RSGM [], aim to alleviate training complexities and enhance model scalability through the introduction of simulation-free objectives. Scaled-RSGM achieves this by focusing on Riemannian symmetric spaces, while RFM constructs conditional flows using premetrics. However, it is important to note that both of the approaches still require the aforementioned approximations during the training and sampling phases, which may potentially introduce inaccuracies in the generated results. Furthermore, whether RFM is expressive enough for intricate data distribution was discussed in [].

In this work, we build upon recent progress in momentum-based optimization [] and sampling [] on Lie groups to develop a highly scalable and effective generative model for data on these manifolds. Our approach departs from prior momentum-based generative models [, ] due to an additional technique called trivialization, which utilizes the additional group structure and enables us to learn score in a fixed flat space, while still encapsulating the curved geometry without any approximation. It results in a novel Trivialized Diffusion Model (TDM),
and our contributions are threefold:

- 1)
We introduce TDM that enables manifold data generative modeling through learning trivialized score function in a fixed flat space, which dramatically improves the generative performance.
- 2)
We leverage a nontrivial Operator Splitting Integrator to stay exactly on the manifold in an accurate and efficient way. The reduction of approximations further improves the generation.
- 3)
We outperform baselines by a large margin on protein/RNA torsion angle datasets. We achieve much higher quality generation on a newly-introduced challenging problem called Pacman. We present the first results on generating $\mathsf{U}(n)$ data corresponding to quantum evolutions, and high dim $\mathsf{SO}(n)$ data too; these results are also appealing.

## 2 Preliminaries

Euclidean Diffusion Model, kinetic Langevin, and Lie group are briefly reviewed in Appendix A.

## 3 Method

In this section, we will discuss how to perform generative modeling of data distribution on a class of smooth Riemannian manifolds, namely Lie groups, by only learning a score function in a fixed Euclidean space. Our goal is to recover the scenario of Euclidean generative modeling to the maximum by leveraging the group structure of the Lie group apart from its Riemannian manifold structure. To achieve this, we explore a specific manifold extension of Kinetic Langevin dynamics [], which contains an additional variable known as momentum. Importantly, a direct introduction of the momentum would not simplify the situation, since the momentum lives in a changing tangent space as the position moves. Fortunately, the group structure of the Lie group enables the design of a trivialized momentum that stays in a Lie algebra for the whole time, which is a simple fixed Euclidean space that suits our needs. In the sequel, we will discuss how the technique of trivialization can help completely avoid challenges posed by the curved geometry in an exact, analytical fashion, without resorting to complicated differential geometry notions such as parallel transport, and certainly no need for approximations, projections and retractions.

In the following, we first introduce a forward process that converges to an easy-to-sample distribution with such trivialized momentum. We derive the time reversal of such a process, which can serve as a backward generative process. We discuss methods to efficiently learn the drift of the backward process. Finally, we introduce a numerical integrator that achieves high accuracy and preserves the manifold structure of the Lie group.

### 3.1 Trivialized Kinetic Langevin Dynamics on Lie Group as Noising Process

Kong and Tao [] appropriately added noise to variational Lie group optimization dynamics [] and constructed the following kinetic Langevin sampling dynamics on Lie groups:

$$ $\begin{cases}\dot{g_{t}}=T_{e}\mathsf{L}_{g_{t}}\xi_{t},\\ {\mathrm{d}}\xi_{t}=-\gamma(t)\xi_{t}{\mathrm{d}}t-T_{g}\mathsf{L}_{g_{t}^{-1} }(\nabla U(g_{t})){\mathrm{d}}t+\sqrt{2\gamma(t)}{\mathrm{d}}W^{\mathfrak{g}}_ {t},\end{cases}$ (1) $$

where $g_{t},\xi_{t}\in G\times\mathfrak{g},\;\forall t\geq 0$, here $G$ denotes a Lie group and $\mathfrak{g}$ denotes its associated Lie algebra, ${\mathrm{d}}W^{\mathfrak{g}}_{t}$ is the Brownian motion on Lie algebra $\mathfrak{g}$, $\nabla U$ is the riemannian gradient of $U$, and $U:G\rightarrow\mathbb{R}$ is a potential function. $\xi_{t}$ is the left-trivialized momentum at time $t$ and $T_{e}\mathsf{L}_{g_{t}}\xi_{t}$ is the true momentum.

They also proved [] that for connected compact Lie groups, which will be our setup, (1) converges, under Lipschitzness of $\nabla U$, exponentially fast to its invariant distribution,
which is

$$ $\displaystyle\pi_{*}(g,\xi)=\frac{1}{Z}\exp\Big{(}-U(g)-\frac{1}{2}\langle\xi, \xi\rangle\Big{)}{\mathrm{d}}g{\mathrm{d}}\xi,$ (2) $$

where ${\mathrm{d}}g$ denotes the Haar measure, ${\mathrm{d}}\xi$ denotes the Lebesgue measure on $\mathfrak{g}$, and $Z$ is the normalizing constant. Dynamic (1) is a generalization of the Euclidean kinetic Langevin equation on $\mathbb{R}^{k}$ to general Lie groups ($\mathbb{R}^{k}$ is a Lie group with vector addition being the group operation).

By Peter-Weyl Theorem, a connected compact Lie group can be represented as closed subgroups of $\operatorname{GL}(n,\mathbb{C)}$ [], i.e. the group of $n\times n$ invertible matrices with entries in $\mathbb{C}$. We want to construct a forward noising process based on (1) by choosing a potential $U$ that corresponds to an easy-to-sample distribution on $G$. In the case of connected compact Lie groups, we pick the natural choice, which is the uniform distribution on $G$, to be the invariant distribution (note in this case, the limiting measure is Haar in $g$ which is easy to sample from; see []). This means that $U(g)=0$ and the corresponding $\nabla U(g)=0$. In this case, we derive the following dynamic as the forward noising process,

$$ $\begin{cases}\dot{g_{t}}=g_{t}\xi_{t},\\ {\mathrm{d}}\xi_{t}=-\gamma(t)\xi_{t}{\mathrm{d}}t+\sqrt{2\gamma(t)}{\mathrm{d }}W^{\mathfrak{g}}_{t},\end{cases}$ (3) $$

Important examples of connected compact Lie Groups include but are not limited to the Special Orthogonal group $\mathsf{SO}(n)$, the Unitary group $\mathsf{U}(n)$, the Special Unitary group $\operatorname{SU}(n)$, etc. Note 1-sphere $\mathbb{S}^{1}$, torus $\mathbb{T}$, and $\mathsf{SO}(2)$ are essentially the same thing (isomorphic). Note also the direct product of any two connected compact Lie groups is still a connected compact Lie group, so in general we can consider the Lie group $G$ of form,

$$ $\displaystyle G=G_{1}\times G_{2}\times\dots\times G_{k},$ (4) $$

where $G_{1},\dots,G_{k}$ are $k$ connected compact Lie groups.

### 3.2 Time Reversal of Trivialized Kinetic Langevin

The following result allows us to revert the time of the forward noising process. Thanks to the introduction of momentum and the fact that it is trivialized, the time reversal looks very much like the Euclidean version [] despite that $g$ lives on a manifold. This pleasant structure is because the forward dynamics (1) has no (direct) noise on $g$ dynamics and therefore no score-based correction is needed for its reversal. One important implication of the trivialization of the momentum variable $\xi$ is that the only score present in the dynamic is $\nabla_{\xi}\log p_{T-t}(g_{t},\xi_{t})$, which now stays in the Lie algebra $\mathfrak{g}$ (a fixed space and also is isomorphic to Euclidean space). This implies that we manage to get rid of $\nabla_{g}\log p_{T-t}$ from the dynamic, which is a much more complicated subject than $\nabla_{\xi}\log p_{T-t}$ due to being a Riemannian gradient and has complicated geometric dependency. Such trivialization has multiple benefits on numerical accuracy and score representation learning, which is not enjoyed by previous works such as RFM[], RDM [] and RSGM[] on Riemannian generative modeling. We discussed in detail the advantages of trivalized dynamic in Section 3.4.

Although a similar time reversal formula has been proved for the Euclidean case in [], their results are not applicable due to the presence of the manifold structure. In fact, we need a non-trivial adaptation of the arguments and the proof relies on the Fokker Planck equation on the manifold $G\times\mathfrak{g}$. For details of the proof of Theorem 1, see Appendix B.

###### Theorem 1 ( Time Reversal of Trivialized Kinetic Langevin on Lie Group ) .

Let $T\geq 0$, $W^{\mathfrak{g}}_{t}$ be a Brownian motion on the Lie algebra $\mathfrak{g}$. Let $\mathbf{X_{t}}=(g_{t},\xi_{t})$ be the trajectory of the forward dynamics (3), with $\mathbf{X}_{t}$ admitting a smooth density $p_{t}(g_{t},\xi_{t})$ with respect to the Haar measure on $G$ and Lebesgue measure on $\mathfrak{g}$. Then, the solution to the following SDE

$$ $\begin{cases}\dot{g_{t}}=-g_{t}\xi_{t},\\ {\mathrm{d}}\xi_{t}=\gamma(T-t)\xi_{t}{\mathrm{d}}t+2\gamma\nabla_{\xi}\log p_ {T-t}(g_{t},\xi_{t}){\mathrm{d}}t+\sqrt{2\gamma(t)}{\mathrm{d}}W^{\mathfrak{g} }_{t}.\end{cases}$ (5) $$

satisfy $\mathbf{Y}_{t}\stackrel{{\scriptstyle d}}{{=}}(\mathbf{X}_{T-t})$ under the notation $\mathbf{Y}_{t}:=(g_{t},\xi_{t})$ and initialization $\mathbf{Y}_{0}=\mathbf{X}_{T}$.

Similar to Diffusion-based Models, dynamic 5 has a corresponding probabilistic ODE counterpart which can be expressed as the dynamic 6:
{graybox}

###### Remark 1 (Probability Flow ODE) .

The following dynamic is has the same marginal as (14):

$$ $\begin{cases}\dot{g_{t}}=-g_{t}\xi_{t},\\ {\mathrm{d}}\xi_{t}=\gamma(T-t)\xi_{t}{\mathrm{d}}t+\gamma\nabla_{\xi}\log p_{ T-t}(g_{t},\xi_{t}){\mathrm{d}}t\end{cases}$ (6) $$

###### Theorem 1 ( Time Reversal of Trivialized Kinetic Langevin on Lie Group ) .

###### Remark 1 (Probability Flow ODE) .

### 3.3 Likelihood Training and Score-Matching for Trivalized Kinetic Langevin

To perform generative modeling of data distribution, we would like to simulate and sample from the stochastic dynamic in (5). However, the score $\nabla_{\xi}\log p_{T-t}(g_{t},\xi_{t})$ is intractable and we want to approximate it with a neural network score model $s_{\theta}(g_{t},\xi_{t},t)$. We denote the sequence of probability distribution $q^{\theta}_{t}$ as the density of $\mathcal{L}(\mathbf{Y}_{t}^{\theta})$ with respect to the reference measure, where $\mathbf{Y}_{t}^{\theta}$ is the trajectory of the following dynamic,

$$ $\begin{cases}\dot{g_{t}}=-g_{t}\xi_{t},\\ {\mathrm{d}}\xi_{t}=\gamma(T-t)\xi_{t}{\mathrm{d}}t+2\gamma s_{\theta}(g_{t}, \xi_{t},t){\mathrm{d}}t+\sqrt{2\gamma(t)}{\mathrm{d}}W^{\mathfrak{g}}_{t},\\ \end{cases}\quad s.t\ \ g_{0},\xi_{0}\sim\pi_{*}.$ (7) $$

In order to generate new data with dynamic (7), we would need $q_{T}^{\theta}\approx p_{0}$, which would require learning a score that is close to the true score $\nabla_{\xi}\log p_{t}(g,\xi)$. A known approach to learn the true score is through Score Matching objective (SM) between $s_{\theta}$ and $\nabla_{\xi}\log p_{t}(g,\xi)$.

Here, we will discuss two classical tractable variants of SM, Denoising Score Matching (DSM) and Implicit Score Matching $(\textbf{ISM})$ for learning the score of the trivialized kinetic Langevin.

#### Denoising Score Matching

We first remark that,
we have the following expansion of $\mathcal{J}_{\operatorname{SM}}(\theta)$ [],

$$ $\displaystyle\mathcal{J}_{\operatorname{SM}}(\theta)=\underbrace{\mathbb{E}_{t ,p_{0}}\mathbb{E}_{p_{t|0}}\Big{[}\big{\|}\nabla_{\xi}\log p_{t|0}(g,\xi)-s_{ \theta}(g,\xi,T-t)\big{\|}^{2}\Big{]}}_{\mathcal{J}_{\operatorname{DSM}}( \theta)}+\,C_{1}$ $$

where $C_{1}$ is a constant independent of $\theta$. Hence, $\operatorname{argmin}_{\theta}\mathcal{J}_{\operatorname{SM}}=\operatorname{
argmin}_{\theta}\mathcal{J}_{\operatorname{DSM}}$, but evaluating $\mathcal{J}_{\operatorname{DSM}}$ only requires knowledge of the conditional transition probability $p_{t|0}$. The question boils down to finding out such condition transition probability induced by the forward dynamic (3).

Note that the Lie algebra $\mathfrak{g}$ is a tangent space of $G$ at the identity, so it’s a vector space that is isomorphic to Euclidean space $\mathbb{R}^{d}$, where $d=\operatorname{dim}(\mathfrak{g})$. For example, the $\mathfrak{so}(2)$ is the Lie algebra of the Special Orthogonal group $\mathsf{SO}(2)$. $\mathfrak{so}(2)$ consists of all the $2\times 2$ skew-symmetric matrices. This implies that, for any $\xi\in\mathfrak{so}(2)$,

$$ $\displaystyle\xi=\begin{bmatrix}0&\theta\\ -\theta&0\end{bmatrix},\,\theta\in\mathbb{R}\implies\mathfrak{so}(2)\cong \mathbb{R}$ $$

Since the Brownian motion on $\mathfrak{g}$ should be understood as ${\mathrm{d}}W_{t}^{\mathfrak{g}}=\sum_{i=1}^{d}{\mathrm{d}}W_{t}^{i}\cdot e_{i}$, where $\{{\mathrm{d}}W_{t}^{i}\}_{i=1,\dots d}$ are independent standard Brownian motions on $\mathbb{R}$ and $\{e_{i}\}_{i=1,\dots,d}$ is an orthogonal basis for $\mathfrak{g}$. Therefore, the forward dynamic (3) with initial condition $g(0)=g_{0},\xi(0)=\xi_{0}$ is equivalent to the following,

$$ $\begin{cases}\dot{g_{t}}=g_{t}\xi_{t},\\ {\mathrm{d}}\xi^{i}_{t}=-\gamma\xi_{t}{\mathrm{d}}t+\sqrt{2\gamma}{\mathrm{d}} W^{i}_{t}\quad\forall 1\leq i\leq d.\end{cases}\quad s.t\ \ g(0)=g_{0},\;\xi^{ i}(0)=\xi_{0}^{i}\quad\forall 1\leq i\leq d.$ (8) $$

Here, without loss of generality, we choose $\gamma(t)$ to be a constant $\gamma>0$. We notice that each $\xi^{i}$ follows is OU process with an explicit solution.

This reduces problem (8) to a matrix-valued initial value problem (IVP) for $g_{t}$, since $\xi_{t}$ can be treated as a known function of time. Then the IVP $\dot{g_{t}}=g_{t}\xi_{t},g(0)=g_{0}$ is just a linear system.

Unfortunately, note even though the linearity ensures linear structure in the solution, namely $g(t)=g_{0}\Phi(t)$ where $\Phi$ is known as a fundamental matrix, $\Phi$ in general may not be analytically available in closed-form because the linear system has a time-dependent coefficient matrix. This differs from the scalar case where $\Phi(t)$ would just be $\exp(\int_{0}^{t}\xi(s){\mathrm{d}}s)$ or the constant coefficient matrix case where $\Phi(t)$ would just be $\operatorname{expm}(\xi t)$. Instead, we can represent the solution using geometric tools, resulting in Magnus expansion [] in the following form

$$ $\displaystyle g(t)=g_{0}\operatorname{expm}(\Omega(t)),\quad\Omega(t)=\sum_{k= 1}^{\infty}\Omega_{k}(t).$ (9) $$

Here $\{\Omega_{k}\}_{k=1,\dots,\infty}$ is called the Magnus series, which is written in terms of integrals of iterated Lie algebra between $\xi(t)$ at different times. The first three terms of the Magnus series are given below to illustrate the idea,

$$ $\displaystyle\Omega_{1}(t)=\int_{0}^{t}\xi(t_{1}){\mathrm{d}}t_{1},\quad\Omega _{2}(t)=\frac{1}{2}\int_{0}^{t}\int_{0}^{t_{1}}[\xi(t_{1}),\xi(t_{2})]{\mathrm {d}}t_{2}{\mathrm{d}}t_{1}$ $\displaystyle\Omega_{3}(t)=\frac{1}{6}\int_{0}^{t}\int_{0}^{t_{1}}\int_{0}^{t_ {3}}\Big{(}\big{[}\xi(t_{1}),[\xi(t_{2}),\xi(t_{3})]\big{]}+\big{[}\xi(t_{3}), [\xi(t_{2}),\xi(t_{1})]\big{]}\Big{)}{\mathrm{d}}t_{3}{\mathrm{d}}t_{2}{ \mathrm{d}}t_{1}$ $$

In general, the solution given in (9) may not be tractable due to the fact that $\Omega(t)$ is an infinite series with increasing intricacy for each term. However, we want to discuss a special yet important case, where the infinite series is reduced to only the first term. In fact, when $G$ is an Abeliean Lie group, for any $\xi,\hat{\xi}\in\mathfrak{g}$, the Lie bracket $[\xi,\hat{\xi}]=0$ vanishes identically, and the solution to IVP in (9) reduces to $g(t)=g_{0}\exp(\int_{0}^{t}\xi(s){\mathrm{d}}s)$.

An important example of Abelian Lie group is the torus $\mathbb{T}$ or special orthogonal group $\mathsf{SO}(2)$, and any of their direct product. In this case, we can compute the conditional transition probability $p_{t|0}(g_{t},\xi_{t})$ exactly due to the capability of solving the IVP exactly. We summarize the results in Theorem 2 and leave the proof in Appendix C.

###### Theorem 2 (Conditional transition probability for Abelian Lie Group) .

Let $G$ be an Abelian Lie group which is isomorphic to $\mathbb{T}$ or $\mathsf{SO}(2)$. In this case, the conditional transition probability can be written explicitly as,

$$ $\displaystyle p_{t|0}(g_{t},\xi_{t}\mid g_{0},\xi_{0})=\operatorname{WN}( \operatorname{logm}(g_{0}^{-1}g_{t});\mu_{g},\sigma^{2}_{g})\cdot\mathcal{N}( \xi_{t};\mu_{\xi},\sigma^{2}_{\xi})$ (10) $$

where $\operatorname{WN}(x;\mu,\sigma^{2})$ is the density of the Wrapped Normal distribution with mean $\mu$ and variance $\sigma^{2}$ evaluate at $x$. For explicit expressions of $\mu_{g},\sigma^{2}_{g}$ and $\mu_{\xi},\sigma^{2}_{\xi}$ as well as formula for the multivariate case, please see Appendix C.

###### Theorem 2 (Conditional transition probability for Abelian Lie Group) .

#### Implicit Score Matching

When $G$ is not Abelian, the conditional transition probability in (10) might not be available. In this case, we resort to another computationally tractable variant of the score-matching loss derived by performing integration by parts, also known as the implicit score-matching objective $\mathcal{J}_{\operatorname{ISM}}$ []. In fact, we can connect $\mathcal{J}_{\operatorname{ISM}}$ and $\mathcal{J}_{\operatorname{SM}}$ by,

$$ $\displaystyle\mathcal{J}_{\operatorname{SM}}(\theta)=\underbrace{\mathbb{E}_{t ,p_{t}}\Big{[}\big{\|}s_{\theta}(g,\xi,t)\big{\|}^{2}+2\operatorname{div}_{\xi }(s_{\theta}(g,\xi,t))\Big{]}}_{\mathcal{J}_{\operatorname{ISM}}(\theta)}+\,C_ {2}$ $$

where $C_{2}$ is a constant independent of $\theta$. Hence, $\operatorname{argmin}_{\theta}\mathcal{J}_{\operatorname{SM}}=\operatorname{
argmin}_{\theta}\mathcal{J}_{\operatorname{DSM}}$. Computing $\mathcal{J}_{\operatorname{ISM}}$ requires evaluating the divergence with respect to $\xi$, which is the trace of the Jacobian. For high dimensional problems, a stochastic approximation of this trace with Hutchinson’s trace estimator [, ] is often employed to improve the computational efficiency.

### 3.4 Numerical Integration and Score Parameterization

To either simulate the forward dynamic for generating trajectories used for evaluating implicit score matching objective $\mathcal{J}_{\operatorname{ISM}}$ or sampling from the backward dynamic for generating new samples, we need to integrate the dynamic. To exploit the Euclidean structure of $\xi$ to achieve higher numerical accuracy, we introduce the Operator Splitting Integrator. Apart from enjoying a better prefactor in terms of numerical errors, such an integrator is also manifold-preserving and projection-free. Details of the integrator can be found in Appendix D.

#### Integrating forward dynamic

In order to numerically integrate the forward dynamic (3), we note that the dynamic can be split into the sum of two much simpler dynamics depicted in (11). This is also the approach considered in the work of Kong et.al. [].

$$ $\begin{array}[]{ccc}A^{\mathcal{F}}_{g}:\Big{\{}\begin{array}[]{ll}\dot{g_{t}} =g_{t}\xi_{t}\\ {\mathrm{d}}\xi_{t}=0\,{\mathrm{d}}t\end{array}\par&+&A^{\mathcal{F}}_{\xi}: \Big{\{}\begin{array}[]{ll}\dot{g_{t}}=0\\ {\mathrm{d}}\xi_{t}=-\gamma\xi_{t}{\mathrm{d}}t+\sqrt{2\gamma}{\mathrm{d}}W^{ \mathfrak{g}}_{t}\end{array}\end{array}$ (11) $$

While the original forward dynamic does not in general have a simple, closed-form solution for non-Abelian groups, the two smaller systems $A^{\mathcal{F}}_{g}$ and $A^{\mathcal{F}}_{\xi}$ are linear and both allow exact integration with closed-form solutions. Therefore, instead of directly integrating the forward dynamic, we can integrate $A^{\mathcal{F}}_{g}$ and $A^{\mathcal{F}}_{\xi}$ alternatively for each timestep. Another notable property of such integration is that the trajectory of this numerical integration scheme will stay exactly on the manifold $G\times\mathfrak{g}$. This avoids the use of projection operators at the end of each timestep to ensure the iterates stay on the manifold. By performing such a manifold-preserving integration technique, we not only get rid of the inaccuracy caused by projections but also greatly reduce the implementation difficulties since such projections in general do not admit a closed form.

#### Integrating backward dynamic

To perform generative modeling and sample from the backward dynamic, we can either directly work with the stochastic backward dynamic in (7) or its corresponding marginally-equivalent probability flow ODE. We discuss mainly the integrators for the stochastic dynamic and defer the discussion of probability flow ODE to Appendix D. Employing
a similar operator splitting scheme, dynamic (7) can be split into the following two simpler dynamics,

$$ $\begin{array}[]{ccc}A^{\mathcal{B}}_{g}:\Big{\{}\begin{array}[]{ll}\dot{g_{t}} =-g_{t}\xi_{t}\\ {\mathrm{d}}\xi_{t}=0\,{\mathrm{d}}t\end{array}&+&A^{\mathcal{B}}_{\xi}:\Big{ \{}\begin{array}[]{ll}\dot{g_{t}}=0\\ {\mathrm{d}}\xi_{t}=\gamma\xi_{t}{\mathrm{d}}t+2\gamma s_{\theta}(g_{t},\xi_{t },t){\mathrm{d}}t+\sqrt{2\gamma}{\mathrm{d}}W^{\mathfrak{g}}_{t}\end{array} \end{array}$ (12) $$

While $A_{g}^{\mathcal{B}}$ still allows exact integration and helps preserve the trajectory on the Lie group, $A_{g}^{\mathcal{B}}$ no longer has a closed form solution due to the nonlinearity in $s_{\theta}$. In this case, we still use exponential integrator to conduct the exact integration of the linear component and discretize the nonlinear component by using a left-point rule, i.e. pretending that $g$ and $\xi$ do not change over short time $h$.

Figure: Algorithm 1 TDM (Trivialized Diffusion Model)

#### Score parameterization

Previous work on manifold generative modeling like RFM [], RDM[], and RSGM [] often requires learning a score that belongs to the tangent space at the input, i.e., $s_{\theta}(g,t)\in T_{g}G$. This means that the score network at each input $g$ needs to adapt individually to the geometric structure at that point. One thus needs to either write explicitly the $g-$dependent isomorphism between $T_{g}G$ and $\mathbb{R}^{d}$ for each $g$, or embed $T_{g}G$ in the euclidean space $\mathbb{R}^{n}$ with $n\gg d$ and apply projections onto $T_{g}G$ to obtain a valid score. Either way, one needs to handle the geometry of $G$ and/or deal with additional approximation errors and computational costs (e.g., incurred by projections), and learn a hard object in a changing space with structural constraints.

On the other hand, since our approach only needs to approximate the score $\nabla_{\xi}\log p_{t}$, which is an element in the Lie algebra $\mathfrak{g}$, we can use a standard Euclidean-valued neural network to universally approximate $s_{\theta}$. Thanks to our use of trivialization technique, we can enjoy the already demonstrated success of learning a score in a fixed Euclidean space, where the non-Euclidean effects stemming from the Riemannian geometry are extracted and represented through the left-multiplied $g$ position variable. The need to parameterize the score function in a geometry-dependent space is completely by-passed, without any approximation in this step. Since we implicitly hardwire the geometric structure constraints in the dynamic, this greatly reduces the implementation difficulty, improves the efficiency of score representation learning, and releases the flexibility to choose score parameterization to users.

Figure: Figure 1: Visualization of Generated $\mathsf{SO}(n)$ data. $\textbf{Left: }\mathsf{SO}(4)$. $\textbf{Middle: }\mathsf{SO}(6)$. $\textbf{Right: }\mathsf{SO}(8)$
Refer to caption: x1.png

## 4 Experimental Results

We will demonstrate accurate generative modeling of Lie group data corresponding to 1) complicated or high-dim distribution on torus, 2) protein and RNA structures, 3) sophisticated synthetic datasets on possibly high-dim Special Orthogonal Group, and 4) an ensemble of quantum systems, sample-characterized by time-evolution operators of Schrodinger equation, such as for quantum oscillator with a random potential, and Random Transverse Field Ising Model (RTFIM). Details of the dataset and training set-up are discussed in Appendix G.

Figure: (a) Pacman Maze
Refer to caption: extracted/5620846/plot/pacman_gt.png

Figure: Figure 3: Visualization of Generated Time-evolution Operator of Quantum Oscillator on $\mathsf{U}(n)$. $\textbf{Left: }\mathsf{U}(4)$. $\textbf{Middle: }\mathsf{U}(6)$. $\textbf{Right: }\mathsf{U}(6)$
Refer to caption: x2.png

Figure: Figure 4: Visualization of Generated data by TDM on $4\times 4$ and $6\times 6$ checkerboard.
Refer to caption: extracted/5620846/plot/checkerboard_4_vs_6.png

Evaluation Methodology:
We adhere to the standard evaluation criterion in Riemannian generative modeling, which is Negative Log Likelihood (NLL). A consistent number of function evaluations is maintained as per prior studies. All datasets were meticulously partitioned into training and testing sets using a 9:1 ratio. Details of NLL estimation procedure are in Appendix F; note this result is not new and only for completeness, but our proof is particularly adapted to Lie group manifolds, intrinsic and independent of the choice of charts and coordinates.

Complicated and High dimensional Torus Data: Our initial comparison entailed contrasting our model with RFM [] on intricate datasets such as the checkerboard and Pacman on $\mathbb{T}^{2}$, which are discontinuous and multi-modal. Here, Pacman is a dataset newly curated by us to test generation on torus in challenging situations. It’s noted in Lou et al. [, Fig.3] that RFM produces less satisfactory results when generating complicated patterns on torus, such as the checkerboard with a size larger than $4\times 4$. We observed that RFM ran into a similar issue when learning the Pacman data, which is arguably more sophisticated. Figure 4 and Figure 2 show that our model consistently exhibited proficiency in generating intricate patterns within the torus manifold. A scalability study shown in Figure 5 confirmed our method’s good scalability to high-dimensional cases with minimal degradation in absolute performance (NLL). For the scalability study, we adopted the same setting considered in RFM and compared with its results.

Figure: Figure 5: Log likelihood $(\uparrow)$ v.s. Dimensions.
Refer to caption: x3.png

Protein/RNA Torison Angles on Torus: We also test on the popular protein [] and RNA [] datasets compiled by Huang et al. []. These datasets correspond to configurations of macro-molecules represented by torsion angles (hence non-Euclidean), which are 2D or 7D.
Results, including generated data of the protein datasets, are presented in Table 1 and Figure 7. The results of RFM were taken from [], where RDM was compared to and results of RSGM were not provided. Notably, our model outperforms the baselines by a significant margin, as evidenced by the visualizations of RNA illustrating the alignment of generated data with ground truth via density plots. The empirical results demonstrating our model’s substantial performance gains are possibly rooted in the proposed simulation-free training, high-accuracy sampling, and reduced number of approximations.

**Table 1: Test NLL $(\downarrow)$ over Protein/RNA datasets**
| Model | General (2D) | Glycine (2D) | Proline (2D) | Pre-Pro (2D) | RNA (7D) |
| --- | --- | --- | --- | --- | --- |
| Dataset size | 138208 | 13283 | 7634 | 6910 | 9478 |
| RDM [] | $1.04\pm 0.012$ | $1.97\pm 0.012$ | $0.12\pm 0.011$ | $1.24\pm 0.004$ | $-3.70\pm 0.592$ |
| RFM [] | $1.01\pm 0.025$ | $1.90\pm 0.055$ | $0.15\pm 0.027$ | $1.18\pm 0.055$ | $-5.20\pm 0.067$ |
| TDM | $\mathbf{0.69\pm 0.14}$ | $\mathbf{1.04\pm 0.27}$ | $\mathbf{-0.60\pm 0.15}$ | $\mathbf{0.52\pm 0.10}$ | $\mathbf{-6.86\pm 0.46}$ |

Figure: Figure 6: Log likelihood and visualization of generated data for $\mathsf{SO}(3)$ with $32$ mixture components.
Refer to caption: x4.png

Figure: Figure 7: Visualization of Generated Protein Torsion Angle
Refer to caption: x5.png

Special Orthogonal Group in High Dimensions: We now evaluate our model’s performance on $\mathsf{SO}(n)$ data. Notably, our model is the first reported one to successfully generate beyond $n=3$. For $\mathsf{SO}(3)$, we generate a difficult mixture distribution in the same way as in []. We also generate data for $\mathsf{SO}(n)$ with $n>3$ in a similar fashion. With trivialization technique, we bypass the need to compute the Riemannian logarithm map used in RFM training or eigenfunctions of the heat kernel on $\mathsf{SO}(n)$, which is needed by RSGM [] but in general does not admit a tractable form. The accuracy of our approach can be seen from both the visualization and NLL metric in Figure 6 and Figure 1.

Learning Time-Evolution Operators for an Ensemble of Quantum Systems: Lastly, we experiment with a complex-valued Lie group, the unitary group $\mathsf{U}(n)$. $\mathsf{U}(n)$ holds critical importance in, e.g., high energy physics [] and quantum sciences []. Our approach, arguably for the first time, tackles the generative modeling of $\mathsf{U}(n)$ data and manages to scale to nontrivial dimensions. Specifically, how quantum system evolves is encoded by a unitary operator, i.e. an element in $\mathsf{U}(n)$, and we consider training data corresponding to an ensemble of quantum systems, and aim at generating more quantum systems that are similar. Two examples are tested, respectively quantum oscillators in random potentials, and Transverse Field Ising Model with random couplings and field strength. (Spatial discretization, if needed, of) the time evolution operator of Schrödinger equation for each system gives one $\mathsf{U}(n)$ data point in the training set. Fig.3 provides marginals’ scatter plots to showcase the fidelity of our generated distributions, for Quantum Oscillators. Fig.8 is for Transverse Field Ising Model.

Figure: Figure 8: Visualization of Generated Time-evolution Operator of Transverse Field Ising Model on $\mathsf{U}(n)$. $\textbf{Left: }\mathsf{U}(4)$. $\textbf{Middle: }\mathsf{U}(8)$. $\textbf{Right: }\mathsf{U}(8)$
Refer to caption: x6.png

## 5 Limitation and Future Possibilities

The technique of trivialization does not work for general manifolds, although it is possible to extend our approach to homogeneous spaces. Meanwhile, there is significant potential for improvement in the training procedure, including the adoption of techniques such as preconditioning and exponential moving average tuning from EDM []. In addition, network architecture has not been the focus of investigation in this work, but we anticipate quantum data, for example, would benefit from specialized score parametrization other than U-Net, which suits images better.

## Appendix A Backgrounds

Diffusion Generative Model in Euclidean spaces: We first review Diffusion Generative Models (sometimes also referred to as Score-based Generative Model, denoising diffusion model, etc.) [, ]. Here, we adopt the Stochastic Differential Equations (SDE) description []. Given samples of $\mathbb{R}^{d}$-valued random variable $X_{0}$ that follows the data distribution $p_{0}$ which we are interested in, denoising diffusion adopts a forward noising process followed by a backward denoising generation process to generate more samples of $p_{0}$. The forward process transports the data distribution to a known, easy-to-sample distribution by evolving the initial condition via an SDE,

$$ $\displaystyle\mathrm{d}X_{t}=f(X_{t},t)\mathrm{d}t+\sqrt{2\gamma(t)}\mathrm{d} W_{t},\quad X_{0}\sim p_{0}.$ (13) $$

In this case, $p_{+\infty}$ will be a standard Gaussian $\mathcal{N}(0,I)$ with appropriate choice of $\gamma(t)$. The backward process then utilizes the time-reversal of the SDE (13) []. More precisely, if one considers

$$ $\displaystyle\mathrm{d}Y_{t}$ $\displaystyle=\big{(}-f(Y_{t},t)+2\gamma(t)\nabla\log p_{T-t}(Y_{t})\big{)}{ \mathrm{d}}t+\sqrt{2\gamma(T-t)}\mathrm{d}W_{t},\quad Y_{0}\sim p_{T}.$ (14) $$

Then we have $Y_{t}\sim p_{T-t}$, i.e. $Y_{t}=X_{T-t}$ in distribution. In particular, the $T$-time evolution of (14), $Y_{T}$, will follow the data distribution $p_{0}$. In practice, one considers evolving the forward dynamics for finite but large time $T$, so that $p_{T}\approx\mathcal{N}(0,I)$, and then initialize the backward dynamics using $Y_{0}\sim\mathcal{N}(0,I)$ and simulate it numerically till $t=T$ to obtain approximate samples of the data distribution. Critically, the score function $s$ needs to be estimated in the forward process.

To do so, the score $\nabla\log p_{t}$ is often approximated using a neural network $s_{\theta}$. For linear forward SDE, it is typically trained by minimizing an objective based on denoising score matching [], namely

$$ $\displaystyle\mathbb{E}_{t}\mathbb{E}_{X_{0}\sim p_{0}}\mathbb{E}_{X_{t}\sim p _{t}(\cdot|X_{0})}\|s_{\theta}(X_{t},t)-\nabla\log p_{t}(X_{t}|X_{0})\|^{2}$ (15) $$

where $\nabla\log p_{t}(X_{t}|X_{0})$ is the conditional score derived from the solution of (13) with a given initial condition.

Kinetic Langevin dynamics in Euclidean spaces, and CLD:
When Einstein first proposed ‘Brownian motion’, he actually thought of a mechanical system under additional perturbations from noise and friction []. This is now (generalized, formalized, and) known as the kinetic Langevin dynamics [e.g., ], i.e.

$$ $\begin{cases}dQ&=M^{-1}Pdt\\ dP&=-\gamma Pdt-\nabla V(Q)dt+\sigma dW_{t}\end{cases}$ (16) $$

which converges, as $t\to\infty$, to a limiting probability distribution $Z^{-1}\exp(-(P^{T}M^{-1}P/2+V(Q))/T)dPdQ$ under mild conditions, where $M$ is mass matrix that can be assumed to be $I$ without loss of generality, and $T=\sigma^{2}/(2\gamma)$ is known as the temperature. If $T$ is fixed and $\gamma\to\infty$, one recovers (in distribution and after time rescaling) overdamped Langevin dynamics

$$ $dQ=-\nabla V(Q)dt+\sqrt{2T}dW_{t}.$ $$

Just like how overdamped Langevin (often with $V$ being quadratic) can be used as the forward process for diffusion generative model, kinetic Langevin can also be used as the forward process. In fact, in a seminal paper, Dockhorn et al. [] used it to smartly bypass the singularity of score function at $t=0$ when overdamped Langevin is employed as the forward process and data is supported on a low dimensional manifold, and called the resulting method CLD. Similar to Equation 14, one can construct the reverse process of Equation 16 as follow,

$$ $\begin{cases}dQ&=-M^{-1}Pdt\\ dP&=\gamma Pdt+\nabla V(Q)dt+\sigma^{2}\nabla_{P}\log p(Q,P,t){\mathrm{d}}t+ \sigma dW_{t}.\end{cases}$ (17) $$

By endowing $P$ with Gaussian initial condition, $p$ is fully supported in $P$ space, and since the score function only takes the gradient with respect to $P$, it no longer has the aforementioned singularity issue when $t$ tends to zero. This benefits score parameterization and learning.

Finally, let us contrast the trivialized Lie group kinetic Langevin dynamics (used as forward dynamics in this work) with the classical Euclidean kinetic Langevin dynamics by setting $\sigma=\sqrt{2\gamma}$ and $M=1$:

$$ $\text{Lie:}\begin{cases}\dot{g}=g\xi,\\ {\mathrm{d}}\xi=-\gamma\xi{\mathrm{d}}t-T_{g}\mathsf{L}_{g^{-1}}(\nabla U(g)){ \mathrm{d}}t+\sqrt{2\gamma}{\mathrm{d}}W_{t}^{\mathfrak{g}},\end{cases}\text{ Euclidean:}\begin{cases}\dot{Q}=P,\\ {\mathrm{d}}P=-\gamma P{\mathrm{d}}t-\nabla U(Q){\mathrm{d}}t+\sqrt{2\gamma}{ \mathrm{d}}W_{t},\end{cases}$ $$

Note the main difference is the 1st line, i.e. the position dynamics; the 2nd line is identical except for conservative forcing, but that has to be different in the manifold case.

Lie group: A *Lie group* is a differentiable manifold that also has a group structure, denoted by $G$. A *Lie algebra* is a vector space with a bilinear, alternating binary operation that satisfies the Jacobi identity, known as the Lie bracket. The tangent space of a Lie group at $e$ (the identity element of the group) is a *Lie algebra*, denoted as $\mathfrak{g}:=T_{e}G$.

## Appendix B Time Reversal Formula

In this section, we will prove the time reversal formula stated in Theorem 1.
We first introduce an important lemma and calculate the adjoint of the infinitesimal generator of a general diffusion process. We then apply the lemma to derive the Fokker-Planck equation for our process of interest and finish the proof.
{graybox}

###### Lemma 1 .

Given $\alpha,\beta:G\times\mathfrak{g}\to\mathfrak{g}$, let $\mathcal{L}$ denote the infinitesimal generator of the following dynamic

$$ $\begin{cases}\dot{g}=T_{e}L_{g}\alpha(g,\xi){\mathrm{d}}t\\ {\mathrm{d}}\xi=\beta(g,\xi){\mathrm{d}}t+\sqrt{2\gamma(t)}{\mathrm{d}}W_{t} \end{cases}$ (18) $$

The adjoint of $\mathcal{L}$ is given by

$$ $\displaystyle\mathcal{L}^{*}p$ $\displaystyle=-\operatorname{div}_{g}(pT_{e}L_{g}\alpha)-\operatorname{div}_{ \xi}(p\beta)+\gamma(t)\Delta_{\xi}p$ $$

###### Proof of Lemma 1 .

We first write down the infinitesimal generator $\mathcal{L}$ for SDE (18). For any $f\in C_{0}^{2}(G\times\mathfrak{g})$, $\mathcal{L}$ is defined as

$$ $\displaystyle\mathcal{L}f(g,\xi):=$ $\displaystyle\lim_{\delta\rightarrow 0}\frac{\mathbb{E}\left[f(g_{\delta},\xi_ {\delta})|(g_{0},\xi_{0})=(g,\xi)\right]-f(g,\xi)}{\delta}$ $\displaystyle=$ $\displaystyle{\left\langle\nabla_{g}f,T_{e}L_{g}\alpha\right\rangle}+{\left \langle\nabla_{\xi}f,\beta\right\rangle}+\gamma(t)\Delta_{\xi}f$ $$

By definition, $\mathcal{L}^{*}:C^{2}\rightarrow C^{2}$ (the adjoint operator of $\mathcal{L}$) satisfies $\int_{G\times\mathfrak{g}}p\mathcal{L}f{\mathrm{d}}g{\mathrm{d}}\xi=\int_{G
\times\mathfrak{g}}f\mathcal{L}^{*}p\,{\mathrm{d}}g{\mathrm{d}}\xi$ for any $f,p\in C_{0}^{2}(G\times\mathfrak{g})$. By the divergence theorem, we have

$$ $\displaystyle\int_{G\times\mathfrak{g}}p\mathcal{L}f\,{\mathrm{d}}g{\mathrm{d} }\xi=\int_{G\times\mathfrak{g}}f\big{(}-\operatorname{div}_{g}(pT_{e}L_{g} \alpha)-\operatorname{div}_{\xi}(p\beta)+\gamma(t)\Delta_{\xi}p\big{)}\,{ \mathrm{d}}g{\mathrm{d}}\xi$ $$

Here $T_{e}L_{g}\xi$ stands for the left-invariant vector filed on $G$ generated by $\xi\in\mathfrak{g}$. As a result, we have

$$ $\displaystyle\mathcal{L}^{*}p$ $\displaystyle=-\operatorname{div}_{g}(pT_{e}L_{g}\alpha)-\operatorname{div}_{ \xi}\left(p\beta\right)+\gamma(t)\Delta_{\xi}p$ $$

∎

We are now ready to show that the backward dynamic (5) is the time reversal process of the forward dynamic (1). Fokker-Planck characterizes the evolution of the density of a a stochastic process: denote the density at time $t$ as $\rho_{t}$, we have $\rho_{t}$ satisfies $\frac{\partial}{\partial t}\rho_{t}=\mathcal{L}^{*}\rho_{t}$. Thm. 1 is proved by comparing the Fokker-Planck equation for Eq. (1) and (5).

###### Proof of Theorem 1 .

By denoting the density for SDE following the forward dynamic (1) as $p_{t}$ and the density for SDE following backward dynamic (5) as $\tilde{p}_{t}$, we only need to prove $p_{t}\equiv\tilde{p}_{T-t}$.

Using Lemma 1, the Fokker-Planck equation of the forward dynamic Eq. (1) is given by,

$$ $\displaystyle\frac{\partial}{\partial t}p_{t}=-\operatorname{div}_{g}(p_{t}T_{ e}L_{g}\xi)+\gamma(t)\operatorname{div}_{\xi}(p_{t}\xi)+\gamma(t)\Delta_{\xi}p _{t}$ $$

and the Fokker-Planck equation for Eq. (5) is given by

$$ $\displaystyle\frac{\partial}{\partial t}\tilde{p}_{t}$ $\displaystyle=-\operatorname{div}_{g}(-\tilde{p}_{t}T_{e}L_{g}\xi)-2\gamma \operatorname{div}_{\xi}\left(\tilde{p}_{t}\nabla_{\xi}\log\tilde{p}_{t}\right )-\gamma(T-t)\operatorname{div}_{\xi}(\tilde{p}_{t}\xi)+\gamma(T-t)\Delta_{\xi }\tilde{p}_{t}$ $\displaystyle=\operatorname{div}_{g}(\tilde{p}_{t}T_{e}L_{g}\xi)-\gamma(T-t) \operatorname{div}_{\xi}(\tilde{p}_{t}\xi)-\gamma(T-t)\Delta_{\xi}\tilde{p}_{t}$ $$

where the last equation holds due to $\operatorname{div}_{\xi}\left(\tilde{p}_{t}\nabla_{\xi}\log\tilde{p}_{t}\right
)=\operatorname{div}_{\xi}\left(\nabla_{\xi}\tilde{p}_{t}\right)=\Delta_{\xi}p
_{t}$.
We can calculate the partial derivative of the reversed distribution $\tilde{p}_{T-t}$ with respect to $t$, which gives

$$ $\displaystyle\frac{\partial}{\partial t}\tilde{p}_{T-t}$ $\displaystyle=-\operatorname{div}_{g}(\tilde{p}_{t}T_{e}L_{g}\xi)+\gamma(t) \operatorname{div}_{\xi}(\tilde{p}_{t}\xi)+\gamma(t)\Delta_{\xi}\tilde{p}_{t}$ $$

Note that this exactly matches the expression for $\frac{\partial}{\partial t}p_{t}$. Together with the same initial condition $\tilde{p}_{0}=p_{T}$, we deduce that $p_{t}\equiv\tilde{p}_{T-t}$ for all $t$.
∎

###### Lemma 1 .

###### Proof of Lemma 1 .

###### Proof of Theorem 1 .

## Appendix C Denoising Score Matching for Abelian Lie group

We first state a detailed version of Theorem 2 in the following,
{graybox}

###### Corollary 1 (Conditional transition probability for Abelian Lie Group) .

Let $G$ be an Abelian Lie group which is isomorphic to $\mathbb{T}$ or $\mathsf{SO}(2)$. In this case, the conditional transition probability can be written explicitly as,

$$ $\displaystyle p_{t|0}(g_{t},\xi_{t}\mid g_{0},\xi_{0})=\operatorname{WN}( \operatorname{logm}(g_{0}^{-1}g_{t});\mu_{g},\sigma^{2}_{g})\cdot\mathcal{N}( \xi_{t};\mu_{\xi},\sigma^{2}_{\xi})$ (19) $$

where $\operatorname{WN}(x;\mu,\sigma^{2})$ is the density of the Wrapped Normal distribution with mean $\mu$ and variance $\sigma^{2}$ evaluate at $x$, $\operatorname{logm}$ is the matrix logarithm with principal root, and $\mu_{g},\mu_{\xi},\sigma^{2}_{g},\sigma^{2}_{\xi}$ are are given by,

$$ $\displaystyle\mu_{g}=\frac{1-e^{-t}}{1+e^{-t}}(\xi_{t}+\xi_{0}),\;\mu_{\xi}=e^ {-t}\xi_{0}$ $\displaystyle\sigma_{g}^{2}=2t+\frac{8}{e^{t}+1}-4,\;\sigma_{\xi}^{2}=1-e^{-2t}$ $$

In this section, we will prove Theorem 2 by proving its detailed version in Corollary 1, under the condition that $G$ is an Abelian Lie group which is isomorphic to $\mathbb{T}$ or $\mathsf{SO}(2)$. Note that this allows us to compute the conditional transition probability for $G$ that is also direct product of these Lie groups. The reason is that, for a Lie group $G$ that is a direct product of $\mathbb{T}$ and $\mathsf{SO}(2)$, we can represent an element in $G$ as $(g^{1},\dots,g^{k})$ where $g^{i}\in\mathbb{T}$ or $\mathsf{SO}(2)$. The corresponding Lie Algebra can be represented as $(\xi^{1},\dots,\xi^{k})$. For each $1\leq i\leq k$, $(g^{i},\xi^{i})$, they follow the following dynamic,

$$ $\begin{cases}\dot{g^{i}_{t}}=g_{t}\xi^{i}_{t},\\ {\mathrm{d}}\xi^{i}_{t}=-\gamma\xi_{t}{\mathrm{d}}t+\sqrt{2\gamma}{\mathrm{d}} W^{i}_{t}\\ g^{i}(0)=g^{i}_{0},\;\xi^{i}(0)=\xi_{0}^{i}\end{cases}$ (20) $$

Note that this will not create any confusion since $\xi^{i}\in\mathbb{R}$ and there’s no need for another superscript to indicate other elements in $\xi^{i}$. Moreover, an important consequence of the factorization of the dynamic of $(g,\xi)$ as $k$ independent smaller dynamic is that we can also factorize the conditional transition probability for $g,\xi$ as a product of $k$ conditional transition probability, each computed from $g^{i},\xi^{i}$. This means the following,

$$ $\displaystyle p_{t|0}(g_{t},\xi_{t}\mid g_{0},\xi_{0})=\prod_{i=1}^{k}p_{t|0}( g^{i}_{t},\xi^{i}_{t}\mid g^{i}_{0},\xi^{i}_{0})$ (21) $$

Based on (21), we manage to compute the conditional transition probability of any general connected compact Abelian Lie group, since they are necessarily isomorphic to a power of $\mathbb{T}$ or $\mathsf{SO}(2)$[]. Therefore, we just need to compute the conditional transition probability for such a base case, which is stated in Theorem 2.

From now on, we will consider $\gamma=1$ for simplicity. Generalization of our results to time-dependent is straightforward.
Let’s stick to the notation that $g_{0}\in\mathsf{SO}(2)$, $\xi_{0}\in\mathbb{R}\cong\mathfrak{so}(2)$. We slightly abuse the notation in the sense that, when considering the dynamic for $\xi$, we are considering a valid SDE on $\mathbb{R}$, while when we are considering the dynamic for $g$, $\xi$ should be understood as its matrix representation in $\mathfrak{so}(2)$, which is a $2\times 2$ skew-symmetric matrix. Let also denote $Y_{t}=\int_{0}^{t}\xi_{s}{\mathrm{d}}s$ for notational simplicity.

Since $G$ is Abelian, $[\xi,\hat{\xi}]=0$ for any $\xi,\hat{\xi}\in\mathfrak{g}$, the Magnus series $\Omega_{k}(t)=0$ in (9) for $k\geq 2$, and the solution to the IVP can be written explicitly as,

$$ $\begin{cases}g_{t}=g_{0}\operatorname{expm}(Y_{t})\\ \xi_{t}=e^{-t}\xi_{0}+\sqrt{2}\int_{0}^{t}e^{-(t-s)}{\mathrm{d}}W_{s}\end{cases}$ (22) $$

Notice that $(g_{t},\xi_{t})$ is a push forward of $(\xi_{t},Y_{t})$. Therefore, to find the condition transition $p_{t|0}(g_{t},\xi_{t}\mid g_{0},\xi_{0})$, we first compute the joint distribution of $(\xi_{t},Y_{t})$ conditioned on the $(g_{0},\xi_{0})$, and derive the desired conditional transition probability by computing the probability change of variable.

Since $Y_{t}$ is the time integral of $\xi_{t}$, $(\xi_{t},Y_{t})$ is a Gaussian process, with mean and covariance stated in the following Lemma,

###### Lemma 2 .

For a given $t$, $(\xi_{t},Y_{t})$ is distributed according to a bivariate Gaussian,

$$ $\displaystyle\begin{pmatrix}\xi_{t}\\ Y_{t}\end{pmatrix}\sim\mathcal{N}\left(\begin{pmatrix}e^{-t}\xi_{0}\\ (1-e^{-t})\xi_{0}\end{pmatrix},\begin{pmatrix}1-e^{-2t}&e^{-2t}(e^{t}-1)^{2}\\ e^{-2t}(e^{t}-1)^{2}&4e^{-t}-e^{2t}+2t-3\end{pmatrix}\right)$ (23) $$

###### Proof of Lemma 2 .

To show that the joint distribution $(\xi_{t},Y_{t})$ as the desired expression, we just need to compute the mean and variance of $\xi_{t}$ and $Y_{t}$ respectively, as well as their covariance.
For $\xi_{t}$,

$$ $\displaystyle\mathbb{E}[\xi_{t}]=\mathbb{E}\Big{[}e^{-t}\xi_{0}+\sqrt{2}\int_{ 0}^{t}e^{-(t-s)}{\mathrm{d}}W_{s}\Big{]}=e^{-t}\xi_{0}$ $\displaystyle\operatorname{Var}(\xi_{t})=\operatorname{Var}\Big{(}\sqrt{2}\int _{0}^{t}e^{-(t-s)}{\mathrm{d}}W_{s}\Big{)}=2\int_{0}^{t}e^{-2(t-s)}{\mathrm{d} }s=1-e^{-2t}$ $$

For $Y_{t}$, since it’s the integration of $\xi_{t}$, it has the following expression,

$$ $\displaystyle Y_{t}$ $\displaystyle=\int_{0}^{t}e^{-s}\xi_{0}{\mathrm{d}}s+\sqrt{2}\int_{0}^{t}\int_ {0}^{p}e^{-(p-s)}{\mathrm{d}}W_{s}{\mathrm{d}}p$ $\displaystyle=(1-e^{-t})\xi_{0}+\sqrt{2}\int_{0}^{t}\int_{s}^{t}e^{-(p-s)}{ \mathrm{d}}p{\mathrm{d}}W_{s}$ $\displaystyle=(1-e^{-t})\xi_{0}+\sqrt{2}\int_{0}^{t}(1-e^{-(t-s)}){\mathrm{d}} W_{s}$ $$

where we use Stochastic Fubini’s theorem to exchange the integration order of ${\mathrm{d}}W_{s}$ and ${\mathrm{d}}p$.
Therefore, we can compute the mean and variance of $Y_{t}$,

$$ $\displaystyle\mathbb{E}[Y_{t}]=\mathbb{E}\Big{[}(1-e^{-t})\xi_{0}+\sqrt{2}\int _{0}^{t}(1-e^{-(t-s)}){\mathrm{d}}W_{s}\Big{]}=(1-e^{-t})\xi_{0}$ $\displaystyle\operatorname{Var}(Y_{t})=\operatorname{Var}\Big{(}\sqrt{2}\int_{ 0}^{t}(1-e^{-(t-s)}){\mathrm{d}}W_{s}\Big{)}=2\int_{0}^{t}(1-e^{-(t-s)})^{2}{ \mathrm{d}}s=4e^{-t}-e^{2t}+2t-3$ $$

Finally, we need to compute $\operatorname{Cov}(\xi_{t},Y_{t})$ to complete the proof, where we use Ito’s isometry,

$$ $\displaystyle\operatorname{Cov}(\xi_{t},Y_{t})$ $\displaystyle=\mathbb{E}\Big{[}2\int_{0}^{t}e^{-(t-s)}{\mathrm{d}}W_{s}\cdot \int_{0}^{t}(1-e^{-(t-s)}){\mathrm{d}}W_{s}\Big{]}$ $\displaystyle=2\int_{0}^{t}e^{-(t-s)}\cdot(1-e^{-(t-s)}){\mathrm{d}}s=e^{-2t}( e^{t}-1)^{2}$ $$

∎

As a corollary of Lemma 2, we can compute the conditional distribution $Y_{t}|\xi_{t}$, here we omit the dependence on $\xi_{0},g_{0}$ for simplicity since all probability considered in this section is conditioned on these two value. The conditional distribution between bivariate Gaussian is equivalent to orthogonal projections,
{graybox}

###### Corollary 2 .

For a give $t$, $Y_{t}|\xi_{t}$ has distribution

$$ $\displaystyle Y_{t}|\xi_{t}\sim\mathcal{N}\Big{(}\frac{1-e^{-t}}{1+e^{-t}}(\xi _{t}+\xi_{0}),2t+\frac{8}{e^{t}+1}-4\Big{)}$ (24) $$

###### Proof.

Let $\boldsymbol{\Sigma}$ and $\boldsymbol{\mu}$ denotes the variance matrix and the mean vector of $(\xi_{t},Y_{t})$. The $Y_{t}|\xi_{t}$ has conditional mean and variance given by,

$$ $\displaystyle\mathbb{E}[Y_{t}|\xi_{t}]=\boldsymbol{\mu}_{Y}+\boldsymbol{\Sigma }_{Y\xi}\boldsymbol{\Sigma}_{\xi\xi}^{-1}(\xi_{t}-\boldsymbol{\mu}_{\xi})$ $\displaystyle\operatorname{Var}[Y_{t}|\xi_{t}]=\boldsymbol{\Sigma}_{YY}- \boldsymbol{\Sigma}_{Y\xi}\boldsymbol{\Sigma}_{\xi\xi}^{-1}\boldsymbol{\Sigma} _{\xi Y}$ $$

Plug in the expression for $\boldsymbol{\Sigma}$ and $\boldsymbol{\mu}$, and the expressions simplify to the desired ones.
∎

We need the distribution of $Y_{t}|\xi_{t}$ due to the following factorization of $p_{t|0}(g_{t},\xi_{t}\mid g_{0},\xi_{0})$,

$$ $\displaystyle p_{t|0}(g_{t},\xi_{t}\mid g_{0},\xi_{0})=p_{t|0}(g_{t}\mid\xi_{t },g_{0},\xi_{0})\cdot p_{t|0}(\xi_{t}\mid g_{0},\xi_{0})$ $$

Here $p_{t|0}(\xi_{t}\mid g_{0},\xi_{0})$ is known due to $\xi_{t}$ being a Gaussian, we need to compute $p_{t|0}(g_{t}\mid\xi_{t},g_{0},\xi_{0})$, which is a hard object since it’s a distribution on the Lie group $G$. However, we can derive its expression by computing the push-forward of $Y_{t}|\xi_{t},g_{0},\xi_{0}$ by the exponential map. The following theorem characterizes such a change of measure given by the exponential map of Lie group as the push-forward,
{graybox}

###### Theorem 3 (Theorem 3.1 in Falosi et al. [ 13 ] ) .

Let $G$ denotes a Lie group and $\mathfrak{g}$ its Lie algebra. Consider a distribution $m$ on $\mathfrak{g}$ with density $r(\xi)$ with respect to the Lebesgue measure on $\mathfrak{g}$, the push-forward of $m$ to $G$, denoted as $\operatorname{exp}_{*}(m)$ is absolutely continuous with respect to the Haar measure on $G$, with density $p(g)$ given by,

$$ $\displaystyle p(g)=\underset{\xi\in\mathfrak{g}:\,\operatorname{expm}(\xi)=g}{ \sum}r(\xi)|J(\xi)|^{-1},\quad g\in G,$ $$

where $J(\xi)=\operatorname{det}\Big{(}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(k+1)!}(
\operatorname{ad}_{\xi})^{k}\Big{)}$.
Moreover, when $G$ is $\mathsf{SO}(2)$ or $\mathbb{T}$, the scenario simplifies to,

$$ $\displaystyle J(\xi)=1,\forall\xi\in\mathfrak{g}$ $\displaystyle\{\xi\in\mathfrak{g}:\,\operatorname{expm}(\xi)=g\}=\{\xi\in \mathfrak{g}:\,\xi=\operatorname{logm}(g)\pm 2k\pi,k\in\mathbb{Z}\}$ $$

Since the exponential map is not injective, computing the density of the push-forwarded measure at $g\in G$ requires summing over density at all the pre-images $\xi$ of $g$ in $\mathfrak{g}$ and weighted by the inverse of the Jacobian $|J(\xi)|$. Fortunately, for our considered case, both the pre-images and the Jacobian can be explicitly characterized and computed.

Applying Theorem 3 to our case, where $r$ is the density of $Y_{t}|\xi_{t},g_{0},\xi_{0}$, we derive the following expression for $p_{t|0}(g_{t}\mid\xi_{t},g_{0},\xi_{0})$,

$$ $\displaystyle p_{t|0}(g_{t}\mid\xi_{t},g_{0},\xi_{0})=\sum_{k=-\infty}^{\infty }p_{Y_{t}|\xi_{t}}\big{(}\operatorname{logm}(g_{0}^{-1}g_{t})+2k\pi\big{)}$ (25) $$

where $p_{Y_{t}|\xi_{t}}$ denotes the density of $Y_{t}|\xi_{t}$, which is the density of a Gaussian with mean and variance defined in (24). This is also known as the Wrapped Normal distribution [], its name comes from the fact that the density is generated by "wrapping a distribution" on a circle. Therefore, we denote such a distribution with notation $\operatorname{WN}(x;\mu,\sigma^{2})$ denotes such as density evaluated at point $x$, where $\mu,\sigma^{2}$ is the mean and variance of the normal distribution being wrapped. This finishes the proof of Theorem 2 and Corollary 1.

###### Corollary 1 (Conditional transition probability for Abelian Lie Group) .

###### Lemma 2 .

###### Proof of Lemma 2 .

###### Corollary 2 .

###### Proof.

###### Theorem 3 (Theorem 3.1 in Falosi et al. [ 13 ] ) .

## Appendix D Proability Flow ODE and Operator Splitting Integrator

In this section, we will first discuss in detail the Operator Splitting Integrator (OSI) and how they help integrate the forward and backward trivialized kinetic dynamics accurately in a manifold-preserving, projection-free manner. We then introduce the probability flow ODE of the backward dynamic, which is an ODE that is marginally equivalent to dynamic (5). We will also introduce the OSI for the probability flow ODE.

### D.1 Operator Splitting Integrator

In this section, we will demonstrate how OSIs are constructed from the dynamics and the benefits they enjoy. We restrict our attention to first-order numerical integrators in the following discussion. However, such an approach can be generalized and we can indeed craft an OSI with arbitrary order of accuracy by following the approach in Tao and Ohsawa [].

Forward Integrator: Recall that the forward dynamic (3) can be split into two smaller dynamics,

$$ $\begin{array}[]{ccc}A^{\mathcal{F}}_{g}:\Big{\{}\begin{array}[]{ll}\dot{g_{t}} =g_{t}\xi_{t}\\ {\mathrm{d}}\xi_{t}=0\,{\mathrm{d}}t\end{array}&+&A^{\mathcal{F}}_{\xi}:\Big{ \{}\begin{array}[]{ll}\dot{g_{t}}=0\\ {\mathrm{d}}\xi_{t}=-\gamma\xi_{t}{\mathrm{d}}t+\sqrt{2\gamma}{\mathrm{d}}W^{ \mathfrak{g}}_{t}\end{array}\end{array}$ $$

While the original dynamic does not admit a simple closed-form solution, $A^{\mathcal{F}}_{g}$ and $A^{\mathcal{F}}_{\xi}$ can be solved explicitly as is shown in the following equations,

$$ $\displaystyle A^{\mathcal{F}}_{g}:\;g(t)=g(0)\operatorname{expm}(t\xi(0)),\; \xi(t)=\xi(0)$ $\displaystyle A^{\mathcal{F}}_{\xi}:\;g(t)=g(0),\xi(t)=\exp(-\gamma t)\xi(0)+ \int_{0}^{t}\sqrt{2\gamma}\exp(-\gamma(t-s)){\mathrm{d}}W_{s}^{\mathfrak{g}}$ $$

Therefore, we can integrate $A^{\mathcal{F}}_{g}$ and $A^{\mathcal{F}}_{\xi}$ alternatively for each timestep $h$ in order to integrate the original forward dynamic. The detailed algorithm can be found in Algorithm 2.

Backward Integrator: Recall that the backward dynamic (12) can be split into two smaller dynamics,

$$ $\begin{array}[]{ccc}A^{\mathcal{B}}_{g}:\Big{\{}\begin{array}[]{ll}\dot{g_{t}} =-g_{t}\xi_{t}\\ {\mathrm{d}}\xi_{t}=0\,{\mathrm{d}}t\end{array}&+&A^{\mathcal{B}}_{\xi}:\Big{ \{}\begin{array}[]{ll}\dot{g_{t}}=0\\ {\mathrm{d}}\xi_{t}=\gamma\xi_{t}{\mathrm{d}}t+2\gamma s_{\theta}(g_{t},\xi_{t },t){\mathrm{d}}t+\sqrt{2\gamma}{\mathrm{d}}W^{\mathfrak{g}}_{t}\end{array} \end{array}$ $$

Again, we can write out explicit solutions to $A^{\mathcal{B}}_{g}$ and $A^{\mathcal{B}}_{\xi}$ in the following equations,

$$ $\displaystyle A^{\mathcal{B}}_{g}:\;g(t)=g(0)\operatorname{expm}(-t\xi(0)),\; \xi(t)=\xi(0)$ $\displaystyle A^{\mathcal{B}}_{\xi}:\;g(t)=g(0),\xi(t)=\exp(\gamma t)\xi(0)+ \int_{0}^{t}\sqrt{2\gamma}\exp(\gamma(t-s)){\mathrm{d}}W_{s}^{\mathfrak{g}}+2 \gamma\int_{0}^{t}\exp(\gamma(t-s))s_{\theta}(g_{s},\xi_{s},s){\mathrm{d}}s$ $$

Note that, the solution to $A^{\mathcal{F}}_{\xi}$, though presented in an explicit form, can’t be implemented in practice since we can not integrate exactly the neural network $s_{\theta}$. We employ an approximation here and discretize the nonlinear component $s_{\theta}$ by using a left-point rule, i.e. pretending that $g$ and $\xi$ do not change over a short time $h$, and still use the exponential integration technique to conduct the exact integration of the rest of the linear dynamic. The detailed algorithm can be found in Algorithm 3.

Figure: Algorithm 2 Forward Operator Splitting Integration (FOSI)

Figure: Algorithm 3 Backward Operator Splitting Integration (BSOI)

Advantages of OSI: The benefits of using an OSI for integration are threefold.

(1) The first benefit of the OSI is high numerical accuracy. In both the forward and backward dynamic, the linear component of the dynamic is integrated exactly due to the use of the exponential integration technique. This implies that, while OSI is still a first-order method in terms of the order of errors, it enjoys a smaller prefactor thanks to the reduction in error source compared with the Euler–Maruyama method (EM).

(2) The second benefit of OSI is that the trajectories generated stay on the manifold $G\times g$ for the whole time, $(g(kh),\xi(kh))\in G\times\mathfrak{g}$ for any $k\geq 0$. If we use the EM scheme to integrate the Riemannian component of the dynamic, which is the $g$ dynamic, we would arrive at iterates $g((k+1)h)=g(kh)+h\cdot g(kh)\xi(kh)$. Note that since $g(kh)\xi(kh)$ is in the tangent space of Lie group G at point $g(kh)$, moving arbitrary short time $h$ along such direction would result in $g((k+1)h)\notin G$. Therefore, to achieve a valid trajectory on $G$, we need to employ a projection $\pi_{G}$ onto the manifold, which causes additional numerical errors apart from the time discretization error. If employing OSI, we will be free from such a concern of leaving the manifold and also the projection errors.

(3) The third benefit of OSI is that the numerical scheme is projection-free. As we have discussed in point (2), EM method does not respect the Riemannian geometry structure of the Lie group and constantly requires the application of projections to achieve valid iterates. Apart from the numerical error, computing such a projection could be problematic. In general, Lie groups live in a nonconvex set, which naturally raises concerns about the existence of the closed-form formula for such projections and more generally, how to implement them in a fast algorithm. For example, $\mathsf{SO}(n)$ is the set of matrices satisfies $\{X\in\mathbb{R}^{n\times n}\mid X^{\top}X=XX^{\top}=I_{n}\}$, which is characterized by nonlinear constraints. Therefore, finding out the projection onto these Lie groups requires heavy work and needs to be investigated on a case-by-case basis. Not to mention the possibility that these projection functions could be difficult to implement and require heavy computational resources, which is certainly not scalable for large-scale applications and high-dimensional tasks. On the contrary, by employing OSI, we can enjoy a projection-free numerical algorithm and reduce the complexity of both training and generation.

### D.2 Probability Flow ODE

In this section, we will introduce the OSI for probability flow ODE. We recall that the probability flow ODE is given by,

$$ $\begin{cases}\dot{g_{t}}=-g_{t}\xi_{t},\\ {\mathrm{d}}\xi_{t}=\gamma(T-t)\xi_{t}{\mathrm{d}}t+\gamma\nabla_{\xi}\log p_{ T-t}(g_{t},\xi_{t}){\mathrm{d}}t\end{cases}$ $$

Similar to the SDE setting, this can be split into two smaller dynamics,

$$ $\begin{array}[]{ccc}A^{\mathcal{P}}_{g}:\Big{\{}\begin{array}[]{ll}\dot{g_{t}} =-g_{t}\xi_{t}\\ {\mathrm{d}}\xi_{t}=0\,{\mathrm{d}}t\end{array}&+&A^{\mathcal{P}}_{\xi}:\Big{ \{}\begin{array}[]{ll}\dot{g_{t}}=0\\ {\mathrm{d}}\xi_{t}=\gamma\xi_{t}{\mathrm{d}}t+\gamma s_{\theta}(g_{t},\xi_{t} ,t){\mathrm{d}}t\end{array}\end{array}$ $$

Similar to the Backward Operator Splitting Integrator (BSOI), we employ an approximation of the $\xi$ dynamic, discretize the nonlinear component $s_{\theta}$ by using a left-point rule, i.e. pretending that $g$ and $\xi$ do not change over a short time $h$, and use the exponential integration technique to conduct the exact integration of the rest of the linear dynamic. The details can be found in Algorithm 4.

Figure: Algorithm 4 Probability Flow ODE

## Appendix E Evolution of KL divergence

In this section, we will discuss the effectiveness of likelihood training in terms of learning the correct data distribution. We will show that the KL divergence between the true data distribution $p_{0}$ and the learned data distribution can be bounded by accumulated score-matching errors up to an additional discrepancy error caused by a mismatch in the initial condition.

Recall that $\mathbf{X_{t}}=(g_{t},\xi_{t})$ is the trajectory of the forward dynamics (3), with $\mathbf{X}_{t}$ admitting a smooth density $p_{t}(g_{t},\xi_{t})$ with respect to the product of Haar measure on $G$ and Lebesgue measure on $\mathfrak{g}$. Let’s denote $(q_{t})_{t\in[0,T]}=(p_{T-t})_{t\in[0,T]}$. Note that by construction, $q_{0}=p_{T}\approx\pi^{*}$ when $T$ is large, where $\pi^{*}$ is the invariant distribution of the forward dynamic (3). Also, $q_{T}=p_{0}$ is the initial condition for the forward dynamic, which in practice is the (partially unknown) joint data distribution on $g$ and $\xi$. Recall that we have denoted the sequence of probability distribution $q^{\theta}_{t}$ as the density of $\mathcal{L}(\mathbf{Y}_{t}^{\theta})$ with respect to the reference measure, where $\mathbf{Y}_{t}^{\theta}$ is the trajectory of the learned backward dynamic in (7).

We have Theorem 4 regarding the KL divergence between the learnt data distribution $q_{T}^{\theta}$ and the true data distribution $p_{0}$,
{graybox}

###### Theorem 4 .

Let $\mathbf{Y}_{t}^{\theta}$ be the trajectory of the learnt backward dynamic (7) under initial condition $\mathbf{Y}_{0}^{\theta}=\pi_{*}$, $\mathbf{Y}_{t}^{\theta}$ has density $(q^{\theta}_{t})_{t\in[0,T]}$. When the score is given by $s_{\theta}(g,\xi,t):=\nabla_{\xi}\log\hat{q}_{T-t}(g,\xi,t)$, where $(\hat{q}_{t})_{t\in[0,T]}$ is the density of the forward dynamic (3) under initial condition $\hat{q}_{0}$ and satisfies $\hat{q}_{T}=\pi_{*}$. We then have

$$ $\displaystyle{D_{\operatorname{KL}}\left(p_{0}\parallel q_{T}^{\theta}\right)} =\int_{0}^{T}\int_{G\times\mathfrak{g}}p_{T-t}(g,\xi){\left\|\nabla_{\xi}\log p _{T-t}(g,\xi)-s_{\theta}(g,\xi,t)\right\|}^{2}{\mathrm{d}}g{\mathrm{d}}\xi{ \mathrm{d}}t+D_{\operatorname{KL}}(p_{T}\parallel\pi_{*})$ $$

In order to prove Theorem 4, we need the following Lemma that characterizes the time derivative of the KL divergence between two sequences of probability distributions that corresponds to the time marginal of the same SDE with different initial conditions.

###### Lemma 3 .

Given $p_{0},q_{0}$ two distributions on $G\times g$. We denote the sequence of probability distributions $(p_{t}){t\geq 0}$ and $(q_{t}){t\geq 0}$ the marginals of the forward dynamic (3) with initial conditions $p_{0}$ and $q_{0}$ respectively. Then, $p_{t}$ and $q_{t}$ satisfies

$$ $\displaystyle\frac{\partial}{\partial t}D_{\operatorname{KL}}(p_{t}\parallel q _{t})=\int_{G\times\mathfrak{g}}\gamma(t)p_{t}{\left\|\nabla_{\xi}\log p_{t}- \nabla_{\xi}\log q_{t}\right\|}^{2}{\mathrm{d}}g{\mathrm{d}}\xi$ $$

where $\nabla_{\xi}$ is the gradient w.r.t. $\xi$.

Lemma 3 relates the time derivative of KL divergence between two distributions to the difference in their score integrated over the manifold $G\times g$. To prove this lemma, we need to derive the Fokker-Planck equation for $p_{t}$ and $q_{t}$ respectively.

###### Proof of Lemma 3 .

We prove a general case and consider the general form of forward dynamic described in (13). The evolution of $p_{t}$ and $q_{t}$ can be characterized by following Fokker-Plank equations,

$$ $\displaystyle\frac{\partial}{\partial t}p_{t}=\mathcal{L}^{*,p}p_{t}=- \operatorname{div}_{g}(p_{t}T_{e}L_{g}\xi)+\gamma(t)\left(\operatorname{div}_{ \xi}(p_{t}\xi)+\Delta_{\xi}p_{t}\right)$ $\displaystyle\frac{\partial}{\partial t}q_{t}=\mathcal{L}^{*,q}q_{t}=- \operatorname{div}_{g}(q_{t}T_{e}L_{g}\xi)+\gamma(t)\left(\operatorname{div}_{ \xi}(q_{t}\xi)+\Delta_{\xi}q_{t}\right)$ $$

where $\operatorname{div}_{g}$ is the divergence of vector fields on $G$ under the left-invariant metric we choose, $\operatorname{div}_{\xi}$ is the divergence in $\mathfrak{g}$ and $\Delta_{\xi}$ is the Laplace operator on $\mathfrak{g}$. They are well-defined since $\mathfrak{g}$ is a linear space. Consequently, we can evaluate the time derivative of KL divergence between $p_{t}$ and $q_{t}$, where the integration is performed over $G\times g$ unless specifically noted,

$$ $\displaystyle\frac{\partial}{\partial t}D_{\operatorname{KL}}(p_{t}\parallel q _{t})$ $\displaystyle=\frac{\partial}{\partial t}\Big{(}\int p_{t}\log\frac{p_{t}}{q_{ t}}{\mathrm{d}}g{\mathrm{d}}\xi\Big{)}$ $\displaystyle=\int\frac{\partial p_{t}}{\partial t}\log\frac{p_{t}}{q_{t}}\,{ \mathrm{d}}g{\mathrm{d}}\xi-\int\frac{\partial q_{t}}{\partial t}\frac{p_{t}}{ q_{t}}\,{\mathrm{d}}g{\mathrm{d}}\xi$ $\displaystyle=\int\left(\log\frac{p_{t}}{q_{t}}\right)\mathcal{L}^{*,p}p_{t}{ \mathrm{d}}g{\mathrm{d}}\xi-\int\left(\frac{p_{t}}{q_{t}}\right)\mathcal{L}^{* ,q}q_{t}{\mathrm{d}}g{\mathrm{d}}\xi$ $$

Using the divergence theorem, we have for any smooth function $f:G\times\mathfrak{g}\to\mathbb{R}$, we have

$$ $\displaystyle\int f\cdot\mathcal{L}^{*,p}p_{t}{\mathrm{d}}g{\mathrm{d}}\xi= \int{\left\langle\nabla_{g}f,p_{t}T_{e}\mathsf{L}_{g}\xi\right\rangle}+{\left \langle\nabla_{\xi}f,p_{t}\gamma(t)\left(\xi+\nabla_{\xi}\log p_{t}\right) \right\rangle}{\mathrm{d}}g{\mathrm{d}}\xi$ $$

Similar results holds for $\mathcal{L}^{*,q}$. As a consequence, applying the previous equation with $f=\log\frac{p_{t}}{q_{t}}$ and $f=\frac{p_{t}}{q_{t}}$ respectively, we can write,

$$ $\displaystyle\frac{\partial}{\partial t}D_{\operatorname{KL}}(p_{t}\parallel q _{t})$ $\displaystyle=\int{\left\langle\nabla_{g}\log\frac{p_{t}}{q_{t}},p_{t}T_{e} \mathsf{L}_{g}\xi\right\rangle}+{\left\langle\nabla_{\xi}\log\frac{p_{t}}{q_{t }},p_{t}\gamma(t)\left(\xi+\nabla_{\xi}\log p_{t}\right)\right\rangle}{\mathrm {d}}g{\mathrm{d}}\xi$ $\displaystyle\qquad-\int{\left\langle\nabla_{g}\frac{p_{t}}{q_{t}},q_{t}T_{e} \mathsf{L}_{g}\xi\right\rangle}-{\left\langle\nabla_{\xi}\frac{p_{t}}{q_{t}},q _{t}\gamma(t)\left(\xi+\nabla_{\xi}\log q_{t}\right)\right\rangle}{\mathrm{d}} g{\mathrm{d}}\xi$ $\displaystyle=\int{\left\langle\nabla_{g}\frac{p_{t}}{q_{t}},q_{t}T_{e}\mathsf {L}_{g}\xi\right\rangle}+{\left\langle\nabla_{\xi}\frac{p_{t}}{q_{t}},q_{t} \gamma(t)\left(\xi+\nabla_{\xi}\log p_{t}\right)\right\rangle}{\mathrm{d}}g{ \mathrm{d}}\xi$ $\displaystyle\qquad-\int{\left\langle\nabla_{g}\frac{p_{t}}{q_{t}},q_{t}T_{e} \mathsf{L}_{g}\xi\right\rangle}-{\left\langle\nabla_{\xi}\frac{p_{t}}{q_{t}},q _{t}\gamma(t)\left(\xi+\nabla_{\xi}\log q_{t}\right)\right\rangle}{\mathrm{d}} g{\mathrm{d}}\xi$ $\displaystyle=\int\gamma(t){\left\langle\nabla_{\xi}\frac{p_{t}}{q_{t}},q_{t} \left(\nabla_{\xi}\log p_{t}-\nabla_{\xi}\log q_{t}\right)\right\rangle}{ \mathrm{d}}g{\mathrm{d}}\xi$ $\displaystyle=\int\gamma(t)p_{t}\big{\|}\nabla_{\xi}\log p_{t}-\nabla_{\xi} \log q_{t}\big{\|}^{2}{\mathrm{d}}g{\mathrm{d}}\xi$ $$

This finishes the proof of the desired Lemma.
∎

We are now ready to present proof for Theorem 4. Note that under the conditions on $s_{\theta}$, in fact $q_{t}^{\theta}=\hat{q}_{T-t}$. We just need to apply Lemma 3 between $p_{t}$ and $\hat{q}_{t}$ to conclude.

###### Proof of Thm. 4 .

Lemma 3 gives

$$ $\displaystyle{D_{\operatorname{KL}}\left(p_{0}\parallel q_{T}^{\theta}\right)}$ $\displaystyle={D_{\operatorname{KL}}\left(p_{0}\parallel\hat{q}_{0}\right)}$ $\displaystyle={D_{\operatorname{KL}}\left(p_{T}\parallel\hat{q}_{T}\right)}+ \int_{0}^{T}\frac{\partial}{\partial t}{D_{\operatorname{KL}}\left(p_{t} \parallel\hat{q}_{t}\right)}{\mathrm{d}}t$ $\displaystyle={D_{\operatorname{KL}}\left(p_{T}\parallel\hat{q}_{T}\right)}+ \int_{0}^{T}\int_{G\times\mathfrak{g}}\gamma(t)p_{t}{\left\|\nabla_{\xi}\log p _{t}-\nabla_{\xi}\log\hat{q}_{t}\right\|}^{2}{\mathrm{d}}g{\mathrm{d}}\xi{ \mathrm{d}}t$ $$

Using the condition $s_{\theta}(g,\xi,t):=\nabla_{\xi}\log\hat{q}_{T-t}(g,\xi,t)$, and $\hat{q}_{T}=\pi^{*}$, with the choice $\gamma(t)=1$, we arrived at the following equation,

$$ $\displaystyle{D_{\operatorname{KL}}\left(p_{0}\parallel q_{T}^{\theta}\right)} =\int_{0}^{T}\int_{G\times\mathfrak{g}}p_{T-t}(g,\xi){\left\|\nabla_{\xi}\log p _{T-t}(g,\xi)-s_{\theta}(g,\xi,t)\right\|}^{2}{\mathrm{d}}g{\mathrm{d}}\xi{ \mathrm{d}}t+D_{\operatorname{KL}}(p_{T}\parallel\pi_{*})$ $$

∎

###### Theorem 4 .

###### Lemma 3 .

###### Proof of Lemma 3 .

###### Proof of Thm. 4 .

## Appendix F NLL Estimation with Intrinsic Proof

In this section, we provide an intrinsic proof of the instantaneous change of variables on a general manifold, which does not depend on local charts in the proof or the formula. While we are now aware that the results are not new and has been discussed in [, , , ], we still provide proof for a self-consistency.

Let $z$ be a random variable whose range is $\mathcal{M}$ and denote its density as $p_{0}\in\mathcal{C}(\mathcal{M})$. Given a smooth time-dependent vector field $X(t,\cdot)$, i.e.,$X(t,\cdot)\in\Gamma^{\infty}(T\mathcal{M})$ for any $t\in[0,T]$(^*^**$\Gamma^{\infty}(T\mathcal{M})$ denotes the set of all smooth vector fields on $\mathcal{M}$). We consider the push forward map generated by the flow of $X$, i.e., $f_{s}^{t}:\mathcal{M}\to\mathcal{M}$ satisfies

$$ $\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}f_{s}^{t}(x)=X(t,f_{s}^{t}(x)), \quad\forall x\in\mathcal{M},0\leq s\leq t\leq T$ $$

with initial condition $f_{s}^{s}$ is the identity map for any $s$.
We define $p_{t}$ as the density of $f_{0}^{t}(z)$. Then we have the following theorem,
{graybox}

###### Theorem 5 .

[Instaneous Change of Variables on Manifold]
Consider $p:\mathbb{R}\times\mathcal{M}\to\mathbb{R}$, such that $p_{t}=p(t,\cdot)$ is the density of $z(t)$, where $z(t)$ is the random variable defined by $z$ pushforward along $X$ for time $t$. Then we have

$$ $\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\log p(t,f_{0}^{t}(x))=- \operatorname{div}X(t,f_{0}^{t}(x)),\qquad\forall x\in\mathcal{M}$ $$

We first review a standard result (see e.g., []) Lemma 4, and then provide a proof for Theorem 5. The following Lemma 4 describes the relationship between the density of the push-forward as well as the determinant of the differential and corresponding points. We will use this result heavily in our proof.
{graybox}

###### Lemma 4 .

For any diffeomorphism $f:\mathcal{M}\to\mathcal{M}$, we have

$$ $\displaystyle f_{\#}p(f(x))=p(x)\left(\det df(x)\right),\quad\forall x\in \mathcal{M}$ $$

$f_{\#}$ is the push forward density. On the right-hand side, $df(x):T_{x}\mathcal{M}\to T_{f(x)}\mathcal{M}$, denoting the differential of $f$, is a linear map. Consequently, $\det df$ is well-defined and independent of choice of coordinate.

With Lemma 4, we are ready to present our proof for Theorem 5.

###### Proof of Thm. 5 .

In this proof, we use the shorthand notation $x_{t}:=f_{0}^{t}(x)$, which also induces $x_{0}=x$. Lemma 4 gives

$$ $\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\log p(t,f_{0}^{t}(x))$ $\displaystyle=\frac{{\mathrm{d}}}{{\mathrm{d}}t}\log\left[(f_{0}^{t})_{\sharp} p\left(f_{0}^{t}(x)\right)\right]$ $\displaystyle=\frac{{\mathrm{d}}}{{\mathrm{d}}t}\log\left[p_{0}(x)\det df_{0}^ {t}(x)\right]$ $\displaystyle=\frac{{\mathrm{d}}}{{\mathrm{d}}t}\log\left[\det df_{0}^{t}(x)\right]$ $$

Since $f$ is the pushforward map, it has the semi-group structure, i.e., $f_{t_{1}}^{t_{2}}\circ f_{t_{2}}^{t_{3}}=f_{t_{1}}^{t_{3}}$ for any $t_{1}\leq t_{2}\leq t_{3}$, which gives $\det df_{t_{1}}^{t_{2}}\cdot\det df_{t_{2}}^{t_{3}}=\det df_{t_{1}}^{t_{3}}$, and immediately

$$ $\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\log\left[\det df_{0}^{t}\right]$ $\displaystyle=\lim_{\epsilon\to 0^{+}}\frac{1}{\epsilon}\left(\log\left[\det df _{0}^{t+\epsilon}(x)\right]-\log\left[\det df_{0}^{t}(x)\right]\right)$ $\displaystyle=\lim_{\epsilon\to 0^{+}}\frac{1}{\epsilon}\log\left[\det df_{t}^ {t+\epsilon}(x_{t})\right]$ $$

Consequently,

$$ $\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\log p(t,f_{0}^{t}(x))$ $\displaystyle=-\frac{\partial}{\partial\epsilon}\log{\left|\det df_{t}^{t+ \epsilon}(x_{t})\right|}\Big{|}_{\epsilon=0}$ $\displaystyle=-\frac{\frac{\partial}{\partial\epsilon}{\left|\det df_{t}^{t+ \epsilon}(x_{t})\right|}\Big{|}_{\epsilon=0}}{{\left|\det df_{t}^{t+\epsilon}( x_{t})\right|}\Big{|}_{\epsilon=0}}$ $\displaystyle=-\frac{\partial}{\partial\epsilon}{\left|\det df_{t}^{t+\epsilon }(x_{t})\right|}\Big{|}_{\epsilon=0}$ $$

where we use $\frac{\partial}{\partial\epsilon}$ to denote the right derivative, and the last equation is because of $f_{t}^{t}$ is identity. Jacobi’s formula gives $\frac{d}{{\mathrm{d}}}\det(A)=\operatorname{tr}\left(\dot{A}\right)$ at $A=I$, which tells

$$ $\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\log p(t,f_{0}^{t}(x))$ $\displaystyle=-\operatorname{tr}\left(\frac{\partial}{\partial\epsilon}df_{t}^ {t+\epsilon}(x_{t})\right)\Big{|}_{\epsilon=0}$ $$

Before we proceed, we define two set of vector fields, $\left\{E_{i}\right\}_{i=1}^{d}$ and $\left\{Y_{i}\right\}_{i=1}^{d}$. $\{E_{i}\}_{i=0}^{d}$ is defined as a set of smooth coordinate frame, defined on the whole manifold $\mathcal{M}$. $\left\{Y_{i}\right\}_{i=1}^{d}$ is a set of vector fields along $x_{t}$ generated by the push forward map $f_{t}^{t+\epsilon}$, i.e., $df_{t}^{t+\epsilon}$ is a map from $T_{x_{t}}\mathcal{M}$ to $T_{x_{t+\epsilon}}\mathcal{M}$, and $Y_{i}$ satisfies

$$ $\displaystyle df_{t}^{t+\epsilon}(Y_{i}(x_{t+\epsilon}))=Y_{i}(x_{t+\epsilon}) ,\forall i=1,\dots,d,\forall t<t+\epsilon\leq T$ $$

with constraint $Y_{i}(x_{t})=E_{i}(x_{t})$. Note that $Y_{i}$ is defined only along the curve $x_{t}$ for $t\in[0,T]$.

Since we are considering a push forward map $f$ along a time-dependent vectorfield $X(t,\cdot)\in\Gamma^{\infty}(T\mathcal{M})$, we consider to make it time-independent by consider the problem in a new space $\tilde{\mathcal{M}}:=\mathbb{R}\times\mathcal{M}$, and a new time-independent vector field $\tilde{X}\in\Gamma^{\infty}(T\tilde{\mathcal{M}})$ defined by

$$ $\displaystyle\tilde{X}(t,x)=\left(1,X(t,X)\right),\quad t\in[0,T],x\in\mathcal {M}$ $$

Since $x_{t}$ is the integral curve generated by $X$, the integral curve of $\tilde{X}$ with initial condition $(0,x)$ is given by $\tilde{x}_{t}=(t,x_{t})$, i.e., $\tilde{x}_{t}$ satisfies $\dot{\tilde{x}}_{t}=X(\tilde{x}_{t})$.

Both $\left\{E_{i}\right\}_{i=1}^{d}$ and $\left\{Y_{i}\right\}_{i=1}^{d}$ can be extended to $\tilde{\mathcal{M}}$. For $\left\{E_{i}\right\}_{i=1}^{d}$, this is defined by $\tilde{E}_{0}\equiv(1,0)$ and $\tilde{E}_{i}=(0,E_{i})$. Similarly, for $\left\{Y_{i}\right\}_{i=1}^{d}$, this is defined by$\tilde{Y}_{i}(\tilde{x}_{t})=(0,Y_{i}(x_{t}))$ and $\tilde{Y}_{0}\equiv(1,0)$.

By choosing an arbitrary linear connection $\nabla$ on $\mathcal{M}$, we can also extend $\nabla$ to $\tilde{\mathcal{M}}$, and we denote such induced linear connection as $\tilde{\nabla}$ [see e.g., , Ex. 1 in Chap. 6]. Since $f$ is the pushforward generated by $X$, $\frac{\partial}{\partial\epsilon}f_{t}^{t+\epsilon}|_{\epsilon=0}=X(t,\cdot)$, and we have

$$ $\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\log p(t,f_{0}^{t}(x))$ $\displaystyle=-\sum_{i=1}^{d}\frac{\partial}{\partial\epsilon}{\left\langle df _{t}^{t+\epsilon}(E(x_{t})),E_{i}(f_{t}^{t+\epsilon}(x_{t}))\right\rangle}|_{ \epsilon=0}$ $\displaystyle=-\sum_{i=1}^{d}\frac{\partial}{\partial\epsilon}{\left\langle Y_ {i}(f_{t}^{t+\epsilon}(x_{t})),E_{i}(f_{t}^{t+\epsilon}(x_{t}))\right\rangle}| _{\epsilon=0}$ $\displaystyle=-\sum_{i=1}^{d}\tilde{\nabla}_{\tilde{X}}{\left\langle\tilde{Y}_ {i},\tilde{E}_{i}\right\rangle}|_{\tilde{x}_{t}}$ $\displaystyle=-\sum_{i=1}^{d}\nabla_{X}{\left\langle Y_{i},E_{i}\right\rangle} |_{x_{t}}$ $$

We choose $\nabla$ to be the Levi-civita connection. Because it is compatible with the metric, we have

$$ $\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\log p(t,f_{0}^{t}(x))$ $\displaystyle=-\sum_{i=1}^{d}{\left\langle\nabla_{X}Y_{i},E_{i}\right\rangle}+ {\left\langle Y_{i},\nabla_{X}E_{i}\right\rangle}$ $$

By the definition of Lie derivative, we have $\mathcal{L}_{X}Y_{i}\equiv 0$. Together with the constraint $Y_{i}(x_{t})=E_{i}(x_{t})$, we have $\nabla_{X}Y_{i}=\nabla_{Y_{i}}X=\nabla_{E_{i}}X$ at $x_{t}$, which gives

$$ $\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\log p(t,f_{0}^{t}(x))$ $\displaystyle=-\sum_{i=1}^{d}{\left\langle\nabla_{E_{i}}X,E_{i}\right\rangle}+ {\left\langle E_{i},\nabla_{X}E_{i}\right\rangle}$ $$

Now we show ${\left\langle E_{i},\nabla_{X}E_{i}\right\rangle}=0$ using local coordinates:

$$ $\displaystyle{\left\langle E_{i},\nabla_{X}E_{i}\right\rangle}=\sum X_{i} \Gamma_{ij}^{j}=\sum X_{i}\frac{\partial}{\partial e_{i}}\ln\sqrt{{\left|g \right|}}$ $$

where $\Gamma$’s are Christoffel symbols corresponding to the local coordinates $E_{i}$, defined by $\Gamma_{ij}^{k}:={\left\langle\nabla_{E_{i}}E_{j},E_{k}\right\rangle}$. Due to our choice of $E_{i}$ as orthonormal frames, we have ${\left|g\right|}\equiv 1$ and ${\left\langle E_{i},\nabla_{X}E_{i}\right\rangle}=0$ for all $i$.

By simplifying the expression, we arrive at the following desired results, which finishes our proof.

$$ $\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\log p(t,f_{0}^{t}(x))=-\sum_{i =1}^{d}{\left\langle\nabla_{E_{i}}X,E_{i}\right\rangle}=-\operatorname{div}X(t ,f_{0}^{t}(x))$ $$

∎

We can choose $E_{i}$ as the left-invariant vector fields generated by $e_{i}$, i.e., $E_{i}(g)=T_{e}L_{g}e_{i}$, where $e_{i}\in\mathfrak{g}$ is a set of orthonormal basis. As a corollary, we can compute dynamic for the log probability as the following,
{graybox}

###### Corollary 3 (NLL estimation on Lie group with left-invariant metric) .

For SDE in Eq. (1), the time-dependent vector field is given by

$$ $\displaystyle X(g,\xi)=(T_{e}L_{g}\xi,-\gamma\xi+\beta(t,g,\xi))$ $$

Using the fact that $\operatorname{div}_{g}(T_{e}L_{g}\xi)=0$, we have

$$ $\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\log p(t,f_{0}^{t}(x))=\sum_{i= 1}^{d}\left(-\gamma+\frac{\partial}{\partial\xi_{i}}\beta_{i}\right)$ $$

###### Theorem 5 .

###### Lemma 4 .

###### Proof of Thm. 5 .

###### Corollary 3 (NLL estimation on Lie group with left-invariant metric) .

## Appendix G Training Set-up, Dataset

### G.1 Hyperparameters

Hardware: All the experiments are running on one RTX TITAN, one RTX 3090 and one 4090.

Architectural Framework and Hyperparameters: We employed the score function $s_{\theta}(g_{t},\xi_{t},t;\theta)$ parameterized by the same network architecture as outlined in the CLD paper [], albeit with varying parameter counts for each task. Throughout our experiments, we maintained the diffusion coefficient $\gamma(t)$ constant at 1, while the total time horizon $T$ varied depending on the task. We use AdamW optimizer for training the neural networks.

### G.2 Dataset Preparation

Protein and RNA Torsion Angles: We access the dataset prepared by Huang et al. [] from the repository of []. We further post-processed it and transformed the data into valid elements of $\mathbb{T}^{n}$.

Pacman: We take the maze of the classic video game Pacman and extract all the pixel coordinates from the image that corresponds to the maze. We post-processed it and transformed the data into valid elements of $\mathbb{T}^{2}$.

Special Orthogonal group $\mathsf{SO}(n)$: For $\mathsf{SO}$, we followed the same procedure as the one described in [] and generate a Gaussian Mixture with $32$ components, uniformly random mean and variance. For $n>3$, we follow a similar procedure with a reduced number of mixture components.

Unitary group $\mathsf{U}(n)$: We considered the unitary group data of the form $\operatorname{expm}(-it\mathcal{H})$, which is the time evolution operator of the following Schrödinger’s equation for a general quantum system, $i\partial_{t}\psi_{t}=\mathcal{H}\psi_{t}$. Here $\psi$ denotes the quantum state vector and $\mathcal{H}$ denotes the Hamiltonian operator of the system. We considered the following two types of Hamiltonians,

- •
For quantum oscillator, the Hamiltonian is given by $\mathcal{H}=\Delta+V$, where $\Delta$ is the Laplacian operator, and $V(x)=\frac{1}{2}\omega^{2}\|x-x_{0}\|^{2}$ is a random potential function, where $\omega$ and $x_{0}$ are random variables. Note that these are infinite dimensional objects. To obtain a valid element in $\mathsf{U}(n)$ for a finite $n$, we perform spectral discretization on the Laplacian operator as well as the random potential to get a finite-dimensional Hamiltonian operator $\mathcal{H}_{h}$, with which the time-evolution operator is computed with. We choose $t=1$ in this case.
- •
For Transverse field Ising Model, the Hamiltonian is given by
$\mathcal{H}=-\sum_{\langle i,j\rangle}J_{ij}\sigma_{i}^{z}\sigma_{j}^{z}-\sum_
{i}g_{i}\sigma_{i}^{x}$
where $\sigma_{i}^{z},\sigma_{i}^{x}$ are the Pauli matrices, $J_{ij}$ is the coupling parameter and $g_{i}$ is the field strength. Here $J_{ij}$ and $g_{i}$ are random variables, which corresponds to the situation of RTFIM. The time-evolution operator is generated with such a Hamiltonian at $t=1$.

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## Peer Reviews

confidence: 4
contribution: 2
initial_rating: 5
presentation: 3
questions: - 280-283: The sentence “Since the Brownian motion…” does not seem to be grammatically correct.
- 292: “Unfortunately, note [that] even though”. Missing “that”.
- 382: $A_g^B$ used twice, probably copy-paste error.
- The authors make several times the claim that the formalism on which they based their work does not required projection on the manifold. I think this statement, as is, is false. Group equation is $dg = g_0 \xi$, which is an equation on the tangent space of the group. Its solution is (neglecting the variation of $\xi$), $g=\exp(g_0 \xi)$. But the exponential map is precisely the projection function on Lie groups, $\exp: TG —> G$. So to stay on the manifold the authors are performing a projection exactly as all other methods too.
soundness: 3
strengths: The topic is very relevant, as diffusion-based models achieved quite successful results in many applications, but very little has been explored in terms how inductive bias can be incorporate into them, and diffusion on manifold is surely an important field. 

The notation is fairly consistent and the math is, to the extend that I checked, correct (I did not carefully checked the proofs in the appendix).
summary: The authors present a method for diffusion on Lie groups. The method is based on the recently published idea of trivialized momentum, that is, diffusion (i.e. noising) is performed on the Lie algebra of the group, and not on the Lie group itself. The advantage of this approach is that the Lie algebra is a vector space isomorphic to $\mathbb{R}^n$, and therefore it simplifies the problem since all the curvature terms pretty much drop out. This diffusion is brought back into the actual Lie Group G by using the map connecting two tangent spaces of G at different point, which is the tangent of the left multiplication map. 

The authors show that this noising dynamics has an inverse dynamics as well, in very familiar fashion as the usual diffusion dynamics, in which a score function appear, which it is possible to learn with neural networks. 

The authors further argue that it is possible to have a dynamics that is exaclty on the curved manifold at all steps without the need of a projection (I do not agree with this statement, please refer below in the "Questions" section). This is done by separating each step into two contributions, i.e., the update of the Lie algebra element $d\xi$ and the update of the lie group element $dg = g \xi$. Note that also this procedure is already known in the literature. 

They provide an explicit formula for the Abelian case, while they point out that for the general case, it is necessary to learn the score with implicit score matching, which unfortunately involves the divergence term.
weaknesses: My two main concerns regard the originality of the work and its practical relevance:

- **Originality**: As far as I can see, the idea and the theory behind trivialized momentum has been developed in previous works, as well as the  splitting discretization technique to exactly integrate the SDEs. The authors prove a theorem for the inverse SDE of the system, but given that the non-trivial equation (about the variation of the Lie algebra) is essentially in flat space, it reduces pretty much to the flat-space formula.
- **Practical Relevance**: The main drawback is in the non-Abelian case, which is the most interesting since the Abelian case is essentially SO(2) (or copies of it). For non-Abelian groups, there's no conditional transition probability available, requiring implicit score matching. This involves computing the divergence with respect to the data space (more precisely, the Lie algebra space, but as dim data = dim $G$ = dim $\mathfrak{g}$ its computationally as expensive). This limitation significantly reduces the method's practicality and makes it unlikely to be widely adopted. 

I think the paper would be significantly more relevant if it provided an analogue of Theorem 2 for non-Abelian groups.

---

confidence: 3
contribution: 3
initial_rating: 8
presentation: 3
questions: Some possible typos in the text:
- line 155, missing brackets for g_t, \xi_t?
- line 325, should be "\sigma_t evaluated at"?
- line 1022, the A_\xi^\F should be A_\xi^\B?
soundness: 4
strengths: The authors present a novel and clever way to address the computational limitations of previous approaches to manifold-valued diffusion models, by exploiting the group structure of Lie groups through the use of Kinetic Langevin dynamics. This allows for learning a score function in a fixed Euclidean space which reduces the numerical errors associated with the approximations arising from scores defined directly on curved manifolds. This approach is well-fleshed out by the authors who provide a detailed methodology for the application of this for generative modelling, for example, by providing an integration scheme which is manifold-preserving and projection-free, and training objectives for both Abelian Lie groups and non-Abelian Lie groups. This framework is further validated by experiments demonstrating the novel scaling of manifold-valued diffusion models to SO(N), N>3.
summary: The paper introduces a new framework for constructing diffusion models on manifold-structured data in the particular case where the manifold is a Lie group. This is achieved through the use of a forward noising process defined by kinetic Langevin dynamics on the Lie group and its tangent space jointly. The authors demonstrate that the group structure of this class of manifolds allows for the corresponding reverse process to rely solely on a Euclidean score function, as well as allowing for the manifold-preserving and projection-free sampling of the forward and reverse processes, circumventing the complexities and multiple approximations required in previous work. The authors then apply this to various tasks presenting convincing results.
weaknesses: The framework can only provide simulation-free training for the manifold SO(2) and its product space which is less flexible than Riemannian Flow Matching. However, this setting is still important for applications.

---

confidence: 3
contribution: 2
initial_rating: 5
presentation: 3
questions: See above.
soundness: 2
strengths: - The paper is overall well-written and easy to follow. The problem the authors tackle is interesting and important.
- The proposed technique is interesting and avoids some of the problems mentioned with other existing approaches, such as projections, requiring specific architectures, etc.
- The mathematical details are explained cohesively and clearly. 
- The results on the torus datasets, as well as the toy tasks for the SO(n) and U(n) groups seem promising.
summary: This paper provides a new framework for performing generative modelling on Lie groups which doesn’t require any geometry-specific projections or approximations, utilizing the notion of “trivialized momentum.” The authors provide the denoising score matching loss which can be used for SO(2)^n, as the conditional transition probabilities are tractable and an implicit score matching loss for other connected compact Lie groups. They also provide an integration scheme for the forward and backward dynamics.
weaknesses: - The main weakness of the paper is the lack of rigorous experiments to demonstrate the effectiveness of the model in comparison with existing approaches. 
- First, the authors didn’t clarify how the experimental setup and training procedures vary across the different benchmarks. The experimental results would be more convincing if details on the setup are clearly provided. 
- Secondly, it wasn’t clear me to what approximations one has to make when doing RFM in the torus settings, especially since going from the Lie algebra to the Lie group element is essentially trivial in that case. Could the authors clarify what approximations are required in RFM for SO(2)^n experiments?
- Lastly, although the results on the SO(n) and U(n) experiments are interesting and show promise, more rigorous experiments are required to show the scalability of the implicit score-matching loss (as it generally doesn’t perform as well). I think the paper could be strengthened if the authors clearly show better performance compared to RFM and RDM on the higher-dimensional SO(n) and U(n) tasks.
- It would be very helpful if the authors provided a more detailed discussion on the approximations made in RFM and RDM which the authors attribute to the worse performance of these models in the torus settings. 
- The paper also provides experiments for the SO(3) setting, however it doesn’t provide a comparison with SE(3) generative models that rely on IGSO(3) distribution. As mentioned in the paper, the special structure of SO(3) can be leveraged, which makes applying TDM in that setting less motivated.

---

confidence: 4
contribution: 4
initial_rating: 8
presentation: 3
questions: Questions for clarification 
- p.2 l.59: the authors bring up numerical integration errors. It would be helpful to the reader to specify this a bit more. I assume this is because mappings like the exponential mapping can be expensive. Is this only the case in high dimensions or even on low-dimensional Lie groups?
    - a bit more nuance would improve the paper a lot as it would make the actual challenge even clearer, which would highlight your contributions even more
- p.2 l.78: the authors write ``while still encapsulating the curved geometry without any approximation.’’ There is an approximation error if there is curvature right? 
    - It would be great if the authors are a bit more nuanced here as well as it should be clear that this method might introduce new errors.
- Section 3: In the Lie algebra, the learned distribution wraps around. So ideally, the score should be periodic or should become zero if this cannot be enforced. Is there some mechanism that enforces this? 
    - If not, I would recommend adding this explicitly to let the reader know that there is possibility for error here.
- p.6 l.337: In the implicit score matching paragraph, it is not very clear to me how one would go about sampling here (does one have to do a forward simulation for this?). Would the authors clarify this?
- Section 5: 
    - It feels natural to also mention something about flow matching in the outlook. Do the authors have any ideas on this they would like to disclose in the conclusions?

Additional feedback
- Section 1: please do a grammar check. A couple of small things that I noticed:
    - p.1 l.43: change ''manifold’’ in … Neural ODE to manifold with maximum … to ''manifolds’’. This reads better
    - p.1 l.44: change ''algorithm’’ in Rozen; Ben-Hamu develop simulation free algorithm but… to ''algorithms for the same reason as above
    - p.1 l.94: add ''a'' right before ''trivialized’’ in …through learning trivialized score function…
- Section 2: It looks a bit odd that section 2 is just one line. Consider integrating this line in section 3 instead.
- Section 3:
    - p.3 l.133: replace ''sequal'' with ''following''. It sounds like this will be discussed in your next paper rather than further down in this one.
    - p.3 l.147: the authors consider the tangent map. I believe that this is the differential? For the reader who is used to other nomenclature and/or notation, I would suggest to also mention that this is often referred to as the differential.
    - In equation 8, shouldn’t there be a superscript i for the xi in the dt term?
    - p.7 l.373: the authors write ``do not admit a closed form.’’ A closed form what?
soundness: 4
strengths: - The authors provide a very nice motivation for the problem and highlight the key challenges that current manifold-valued models are dealing with.
- The mathematics is displayed in a very clear way and adequately addresses the challenge discussed in the motivation, i.e., how to set up manifold diffusion models on the Lie algebra. They also address different types of Lie groups well. In particular, it is nice that they highlight the special case of Abelian Lie groups learning, where training becomes easier than on non-Abelian ones
- The numerical experiments nicely display that the method performs well on high-dimensional Lie groups (which is where one would expect issues due to numerical inaccuracies from projections and manifold mappings), but also showcase that for low-dimensional Lie groups performance is superior to baselines.
summary: - The paper is concerned with tackling the challenge of incorporating diffusion models into a manifold setting
- Their key takeaway is that on Lie groups for both training and sampling diffusion can take place in the Lie algebra
- They show that their method performs well on various Lie groups of various dimensions that arise in different applications
weaknesses: - I feel that for the introduction, there are some mistakes in terms of English grammar 
- Some parts could have been explained in more detail, but given the page limit restrictions it is natural that the authors aimed to be as concise as possible

Meta-review: The paper studies diffusion processes on Lie groups by introducing the diffusion process on corresponding Lie algebra. This allows introducing the diffusion processes and training corresponding generative models on SO(n) and U(n) manifolds. All the reviewers have recognized the theoretical contribution of the paper. Given the high interest in generative modelling, the paper will be of high interest to the ICLR audience.

Decision: Accept (Poster)