Underdamped Diffusion Bridges with Applications to Sampling

Dear Reviewer znkS.

Thank you very much for your review. We appreciate that you value our generalization of diffusion-based sampling methods to the underdamped setting as novel. Thanks also for highlighting that we can reach improved numerical performance compared to alternative methods. Let us address your questions and comments in the sequel.

Novelty and theoretical significance

We thank the reviewer for expressing their concern. We kindly refer them to the general response, where we briefly reiterate the novelty and significance of our work.

Numerical experiments for generative modeling

Thank you for bringing up this point. We acknowledge that our work does not include experiments on the generative modeling scenario, where samples from the target distribution are available but not its density. We chose not to consider this case in our numerical experiments to maintain a clear focus on the (for us, more compelling) problem of sampling, as reflected in the paper’s title.

(Non-)Dependency of the ELBO on the target distribution

Thank for for asking this important question. We note that even though $\tau$ appears in the formula of the ELBO, namely $\mathbb{E}_{Z\_0 \sim \tau}\left[\log \tau(Z\_0) \right] - D\_\mathrm{KL}(\overset{\rightarrow}{{\mathbb{P}}}^{v, {\tau}} | \overset{\leftarrow}{{\mathbb{P}}}^{u,{\pi}})$, it actually does not depend on $\tau$ since the term $\mathbb{E}\_{Z\_0 \sim \tau}\left[\log \tau(Z\_0) \right]$ lets the dependency of $\tau$ in $D\_\mathrm{KL}(\overset{\rightarrow}{{\mathbb{P}}}^{v, {\tau}} | \overset{\leftarrow}{{\mathbb{P}}}^{u,{\pi}})$ disappear, cf. the formula for the Radon-Nikodym derivative stated in Proposition 2.3. We made this more explicit in the revised version of the manuscript.

Please do not hesitate to ask further questions in case something is still unclear.

Dear Reviewer xTDP.

Thank you very much for your review. We appreciate that you value our generalization of diffusion-based sampling methods to the underdamped setting as novel. Thanks also for highlighting that we can reach improved numerical performance compared to alternative methods. Let us address your questions and comments in the sequel.

Novelty and theoretical significance

We thank you for expressing your concern and agree that our contribution builds on the work [1] and related paper, as also made clear in the text. We kindly refer you to the general response, where we briefly reiterate the novelty and significance of our work.

Challenges for underdamped dynamics

We agree that splitting schemes such as OBAB, BAOAB, and OBABO have been extensively studied in, e.g., [2] for uncontrolled Langevin dynamics. The setting we consider, i.e., controlled Langevin dynamics, indeed comes with additional challenges as it necessitates careful selection of the forward and backward transition kernels $\overset{\rightarrow}{p}$ and $\overset{\leftarrow}{p}$, respectively (see Appendix A.7). This is crucial for deriving the (discrete-time) Radon-Nikodym derivative required for optimizing both the control functions and the associated hyperparameters. Such transition kernels have, to the best of our knowledge, only been considered for uncontrolled forward processes and non-symmetric splitting methods (i.e., OBAB) in [3, 4]. In contrast, we (a) are the first to approach general bridges using underdamped dynamics, i.e., controlled forward and backward processes, and (b) use symmetric splitting methods, which are theoretically proven to yield lower discretization errors [2] and demonstrate superior numerical performance (see Figure 3).

Computational costs of splitting schemes

Thank you for your question regarding the computational costs of the numerical schemes we compared, as this is indeed a crucial consideration for practitioners. You are correct that the proposed splitting schemes can introduce additional computational effort per gradient step, contributing to the slightly longer wall-clock times for the underdamped methods observed in Figure 4.

To compare the performance of the different schemes with respect to actual computational time (wall-clock time), we conducted an additional experiment. This new experiment has been included as Figure 7 in the Appendix of the revised version of our article. Our findings indicate that splitting schemes offer significantly better performance within the same computational budget.

References:

[1] Francisco Vargas, Shreyas Padhy, Denis Blessing, and Nikolas Nüsken. Transport meets variational inference: Controlled monte carlo diffusions. In The Twelfth International Conference on Learning Representations, 2024.

[2] Pierre Monmarche. High-dimensional MCMC with a standard splitting scheme for the underdamped Langevin Diffusion. Electronic Journal of Statistics, 15(2):4117–4166, 2021.

[3] Tomas Geffner and Justin Domke. Langevin diffusion variational inference. arXiv preprint arXiv:2208.07743, 2022.

[4] Doucet, A., Grathwohl, W., Matthews, A. G., & Strathmann, H. (2022). Score-based diffusion meets annealed importance sampling. Advances in Neural Information Processing Systems, 35, 21482-21494.

Dear Reviewer Qe9N.

Thank you for your extensive review. We are happy that you value our novel unified framework and find it clearly understandable and enjoyable to read. We aim to address your questions and comments in the sequel.

Addition compared to previous work

To the best of our knowledge, our work is the first rigorous treatment of the underdamped setting for (general) diffusion bridges. While frameworks for diffusion bridges have already been presented in [1,2], they do not incorporate degenerate diffusion coefficients. We have extended those frameworks to this setting using a general path space measures perspective. We highlight that the formulas and their assumptions differ from the known formulas in a subtle but delicate way. For instance, for the proof of the Radon-Nikodym derivative (see Prop. 2.3), it is non-trivial to rigorously leverage the Girsanov theorem, backward Ito integration, and reversible reference processes in our underdamped setting. In the revised version of our paper (in particular in Appendix A.2), we have added additional discussions on our assumptions, e.g., regarding the Girsanov theorem, Nelson's identity, and the existence and uniqueness of solutions to Fokker-Planck equation (see also our responses to your questions below). If you have further questions or doubts regarding the level of rigorousness, please don't hesitate to let us know.

Moreover, we want to highlight that developing the splitting schemes for the setting of controlled diffusions is non-trivial and can be considered another major contribution of our work.

Encompassing previous work as special cases

You are right that previous methods, of course, also bring some details that we do not study in our generalized framework -- thank you for pointing this out. We have modified the corresponding sentence at the end of Section 1 in the revised version of the paper. Please let us know if we should add some more specific information.

Questions

1. That's a good question, thank you. The differentiation between $f$ and $u$ (or $v$, respectively) is needed because we may only control in dimensions where Brownian motion is active. The diffusion matrix $\eta$, mapping the $d$-dimensional control $u$ (or $v$, respectively) to the entire space dimension $D$, takes care of this. The function $f$, on the other hand, is fixed and may therefore appear in all dimensions. Of course, one could absorb the components in the $d$ dimensions where the Brownian motion is active into the controls. However, it can be advantageous to decouple the effects: For suitable integrators, $f$ cancels in the discrete-time Radon-Nikodym in Section 3 (see Appendix A.8 and we also added a comment in Section 3). Thus, $f$ only affects the simulated process and can be chosen to facilitate good initializations, e.g., as Langevin dynamics (see Appendix A.10.3).

2. The divergence-based formulation of the Radon-Nikodym derivative in Proposition A.4 is needed for the derivation of the score-matching objective stated in Proposition 2.5 (see the proof) and can be readily related to connected PDEs, see, e.g., [3]. Further, it might exhibit lower variance and, noting that $\eta \eta^+ = \begin{pmatrix} \mathbf{0}\_d & \mathbf{0}\_d \\\ \mathbf{0}\_d & \operatorname{Id}\_{d \times d} \end{pmatrix}$, can be readily written down without needing the concept of a pseudoinverse. We note, however, that the divergence-based formulation does not provide a discrete ELBO and does not scale well computationally in high dimensions, which is why we chose to focus on the alternative formulation.

3. Thank you. We renamed the assumption to a global Lipschitz condition (uniformly in time). However, we want to note that this condition implies a linear growth condition (at least for the global Lipschitz condition we are assuming): $\\|g(z,t)\\| \le \\|g(z,t) - g(0,t)\\| + \\|g(0,t)\\| \le K\\|z\\| + \max_{s \in [0,T]} \\|g(0,s)\\| \le \widetilde{K}(1+\\|z\\|)$, where $\widetilde{K}=\max\\{K, \max_{s \in [0,T]} \\|g(0,s)\\|\\}$ and the maximum exists since $[0,T]$ is compact and $g$ is assumed to be continuous. Since $\sigma$ does not depend on the spatial variable, it only needs to be continuous (which we have assumed) for the existence of a strong solution with pathwise uniqueness; see, e.g., [4, Section 8.2]. We added a corresponding comment in Appendix A.2.

Questions (cont.)

4. We added additional comments to our assumptions in Appendix A.2, specifying for which statements they are used.

   • Girsanov: The Girsanov theorem requires that the Doléans-Dade exponential is a uniformly integrable martingale (which is often shown via Novikov's, Kazamaki’s, or Beneš's condition). In our setting, this follows from the continuity and global Lipschitz condition (see, e.g., [5], which we now cited in Appendix A.2).
   • Existence and uniqueness of solutions to FPEs: A standard result by Kolmogorov establishes that the FPE is satisfied if a sufficiently smooth density exists [6]. In the non-degenerate case, sufficient conditions for the existence and uniqueness of such densities have been developed by Friedman based on the parametrix method [7]. In the degenerate case, such conditions become more technical, and we refer to [8] for corresponding results. We added a footnote with further details in Appendix A.2 and note that we only use the FPE to motivate our splitting schemes in Section 3.

5. The equation follows from the definition of the process $Z$, as stated in (5). The difference between $\overset{\leftarrow}{\mathrm d}$ and $\overset{\rightarrow}{\mathrm d}$ only appears in the stochastic integration since time-reversal does not affect the integration of the deterministic part. In other words, integration against $\overset{\leftarrow}{\mathrm d} t$ equals integration against $\overset{\rightarrow}{\mathrm d}t$ by viewing $t \mapsto t$ as a finite variation process (in fact, the statement holds for all finite variation processes, which follows directly from the definition of the stochastic integrals in (22) and (23) and the properties of finite variation processes, see, e.g., [4, Proposition 4.2 and Proposition 4.21]).

6. For diffusion-based samplers, Nelson's identity (see Lemma 2.2) shows that $u^*(\cdot, T) + v^*(\cdot, T) = \eta^\top(T) \nabla \log \tau$, i.e., the sum of the optimal controls equals the scaled score at time $T$. The overdamped case corresponds to the special case, where $\tau = p_{\mathrm{target}}$ and $\eta$ has full rank. However, the score of the target distribution $\nabla \log p_{\mathrm{target}}$ can attain large values or potentially be unbounded, in particular for highly concentrated distributions (see [9]). This leads to numerical issues in objectives for learning $u$ and $v$. In the underdamped case, these issues do not exist: Since we choose $\eta = (\mathbf{0},\sigma)^\top$ and $\tau = p_{\mathrm{target}}(x)\mathcal{N}(y; 0, \mathrm{Id})$ (see Section 3), the right-hand-side of Nelson's identity simplifies to $\eta^\top(T) \nabla \log \tau(x,y) = \sigma^\top(T)\nabla_y \log \mathcal{N}(y; 0, \mathrm{Id})$. Thus, in the underdamped case, the sum of controls equals the (scaled) score of a Gaussian, which is a well-behaved function with linear growth. We mention this in Remark 3.3 in our paper. We hope this clarifies the question.

7. Thanks for asking about the augmented prior and target distributions, $\pi(x,y) = p_{\mathrm{prior}}(x) \mathcal{N}(y; 0, \operatorname{Id})$ and $\tau(x,y) = p_{\mathrm{target}}(x) \mathcal{N}(y; 0, \operatorname{Id}).$ First, note that by choosing underdamped processes, we have extended the dimension of the state space from $d$ to $2d$, where the idea is to use the first $d$ components (relating to the $x$-variable) for the actual sampling goal. The other $d$ components (relating to the $y$-variable) can be understood as auxiliary variables, incorporating the Brownian motion, such that the $x$-part may be easier to integrate (see also the explanations above). While the prior and target marginals of the $x$-part are specified by the task, the $y$-marginals can, in principle, be chosen more or less arbitrarily - with the constraint that we need to be able to sample from the prior of the $y$-marginal and need to be able to evaluate both the prior and the target (up to normalization constant). Building on prior work (e.g., [10]), we choose those marginals to be Gaussians. We hope this answers your question. Please let us know in case we misunderstood your question.

Dear Reviewer 5Kic.

Thank you very much for your review. We are happy that you consider our work as clear, well-organized, and easy to follow and that you value our contribution.

Notation

Thank you for pointing out that we should add additional explanations for our notation -- this is valuable feedback for improving our paper. Based on your feedback, we have extended the section on our notation in Appendix A.1 and added additional explanations, footnotes, as well as references to the appendix in the main part. We believe that all of our notation is now explained in the paper (when it shows up for the first time), but please let us know if you think that additional explanations are required.

Benefits of underdamped diffusion

There are several crucial theoretical benefits (and we also refer to the general response for further details):

1. Splitting schemes: Underdamped diffusions allow us to develop more sophisticated SDE integrators based on splitting schemes that reduce discretization errors which, when using symmetric splitting, yields provably lower discretization errors.; see Section 3 and Appendix A.8. The benefit of such integrator (in particular, our the OBABO method) is also clearly visible in our experiments in Figs. 4, 6, 7, & 8, as well as in Tab. 6.

2. Accelerated theoretical convergence: As mentioned in Section 1, underdamped Langevin dynamics can yield a quadratic improvement in the required number of steps over its overdamped version. Specifically, for smooth and log-concave targets, the number of steps to obtain KL divergence $\varepsilon$ can be reduced from $\tilde{\mathcal{O}}(d/\varepsilon^2)$ to $\tilde{\mathcal{O}}(\sqrt{d}/\varepsilon)$ [1]. On a high level, the benefit comes from the fact that the score only depends on the position variable $X$, which has provably smoother trajectories in the underdamped case. While further research is required for general diffusion bridges (see [2], and our discussion in Section 5), they still seem to profit from better performance (see Figs. 3, 4, 6, 7 & 8 and Tab. 1 & 6). At least at initialization, these theoretical results carry over since our samplers are initialized as Langevin dynamics, and we added this to our discussion in Section 4. This might also be a reason for the accelerated convergence during training; see our new Fig. 7 in the appendix.

3. Beneficial properties of the score: For diffusion-based samplers, Nelson's identity (see Lemma 2.2) shows that $u^*(\cdot, T) + v^*(\cdot, T) = \eta^\top(T) \nabla \log \tau$, i.e., the sum of the optimal controls equals the scaled score at time $T$. The overdamped case corresponds to the special case, where $\tau = p_{\mathrm{target}}$ and $\eta$ has full rank. However, the score of the target distribution $\nabla \log p_{\mathrm{target}}$ can attain large values or potentially be unbounded, in particular for highly concentrated distributions (see [2]). This leads to numerical issues in objectives for learning $u$ and $v$. In the underdamped case, these issues do not exist: Since we choose $\eta = (\mathbf{0},\sigma)^\top$ and $\tau = p_{\mathrm{target}}(x)\mathcal{N}(y; 0, \mathrm{Id})$ (see Section 3), the right-hand-side of Nelson's identity simplifies to $\eta^\top(T) \nabla \log \tau(x,y) = \sigma^\top(T)\nabla_y \log \mathcal{N}(y; 0, \mathrm{Id})$. Thus, in the underdamped case, the sum of controls equals the (scaled) score of a Gaussian, which is a well-behaved function with linear growth. We mention this in Remark 3.3 in our paper.

We hope this answers your remaining questions. Please let us know if you have any further questions. Otherwise, we would be glad if you would consider updating your score.

References:

[1] Yi-An Ma, Niladri S Chatterji, Xiang Cheng, Nicolas Flammarion, Peter L Bartlett, and Michael I Jordan. Is there an analog of nesterov acceleration for gradient-based MCMC?, 2021.

[2] Sitan Chen, Sinho Chewi, Jerry Li, Yuanzhi Li, Adil Salim, and Anru R Zhang. Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions. arXiv preprint arXiv:2209.11215, 2022

References:

[1] Tomas Geffner and Justin Domke. Langevin diffusion variational inference. arXiv preprint arXiv:2208.07743, 2022.

[2] Doucet, A., Grathwohl, W., Matthews, A. G., & Strathmann, H. (2022). Score-based diffusion meets annealed importance sampling. Advances in Neural Information Processing Systems, 35, 21482-21494.

[3] Pierre Monmarche. High-dimensional MCMC with a standard splitting scheme for the underdamped Langevin Diffusion. Electronic Journal of Statistics, 15(2):4117–4166, 2021.

[4] Tim Dockhorn, Arash Vahdat, and Karsten Kreis. Score-based generative modeling with critically damped langevin diffusion. arXiv preprint arXiv:2112.07068, 2021.

[5] Chen, T., Liu, G. H., & Theodorou, E. A. (2021). Likelihood training of Schrödinger bridge using forward-backward SDEs theory. arXiv preprint arXiv:2110.11291

[6] Schauer, Moritz, Frank Van Der Meulen, and Harry Van Zanten. "Guided proposals for simulating multi-dimensional diffusion bridges." (2017): 2917-2950.

[7] Heng, Jeremy, et al. "Simulating diffusion bridges with score matching." arXiv preprint arXiv:2111.07243 (2021).

[8] Lorenz Richter and Julius Berner. Improved sampling via learned diffusions. In International Conference on Learning Representations, 2024.

[9] Valentin De Bortoli, James Thornton, Jeremy Heng, and Arnaud Doucet. Diffusion Schrödinger bridge with applications to score-based generative modeling. Advances in Neural Information Processing Systems, 34:17695–17709, 2021.

[10] Jingtong Sun, Julius Berner, Lorenz Richter, Marius Zeinhofer, Johannes Muller, Kamyar Azizzadenesheli, and Anima Anandkumar. Dynamical measure transport and neural PDE solvers for sampling. arXiv preprint arXiv:2407.07873, 2024.

[11] Guan-Horng Liu, Tianrong Chen, Oswin So, Evangelos A. Theodorou. Deep Generalized Schrödinger Bridge. NeurIPS, 2022.

[12] Takeshi Koshizuka, Issei Sato. Neural Lagrangian Schrödinger Bridge: Diffusion Modeling for Population Dynamics. ICLR, 2023.

[13] Guan-Horng Liu, Yaron Lipman, Maximilian Nickel, Brian Karrer, Evangelos A. Theodorou, Ricky T. Q. Chen. Generalized Schrödinger Bridge Matching. ICLR, 2024.

[14] Denis Blessing, Xiaogang Jia, Johannes Esslinger, Francisco Vargas, and Gerhard Neumann. Beyond ELBOs: A large-scale evaluation of variational methods for sampling. ICML, 2024.

Dear Reviewer nE8S.

Thank you very much for your extensive review. We are happy that you appreciate that we achieved "record performance" with our novel diffusion-based sampling method, which is indeed remarkable due to the many recent improvements in the field. It is, as you mention, partly due to the novel way of automatically tuning hyperparameters. We also agree with you that an essential advantage of underdamped diffusions is the ability to use splitting methods, which have so far been "underexplored" in diffusion-based sampling and, therefore, add a valuable addition to the community. Let us address your points in the sequel.

End-to-end hyperparameter tuning not well explained

Thanks for bringing up this point, as it is indeed an essential contribution of our work. Our path space measure perspective allows us to derive loss functionals (or divergences between path space measures) that depend on the measures relating to the forward and backward process, respectively. Essentially, those measures not only depend on the control functions $u$ and $v$ but also on additional (hyper-)parameters, such as, e.g., the noise coefficient $\sigma$, the mass matrix $M$, the time horizon $T$, and the prior distribution $\pi$. We can, therefore, not only minimize the loss w.r.t. $u$ and $v$ but also w.r.t. the hyperparameters, e.g., in an end-to-end fashion via stochastic gradient descent. As our experiments show, this brings significantly improved performance, see, e.g., Figures 5 and 9, and is therefore also relevant for most other works in the field. We have previously outlined the algorithm and corresponding parametrizations in the appendix. However, we agree with your suggestion and added a revised version of our algorithm to the main part (see Alg. 1), which explicitly specifies how the hyperparameters are learned.

Finally, while end-to-end training of hyperparameters is an important ingredient in our algorithm, we want to emphasize that we use the same strategy also to improve our baselines. In particular, three other crucial ingredients for our strong performance are (1) the diffusion bridge sampler (DBS) (i.e., learning both forward and backward controls), (2) the underdamped version of the DBS (i.e., learning the acceleration instead of the velocity), and (3) improved integrators for underdamped diffusions (i.e., OBABO). This is evidenced by our ablation studies in Figs. 3, 4, 6, 7, & 8 and Tab. 1 & 6 (benefit of underdamped versions), Figs. 4, 6, 7, & 8 and Tab. 6 (benefit of integrators), and Figs. 5 & 9 (benefit of end-to-end training).

Relevance of degenerate case

The degenerate case allows for underdamped diffusions and has so far, as you mention, not been extensively considered for diffusion-based generative modeling and sampling. To the best of our knowledge, we are the first to provide a rigorous theoretical framework, including the Radon-Nikodym derivative for the degenerate case. As outlined in our paper, this can be used to easily derive objectives for generative modeling and sampling in the underdamped case. Such underdamped methods offer several appealing advantages, including the usage of integrators with better theoretical properties and numerical behavior, leading to significantly improved performance (Table 1 and Figure 3). We elaborate on the above advantages in our general response.

Splitting methods for sampling have been used before

To the best of our knowledge, existing works on underdamped Langevin processes focus exclusively on uncontrolled forward processes, such as LDVI [1] and MCD [2]. Additionally, these works primarily employ non-symmetric splitting methods, whereas we (a) are the first to approach general bridges using underdamped dynamics, i.e., controlled forward and backward processes, and (b) use symmetric splitting methods, which are theoretically proven to yield lower discretization errors [3] and demonstrate superior numerical performance (see Figure 3). While symmetric splitting is well-studied in the context of uncontrolled Langevin diffusions, it has not, to our knowledge, been applied in controlled settings.

It is important to note that employing different splitting schemes for controlled Langevin diffusions, unlike the uncontrolled case, necessitates careful selection of the forward and backward transition kernels $\overset{\rightarrow}{p}$ and $\overset{\leftarrow}{p}$, respectively (see Appendix A.7). This is crucial for deriving the (discrete-time) Radon-Nikodym derivative required for optimizing both the control functions and the associated hyperparameters.

Thank you for your response.

note that [6] does not consider a learning task, i.e., no loss functions are defined that can be used for learning a bridge

I am aware the methods proposed in [6,7] do not target sampling marginals but instead sampling the bridge. However the technique of [7] for training appears to be quite similar. See algorithm 1 on page 32, as well as description of the learning task on page 6, in section 1.3 (https://arxiv.org/pdf/1311.3606).

Here they learn just a single parameter but it appears to be a similar procedure. I do not see why it would also not work for learning more than 1 parameter such as a neural network.

end to end training

Thank you for the explanation, I think I understand the method better now. Essentially you take gradients through the whole simulation / integration of the SDE to perform gradient descent on the hyper-parameters? This will likely suffer memory issues for larger models, unlike the paradigm in diffusion generative models where gradients are not taken through time, similar to Diffusion Normalizing Flows https://arxiv.org/abs/2110.07579 or the first algorithm 2 of https://arxiv.org/abs/2110.11291. Or do you use stop gradient operations at each step? If using stop gradient will this not affect the ability to perform hyper parameter tuning?

It is not clear to me why likelihood would be a valid loss for learning some of the hyper parameters like T. What is the learnt T in the cases where this is tried? Can you explain the intuition here?

Reason why underdamped setting I appreciate your response explaining why underdamped is useful, this is primarily empirical? The author response mentions theoretical reasoning too but I cannot find it in the paper quickly can you point me to this and is there any intuition behind the empirical usefulness?

Minor Please note a minor typo in line 496, "Haperparameters"

I have increased my score.

To indicate clearly what has changed, we color-code the modifications according to the reviewer who suggested it as follows:

• orange: reviewer 5Kic
• magenta: reviewer Qe9N
• olive: reviewer nE8S
• red: reviewer xTDP
• blue: reviewer znkS

References

[1] Yi-An Ma, Niladri S Chatterji, Xiang Cheng, Nicolas Flammarion, Peter L Bartlett, and Michael I Jordan. Is there an analog of nesterov acceleration for gradient-based MCMC? 2021.

[2] Tim Dockhorn, Arash Vahdat, and Karsten Kreis. Score-based generative modeling with critically damped Langevin diffusion. arXiv preprint arXiv:2112.07068, 2021.

[3] Richter and Julius Berner. Improved sampling via learned diffusions. In International Conference on Learning Representations, 2024.

[4] Francisco Vargas, Shreyas Padhy, Denis Blessing, and Nikolas Nüsken. Transport meets variational inference: Controlled Monte Carlo diffusions. In The Twelfth International Conference on Learning Representations, 2024.

[5] Pierre Monmarche. High-dimensional MCMC with a standard splitting scheme for the underdamped Langevin Diffusion. Electronic Journal of Statistics, 15(2):4117–4166, 2021.