We sincerely thank the reviewer for the thoughtful and constructive feedback. We particularly appreciate the recognition of our paper’s originality, clarity, and experimental design. Below, we address the two major questions, which both touch on the foundational motivations behind this work. We also answer the additional concerns of the reviewer.

• Q1: On the restriction to conformal metrics: This is an excellent and timely question. Our initial experiments were not limited to conformal metrics. We tested the Stein metric $s_\theta(x).s_\theta(x)^T$, s.t. $s_{\theta}(x) = \nabla_x E_{\theta}(x)$, which has been proposed as a valid Riemannian metric for EBMs [25], and also explored the Hessian-based metric in early trials. While both define valid geometries, we found that conformal metrics matched the performance of the Stein metric while offering clearer interpretability and significantly lower computational cost. Furthermore, for the Gaussian case, the Hessian of the energy yields geodesics that are straight lines—ignoring mass concentration around the mean—which we found undesirable. Finally, computing and storing the full metric tensor incurs a quadratic cost in dimension. Conformal metrics align with density level sets and produce isotropic, interpretable geodesics. In contrast, non-conformal metrics can introduce anisotropy and directional bias, offering flexibility but at the cost of control and interpretability. For these reasons, we chose to focus on the conformal case. To clarify this design choice, we propose adding the following sentence to the conclusion (line 346): “While more complex non-conformal metrics (e.g., the Stein metric) are accessible from the EBM score, we found that conformal metrics offer comparable performance with simpler interpretation and reduced computational cost, justifying our focus in this work.”

• Q2: On envisioned applications and cost-benefit tradeoff: We agree that assessing the cost-benefit tradeoff is essential when evaluating practical impact. Training EBMs is indeed computationally expensive, but we believe the cost is justified in domains where (i) data are high-dimensional and sparse, and (ii) precise control over the geometry of the latent space is critical. In these domains, high-fidelity geodesics ensure that interpolated paths (e.g., geodesics) remain meaningful and do not cut through implausible regions. A prime example (and the original motivation for this work) is the modeling of neural manifolds: the geometry traced by high-dimensional population neural activity recorded over time. In this context, a low-fidelity approximation of the data geometry is insufficient, as downstream interpretations rely on fine-grained geometric structure. We are exploring this in an ongoing work in which we show that geodesic length in learned neural manifolds correlates with human reaction times in object recognition tasks, suggesting that energy-derived geometry can capture behaviorally relevant structure. This paper lays the theoretical and empirical foundation for such applications by validating the method on controlled benchmarks. To better highlight the cost-benefits tradeoff of our method, we propose adding the following sentence to the discussion (l. 356): “This approach is particularly relevant for neuroscience, where datasets are high-dimensional, often sparsely sampled, and where understanding the geometry of neural population activity is central to scientific insight. In such settings, high-fidelity geodesics are essential for capturing the true structure of neural trajectories—approximations may distort the manifold and lead to misinterpretation of brain dynamics. While training EBMs is costly, the benefits in terms of interpretability and geometric accuracy make this approach compelling for applications where precision is critical.”

• Q3: On geodesic behavior with manifold gaps. This is an important and insightful question. When the data manifold contains significant gaps—such as missing intermediate orientations—our method does not impose a fixed strategy but instead lets the learned energy landscape shape the geodesic. Since the Riemannian metric is derived from the EBM and inversely related to data density, geodesics are encouraged to follow high-density (i.e., plausible) regions. However, if no high-density path exists between two endpoints, the geodesic may cut across a low-density gap if doing so yields lower accumulated energy than detouring through unrelated regions (e.g., another character’s manifold). This behavior reflects a key strength of our framework: geodesics adapt to the structure learned by the EBM and naturally trade off between identity preservation and plausibility. In ambiguous cases, the path reflects the most efficient solution under the current energy landscape. We agree that this behavior would be best illustrated with a concrete simulation. While this was beyond the scope of our initial evaluation, we will include a follow-up analysis in the supplementary material to examine geodesic behavior on discontinuous or sparse manifolds. This will help clarify when and how shortcuts or detours emerge under different density configurations.

• Q4: Clarification on Figures 1 and 3: Thank you for pointing this out. We will revise the captions to improve clarity. For Figure 1, we will explicitly mention that the trajectories are projected into the first two PCA components of the latent space. For Figure 3, the x-axis represents optimization steps, and the y-axis is the Euclidean norm of the geodesic velocity vector at each step. We will also expand the paragraph describing Figure 3 to clarify the stability behavior and interpret the drop in step size over training.

• Q5: On finite difference stability: We observed that computing velocities via finite differences was more stable and accurate (and also much faster) than using full auto-differentiation. This is likely due to the fact that finite differences avoid computing second-order gradients through the geodesic interpolant network, which can be noisy or unstable in practice. Additionally, the network typically takes the time in input with a time embedding layer (with high frequency sine functions, a la flow matching), whose first order derivative can be ill-conditioned. Finally, finite differences act as a form of local smoothing, making the velocity estimate less sensitive to local curvature irregularities. This contributes to both numerical stability and faster convergence.

• Q6: On grayscale robustness of plots: Thank you for the suggestion. We will update key visualizations to improve grayscale robustness by using colorblind-friendly palettes, distinct line styles, and/or markers to ensure readability when printed in black and white.

We hope these clarifications address your concerns and will contribute to a positive reassessment of the paper. We remain fully open to further discussion and sincerely thank you again for your thoughtful and constructive review.

Thank you for your detailed responses.

I agree that including the additional application areas you mentioned would strengthen the paper by better justifying the method's cost-benefit tradeoff. Thank you also for clarifying how positive-definiteness is guaranteed by construction.

Regarding your response to Q3, I am still not fully convinced that allowing geodesics to cross low-density gaps is an inherent advantage. In situations with insufficient data, finding a principled geodesic is fundamentally challenging for any method. The behaviour of your model is a direct consequence of its objective, but whether this outcome is desirable is highly context-dependent. While this is not a flaw of your method (as no approach can perfectly resolve data sparsity), I would be hesitant to frame this specific adaptive behaviour as a definitive strength.

Overall, I believe this is a very nice work, and I will update my score to reflect this positive assessment.

We thank the reviewer for the feedback.

• About the “lack of theoretical support of the proposed method” (weakness 1): We would like to clarify that the notion of a “data manifold” is not an observable ground truth but a modeling assumption introduced by the experimenter. Consequently, there can be no general theoretical guarantee on the optimality of any chosen Riemannian metric, since manifold recovery from finite, noisy data is an inherently underdetermined problem. Defining a metric requires imposing structure—just as in Topological Data Analysis, where researchers rely on persistent homology. Our approach follows a well-established principle [13, 24, 33, 38]: that data density can guide the design of a meaningful Riemannian geometry. We show that this modeling choice leads to empirically useful geodesics that capture structure in the data. The strength of our empirical validation supports the relevance of this choice. As noted by other reviewers: Reviewer yiCX described our evaluation as “convincing,” Reviewer z3pm highlighted the “impressive performance” of our method, and Reviewer RvU6 acknowledged both its novelty and relevance.

• About the experiment on the toy dataset (weakness 3): In the toy experiment (Section 4.2), the data are sampled from a 2D mixtures of Gaussian arranged along a semi-circle. While we agree that the shortest path lies along a 1D structure (lying on the $\mathcal{S}^1$), we emphasize that our geodesic optimization takes place in the full ambient space ($\mathbb{R}^2$), and the initial trajectory is a straight line between the endpoints—thus not aligned with the manifold $\mathcal{S}^1$. The optimization must therefore rely entirely on the learned Riemannian metric to steer the path toward the data manifold, without any explicit constraints enforcing manifold adherence. This setup makes the problem more challenging than optimizing directly on $\mathcal{S}^1$, and demonstrates the method’s ability to recover geodesics that align with the data structure from off-manifold initializations.

• Q1-2 (benchmarking geodesics): As stated previously, the notion of a "data manifold" is not an observable ground truth but a modeling assumption introduced by the experimenter. In this work, we adopt a common and well-supported choice: equipping the ambient space with a density-based Riemannian metric, following prior work [13, 24, 33, 38]. This assumption defines both the manifold’s geometry and the corresponding geodesics. Under this modeling framework, our evaluations are internally consistent. In Section 4.1, we construct toy datasets with known density functions, which allows us to define a ground-truth metric and geodesics analytically. This setup enables a principled evaluation of geodesic accuracy using RMSE—not merely of density estimation, but of how well the learned metric captures the induced geometry. We agree that this modeling assumption could have been more clearly stated in the paper. To address this, we will add the following sentence at line 165: “Throughout this work, we adopt the common choice of equipping the data space with a density-based Riemannian metric, thereby defining the geometry of the manifold in terms of data concentration.”

• Q3 (related to weakness 2): Section 4.3 was specifically designed to address this concern. In high-dimensional settings—as in AFHQ (Section 4.4)—it is indeed difficult to obtain ground-truth geodesics. To mitigate this, we introduced the rotated character setup, where we can define a meaningful proxy for geodesic paths. The latent space is regularized using a triplet loss to ensure that small rotations of the same letter are mapped to nearby points. As a result, the shortest path between two rotated versions of the same character corresponds to the smoothest in-plane rotation, which serves as a valid approximation of the true geodesic in this space. The $\gamma^{\star}$-RMSE metric then quantifies how closely our predicted geodesics align with this ground-truth proxy. To clarify this in the manuscript, we propose adding the following sentence at line 273: “This setup provides a unique middle ground: although the underlying Riemannian metric is unknown, we can treat the smooth in-plane rotation between two instances of the same character as a proxy for the ground-truth geodesic. Thanks to the triplet loss, the latent space is structured so that nearby points correspond to slight rotations of the same character, making the shortest path between two orientations a meaningful approximation of the true geodesic in the task-relevant transformation space.”

• Q4 (about the Dijkstra baseline): We agree that Dijkstra’s algorithm can serve as a strong baseline in low-dimensional settings, and we will include it in Section 4.2. However, it’s important to note that Dijkstra operates on a discretized graph and must rely on the Riemannian metric derived from our EBM to reflect manifold structure. As such, it does not directly assess the quality of the learned metric, but rather the quality of geodesics we compute. Extending Dijkstra to high-dimensional spaces is computationally infeasible due to the curse of dimensionality. To approximate a geodesic with ( $T = 100$) discrete steps, a naive discretization requires evaluating a graph with ( $\mathcal{O}(d^T)$) nodes, where ( $T$) is the number of discretization points per dimension. Even if the intrinsic manifold has dimension ( $d$ ), the search frontier at each step spans a $(d - 1)$-dimensional surface, resulting in memory costs of $\mathcal{O}((d - 1)^T)$. This makes Dijkstra intractable beyond very low-dimensional settings (see, e.g., [87, 88] for examples in 3D).

We hope this clarifies our methodological choices and reinforces the coherence of our evaluation framework, demonstrating that our work offers a meaningful contribution to the community.

Dear Reviewer RvU6,

All reviewers are supposed to engage with author responses to reviews. Please respond to the authors' response directly as well as in our internal discussion. Given the lack of response so far, I may lower the weight of your review in my assessment.

Best,

Your AC

We thank the reviewer for the thoughtful and constructive feedback. Below, we address the main concerns regarding the clarity of our hyperparameter choices, the scalability of our approach, and the stability of EBM training.

• Q1: On the selection of the $\alpha$ and $\beta$ parameters: Thank you for pointing this out. The parameters $\alpha$ and $\beta$ are used to scale the energy (and likelihood) derived Riemannian metrics for comparability across methods. Specifically, we set $\alpha$ and $\beta$ so that all tested metrics share the same minimum value on the data manifold (normalized to 1) and a maximum value of $10^3$ off the manifold. This normalization ensures that observed differences in performance (e.g., FID, smoothness) are not due to scale differences between metrics, but rather to the structure they induce. Empirically, we found that performance was robust to a wide range of $\alpha$ and $\beta$ values, as long as this normalization was preserved. To clarify this in the manuscript, we propose adding the following sentence at line 194: “We normalize all metrics by choosing $\alpha$ and $\beta$ such that the minimum Riemannian energy on the data manifold is 1, and the maximum energy in off-manifold regions is $10^3$. This allows fair comparison across metric choices without introducing significant sensitivity to hyperparameter tuning. Note that only the ratio alpha/beta influences the geodesics.”

• About the training of the EBM: We first emphasize that the core contribution of our paper is primarily methodological, and serves as an additional justification for the development of EBMs. Our contribution is not only to propose interpolants while using EBM and metrics, it is also to demonstrate that EBM can be used to define a geometry over the data manifold. Geodesics interpolants are merely a consequence of this modeling choice. We are optimistic that further developments in EBM training will enhance our results (backed by the elements below). We agree that training EBMs can be challenging, particularly in high-dimensional settings. However, recent advances—some of which we leverage in this paper—have made EBM training significantly more stable and scalable:

  • Latent space modeling: Rather than training EBMs on raw pixels, we operate in a pretrained latent space that is smoother, lower-dimensional, and semantically structured—greatly improving training stability and efficiency. For our high-dimensional experiment (AFHQ), we used the same VAE encoder as Stable Diffusion, which maps natural images into a 1024-dimensional latent space. This latent space has been shown to capture rich, fine-grained structure in large-scale datasets like LAION-5B. As such, the use of a 1024-dimensional latent does not reflect a limitation of our method, but rather a design choice aligned with modern generative modeling practices.
  • Denoising Score Matching regularization: While not widely documented in the literature, we found that adding a denoising score matching (DSM) term to the contrastive divergence loss significantly improves training stability and convergence of the EBM. This regularization smooths the energy landscape and reduces sensitivity to small-scale noise in the latent space—without compromising the model’s expressiveness. Importantly, it does not impair the key advantage of contrastive divergence: the learned energy landscape remains global (i.e., not tied to a specific noise level), which is essential when using it to define a Riemannian metric. Empirically, this regularization appears to improve both performance and robustness. We are currently preparing a dedicated article that systematically benchmarks its impact across tasks and architectures. To make this point explicit in the manuscript, we propose adding the following sentence at line 185: “To improve stability, we regularize the contrastive divergence loss with a denoising term, which preserves the global structure of the energy landscape while enhancing convergence—a technique we find both effective and broadly applicable.”
  • Established EBM training techniques: In addition to the above, we apply well-known stabilization techniques from the EBM literature [16], including support clipping (to constrain latent codes during Langevin dynamics) and replay buffers (to initialize Langevin sampling from previously generated examples, with periodic resets). These practical tricks further contribute to stable training dynamics.

  Together, these strategies allow us to train stable and expressive EBMs at scale—without compromising the quality of the learned energy landscape. We believe that these practical insights will facilitate the adoption of our method in more demanding applications. We thank the reviewer again for these valuable comments, which led us to clarify key aspects of our approach and improve the overall quality of the paper.

We thank the reviewer for continuing the discussion and appreciate the opportunity to clarify our response regarding scalability. There may have been a misunderstanding, as our initial rebuttal already addressed this point in the paragraph titled ‘Latent space modeling’. Nonetheless, we are happy to take this opportunity to further clarify and elaborate on it.

Our method ensures scalability by operating in pretrained latent spaces, rather than directly training EBMs on high-dimensional pixel data—a task known to be challenging. For the AFHQ dataset (composed of high-resolution natural images of size 512x512), we use the latent space of Stable Diffusion’s (i.e. a pretrained VAE, which is 1024-dimensional). This latent space has been shown to capture rich semantic structure across large-scale, high-resolution datasets such as LAION-5B [18]. This design choice aligns with standard practice in modern generative modeling and, as demonstrated by latent diffusion models [18], improves the geometric smoothness of the latent space and enhances the scalability of training by reducing the dimensionality and complexity of the data representation. As a result, it mitigates the typical issues of instability and slow convergence in high-dimensional EBM training and enables our approach to scale effectively. To further enhance scalability and robustness, we incorporate several stabilization techniques, including denoising score matching (DSM), replay buffers, and support clipping. These techniques improve training dynamics without compromising model expressiveness. Importantly, our goal in this paper is not to push toward the largest possible models, but to validate the method in a regime that is already aligned with modern generative pipelines. Given sufficient compute resources, our approach can be straightforwardly scaled to more complex datasets or higher-dimensional latent spaces.

We acknowledge the reviewer’s feedback that these points could be emphasized more clearly. We propose highlighting explicitly in the manuscript that operating within pretrained latent spaces is not merely a convenience but a deliberate methodological choice to ensure scalability to high-dimensional, complex datasets.

We thank the reviewer for the thoughtful and constructive feedback. We greatly appreciate the recognition of the paper’s originality, clarity, and mathematical formulation. Below, we address each question and concern in detail, while also proposing concrete edits to clarify and strengthen the manuscript.

• Q1 : About the form of the Riemannian metric: Thank you for raising this important question. The conformal metric we use, defined as $G(x) \propto p(x)^{-1}⋅I$, indeed assumes isotropy at each point in space—it scales all directions equally, without modeling directional preferences or anisotropies. While this does not imply full independence of latent factors, it does neglect directional correlations that could be captured by a full, non-conformal metric tensor. This modeling choice is grounded in a long-standing tradition in Riemannian geometry and manifold learning, where density-based conformal metrics are used to contract high-density regions and expand low-density regions (e.g., [13, 24, 33, 38]). This heuristic offers a compelling balance between expressiveness and interpretability. In our early experiments, we also explored non-conformal alternatives, particularly the metric explored in the blog of Perone [25] ( $s_{\theta}(x).s_{\theta}(x)^{T}$ with $s_{\theta}(x) = \nabla_x E_{\theta}(x)$). However, we found that (i) it offers no performance gain, (ii) it is harder to interpret, and (iii) it is significantly more expensive to compute due to nested gradients. For these reasons, we focused on the conformal case. We propose to clarify this in the conclusion by adding: “While more complex non-conformal metrics (e.g., the Stein metric) are accessible from the EBM score, we found that conformal metrics offer comparable performance with simpler interpretation and reduced computational cost, justifying our focus in this work.”

• Q2: Code release: We fully agree that open-sourcing the code improves transparency and reproducibility. Although the codebase was not ready for public release at the time of submission, we have since prepared a clean version. Due to NeurIPS rebuttal guidelines, we are unfortunately not allowed to share any links at this stage, including an anonymous GitHub repository. Nevertheless, we want to reassure the reviewer of our strong commitment to open science, and we will publicly release the full code upon acceptance.

• Q3 : About the RMSE in Fig 2: Thank you for this comment. In Figure 2, the RMSE is computed with respect to the ground-truth geodesics defined by a known Riemannian metric, which in turn is derived from the known data density used to generate each synthetic dataset (e.g., $G(x)\propto p(x)^{-1}.I$). All models—including LAND and RBF—are evaluated against this same reference metric to ensure fair and consistent comparison. We will clarify this point in the caption and main text.

• Q4 : About the latent space (also related to weaknesses 1): This is an insightful question. While it is theoretically possible to train an EBM directly in pixel space, doing so is both computationally intensive and geometrically problematic: Euclidean distances in image space are not perceptually meaningful. For this reason, we project images into a learned latent space where distances are more semantically aligned with task-relevant transformations. In the rotated character experiment (Section 4.3), we specifically use a triplet loss to regularize this latent space. The goal is to ensure that the shortest path between two instances of the same character with different orientations corresponds to a smooth in-plane rotation. The triplet loss enforces local structure such that small angular changes are reflected by small latent distances, while large rotations and identity changes are discouraged. This gives us a meaningful approximation of ground-truth geodesics that we have leveraged to benchmarked the different metrics. In general, the use of a latent space makes EBM training significantly easier, as the latent space is lower-dimensional and smoother than pixel space. To clarify this in the manuscript, we propose adding the following sentence at line 273: “This setup provides a unique middle ground: although the underlying Riemannian metric is unknown, we can treat the smooth in-plane rotation between two instances of the same character as a proxy for the ground-truth geodesic. Thanks to the triplet loss, the latent space is structured so that nearby points correspond to slight rotations of the same character, making the shortest path between two orientations a meaningful approximation of the true geodesic in the task-relevant transformation space.”

• A word about scalability of our method: We respectfully disagree with the concern that our method lacks scalability. While the latent dimensionality used in our AFHQ experiment is indeed $1024$, this corresponds to a spatial code of size 16×16×4—identical to the pretrained autoencoder used in Stable Diffusion, which is trained on the LAION datasets. This dimensionality has been shown to capture rich visual features across highly diverse image distributions, indicating that it is not a limiting factor. Importantly, our method operates directly on this latent space without architectural changes, demonstrating that it is readily compatible with state-of-the-art, large-scale generative pipelines. Our goal here was not to push to the largest possible models, but to validate the method in a regime already relevant to modern generative pipelines. The method itself imposes no architectural limitations and can easily be scaled to more complex datasets or higher-dimensional latents given appropriate compute resources..

• Concerning the geodesic interpolant: Our geodesic interpolant follows the formulation introduced in prior work [28], which decomposes the trajectory into two components. The first is a fixed linear interpolation, $(1-t)x_0 + t x_1$, which ensures that the curve starts at $x_0$​ and ends at $x_1$​. However, this linear path is only a geodesic in Euclidean space, which does not apply in our setting. To account for the underlying geometry, we add a learnable deformation term, $2 t (1-t) \phi(x_0,x_1, t)$, which vanishes at the endpoints and peaks at $t=0.5$, allowing the network $\phi$ to flexibly bend the curve while preserving boundary conditions. This design balances geometric expressiveness with stability and ensures that the interpolant satisfies geodesic constraints in the learned Riemannian space. To give more intuition about the geodesic interpolant, we propose adding the following sentence (l. 142): "Intuitively, our geodesic interpolant begins with a straight line between the endpoints and uses a neural network to compute a smooth curvature relative to this baseline—bending the path toward regions of higher data density, much like pulling a string taut over a curved surface that reflects the geometry of the data."

• About the limitations section: We agree with the reviewer. The limitations of our approach (e.g., assumptions about smoothness, dependence on latent space quality) are important and deserve more visibility. We will move the content of the Appendix Section 7 into the main paper and streamline the discussion to make it fit within the page limit.

We thank reviewer yiCX for the positive feedback, and we are particularly grateful for the recognition of the clarity of our research question, the novelty of our contribution, and the strength of our experimental evaluation.

• Q1: Our method involves two main stages: (i) training an Energy-Based Model (EBM), and (ii) training a geodesic interpolant network. The primary computational cost lies in stage (i). To mitigate the expense of training EBMs in pixel space, we follow a common strategy from generative modeling (e.g., Stable Diffusion) and operate in the latent space of a pretrained autoencoder. This significantly reduces dimensionality (e.g., 1024 for AFHQ) and lowers overhead. Stage (ii) (i.e. training the geodesic interpolant) is efficient and converges quickly. In particular, the use of finite difference to estimate velocities in Algorithm 1 (instead of full auto differentiation) significantly reduces computational cost while maintaining accuracy. Overall, we found the computational effort of the method to be reasonable and not limiting in practice. To clarify this in the manuscript, we propose adding the following sentence at line 343: “To keep computational cost manageable, we train the EBM in the latent space of a pretrained autoencoder and use finite differences for geodesic optimization, both of which significantly reduce complexity without compromising performance.”

• Q2 (related to Weakness 2): The main hyperparameters in our method are the energy scaling ratio alpha/beta and the number of discretization steps along the geodesic. We found the model to be highly stable across a wide range of scaling values. These constants are introduced primarily to ensure fair comparison across metrics by matching their dynamic range; they have negligible impact on convergence speed or geodesic quality. Regarding the number of steps, smaller values lead to faster convergence but coarser geodesics, while larger values improve geodesic fidelity at the cost of training time. Our choice of T=100 offers a good trade-off and is consistent with prior work (e.g., [13, 24]). To clarify this in the paper, we will include convergence plots for different values of T in the appendix and add the following sentence at line 230: “The scaling constants (\alpha, \beta) are introduced to ensure consistent dynamic range across metrics and have minimal impact on convergence or geodesic quality; the number of discretization steps is chosen as a trade-off between efficiency and accuracy, consistent with prior work.”

• Q3 (related to Weakness 1): We agree that our high-dimensional experiments lack access to ground-truth geodesics. This is a deliberate choice: unlike prior work that evaluates on synthetic manifolds with known metrics, our aim is to learn the Riemannian metric—via an Energy-Based Model—in complex domains where no closed-form metric exists, such as natural images. While this makes evaluation more challenging, it also highlights the originality and applicability of our approach. To partially mitigate this limitation, we include the rotated letter experiment, where the transformation space is well understood. The latent space, shaped by a triplet loss, ensures that nearby points correspond to small in-plane rotations of the same character. As a result, the smoothest rotation between two character poses serves as a meaningful proxy for the geodesic. To clarify this in the manuscript, we propose adding the following sentence at line 273: “This setup provides a unique middle ground: although the underlying Riemannian metric is unknown, we can treat the smooth in-plane rotation between two instances of the same character as a proxy for the ground-truth geodesic. Thanks to the triplet loss, the latent space is structured so that nearby points correspond to slight rotations of the same character, making the shortest path between two orientations a meaningful approximation of the true geodesic in the task-relevant transformation space.”

Please note that limitations are stated in appendix G. We sincerely thank reviewer yiCX for the constructive feedback, which helped us improve the clarity and completeness of the manuscript.