Note of Gratitude

We thank the reviewer for the response and the insights provided that can help strengthen the paper.

1. Do you have evaluation metrics to measure the trajectory smoothness? I am concerned about whether the signature kernel will result in sudden accelerations or sharp turns.

The strong repulsive force provided by the signature kernel may at first seem like a recipe for producing rough and non-smooth trajectories, as producing rough trajectories is one way to boost diversity. However, we found that this did not have an adverse effect on the smoothness. We include a smoothness penalty in the cost/loss function $\mathcal J$ to enforce trajectory regularity (see Sections 2 & 3 and Appendix J and the resulting trajectories are visualized in Figure 1, Section 7 and Appendix I) and SVES guarantees convergence to a locally optimal trajectory with mild conditions on the chosen kernel (that signature kernel satisfies). The proof of convergence is provided in Appendix A of [1]. So, the trajectories should converge to a set of smooth trajectories provided that there is a smoothness penalty in the cost/loss function.

After the rebuttal, an ablation study that tunes the trajectory smoothness and compares the performance of the kernels with respect to the trajectory smoothness will be provided along with the proof of convergence.

2. Could you explain in more detail how the signature kernel differs from the RBF kernel when it comes to comparing two similar trajectories?

We provide a brief mathematical reasoning for why the signature kernel makes a good kernel, it is discussed in lines 154-158 & 768-775 (the reviewer may also refer to references from [11,14,15,19,61,62] from our paper). A short and oversimplified answer is that the signature transform is injective on all non-tree-like paths, meaning no two such paths share the same signature. But this is not the case for the RBF kernel. Consider paths:

1. $p_1(t) = (x_1(t),y_1(t)) = (t, 0) \quad t \in [0, 2\pi]$
2. $p_2(t) = (x_2(t),y_2(t)) = (t, sin(t)) \quad t \in [0, 2\pi]$
3. $p_3(t) = (x_3(t),y_3(t)) = (t, sin(t + \pi)) \quad t \in [0, 2\pi]$

It is easy to see that $k^{RBF}(p_1,p_2) = k^{RBF}(p_1,p_3)$. But this not the case with the signature kernel due the the signature transform being invective up to “tree‑like” equivalence.

Weakness

Weakness 2:

This is briefly discussed in Section 1 of our paper. In short, classical ergodic planners focus on steering a single trajectory (or a centrally controlled fleet) to match a target spatial distribution. In contrast, our SVES‐diversification methods generate an set of distinct or diverse ergodic trajectories in parallel, leveraging repulsive interactions or signature‐based seeding to ensure both high coverage and trajectory diversity in a unified optimization. Refer to Section 1 & 2 of [1] and Section 1 and Appendix A of our paper for a more formal take on this. The reviewer can also choose to refer the response provided to reviewer KSg3.

Weakness 3:

Because our primary goal is to demonstrate diversification within SVES, we intentionally replicated the ergodic search experiments from the original Stein Variational Ergodic Search paper. Throughout the paper, we also also include references to prior successes of the signature kernel in robotics applications, for example:

1. reference [11] from our paper used the signature kernel to produce a diverse set of solutions in motion planning tasks such as manipulation.
2. reference [17] from our paper used the signature transform for Human Action Recognition.
3. reference [18] from our paper used the signature kernel for trajectory following/stabilization.

In addition, the reviewer may also refer to [2], where the authors used the signature transform in data driven control.

Typo:

Clarification of the Constrained vs. Penalized Formulation

In the main text we first posed the constrained optimization problem as ${\text{minimize}} \quad \mathcal{E}_\pi(x(t)), \qquad \text{subject to} \quad h_1(x) = 0, \quad h_2(x) \le 0, \quad \forall t \in [0, T]. \qquad \quad (1)$

There is no typo in Eq 3 of our paper; instead, we switch to an unconstrained, penalty‑augmented version of (1) by adding a hinge‐loss term that penalizes any violation of the inequality constraint. Concretely, we define

=\mathcal E_\pi(x) +\rho(x) +c_1\,h_1(x)^2 +c_2\max (0,h_2(x)). \qquad \quad (2)$$ ## Why this is correct and not a typo: 1. The hinge function $\max(0,h_{2}(x))$ is $0$ whenever $h_{2}(x)\le0$, i.e. the original constraint is satisfied. 2. If $h_{2}(x)$ becomes positive (constraint violation), $\max(0,h_{2}(x))=h_{2}(x)>0$, so the penalty term $c_{2}\,h_{2}(x)$ grows linearly, discouraging infeasible solutions. 3. Thus (2) is a standard “soft‐constraint” or “penalty” reformulation of the hard constraint in (1), not a mistake in the sign or definition. # References [1] Darrick Lee, Cameron Lerch, Fabio Ramos, and Ian Abraham. Stein Variational Ergodic Search. arXiv preprint arXiv:2406.11767, 2024. https://arxiv.org/abs/2406.11767 [2] Anna Scampicchio and Melanie N. Zeilinger. On the role of the signature transform in nonlinear systems and data-driven control. arXiv preprint arXiv:2409.05685, 2025. https://arxiv.org/abs/2409.05685

1. How do the computational costs of each choice of kernel compare? What about the cost for SV-CMA-ES over SVGD? I know this depends a lot on the choice of hardware, but some numbers would help give a ball-park idea.}

We thank the reviewer for pointing out the lack of computational benchmarks in our original submission. We agree that providing concrete estimates of computational costs for different kernel choices, as well as comparing the runtime complexity of SV-CMA-ES versus SVGD, would help clarify the practical implications of our method. We provide the following analysis in response.

Theoretical Computational Cost Comparisons

Kernels Computational cost

We provided the computational costs for some kernels in Section 5 and Appendix F.

1. The time complexity of the RBF kernel is $O(Td)$, but in practice can be as low as $O(d)$ on modern hardware with T threads.

2. The time complexity of MRBF is $O\bigl(|\mathcal G| \cdot O(\mathrm{RBF})\bigr)$.

3. The signature kernel, the global alignment kernel and the kernelized DTW can be computed in $O(Td)$ with modern parallelized hardware.

SV-CMA-ES vs SVGD Computational cost

The original Stein Variational Gradient Descent (SVGD) paper (reference [10] from our paper) identifies the gradient of the log‑density as its primary computational bottleneck. Comparing the costs of SVGD and SV‑CMA‑ES is therefore nontrivial, since both methods require evaluating gradients of the kernel and of the log‑density at each update. The key difference lies in the cost of obtaining the log‑density gradient itself. When that gradient is easy to compute or available analytically, SVGD will generally be the more efficient choice. By contrast, SV‑CMA‑ES truly shines when the log‑density is nondifferentiable or expensive to evaluate.

In practice, computing the log‑density gradient typically entails two passes through the density function (a forward and a backward pass). Evolution‐strategy (ES) updates, however, are highly parallelizable on GPUs. The ES updates require multiple parallel forward passes of the log density, and computing the weighted recombination step for each particle can all be batched on modern hardware (with say Cholesky decomposition). Thus, even though SV‑CMA‑ES performs more sampling “work,” its overhead beyond objective evaluations is often quite small. If the log density is non‑differentiable or costly to backpropagate, SV‑CMA‑ES can actually deliver faster wall‑clock convergence by avoiding gradient calls altogether. Moreover SV-CMA-ES can have faster convergence rates when compared to SVGD with well-tuned hyperparameter(see reference [25] from our paper).

Experimental Benchmarks

In the following, we report mean run-times (over 10 independent runs) for SVGD and SV‑CMA‑ES on a common hardware platform, and discuss implementation caveats.

Experimental Setup

The SVGD implementation was fully JIT‑compiled, whereas SV‑CMA‑ES was only partially JIT‑compiled. No optimizations were applied to the signature kernel, so absolute timings may differ under a more optimized implementation. All experiments were run using the following hardware:

1. CPU: AMD Ryzen 7 7800X3D (8 cores @ 4.2 GHz)
2. Memory: 32 GB (@ 5600 MT/s)
3. GPU: NVIDIA Geforce RTX 5060 Ti

Results

1. Please provide runtime analysis on a common hardware platform. Ideally benchmark with different number of particles, and problem dimensions?

We thank the reviewer for requesting a detailed runtime comparison. In the following, we report mean runtimes (over 10 independent runs) for SVGD and SV‑CMA‑ES on a common hardware platform, and discuss implementation caveats.

Experimental Setup

The SVGD implementation was fully JIT‑compiled, whereas SV‑CMA‑ES was only partially JIT‑compiled. No optimizations were applied to the signature kernel, so absolute timings may differ under a more optimized implementation. All experiments were run using the following hardware:

1. CPU: AMD Ryzen 7 7800X3D (8 cores @ 4.2 GHz)
2. Memory: 32 GB (@ 5600 MT/s)
3. GPU: NVIDIA Geforce RTX 5060 Ti

Results

SVGD vs SV-CMA-ES

The original Stein Variational Gradient Descent (SVGD) paper (see reference [10] from our paper) identifies the gradient of the log‑density as its primary computational bottleneck. Moreover, comparing the costs of SVGD and SV‑CMA‑ES is nontrivial, since both methods require evaluating gradients of the kernel and of the log‑density at each update. The key difference lies in the cost of obtaining the log‑density gradient itself. When that gradient is easy to compute or available analytically, SVGD will generally be the more efficient choice. By contrast, SV‑CMA‑ES truly shines when the log‑density is nondifferentiable or expensive to evaluate.

In practice, computing the log‑density gradient typically entails two passes through the density function (a forward and a backward pass). Evolution‐strategy (ES) updates, however, are highly parallelizable on GPUs. The ES updates require multiple parallel forward passes of the log density, and computing the weighted recombination step for each particle can all be batched on modern hardware (with say Cholesky decomposition). Thus, even though SV‑CMA‑ES performs more sampling “work,” its overhead beyond objective evaluations is often quite small. If the log density is non‑differentiable or costly to backpropagate, SV‑CMA‑ES can actually deliver faster wall‑clock convergence by avoiding gradient calls altogether. Moreover SV-CMA-ES can have faster convergence rates when compared to SVGD with well-tuned hyperparameter (see reference [25] from our paper).

2. Are there other efficient popular heuristics other than the median heuristic for selecting bandwidth of kernels? Might be interesting to benchmark these.

We appreciate the reviewers request to benchmark the different heuristics and an ablation study on the choice of heuristics will be included in the final revision of the paper. The reviewer may refer to the response provided to the 4th question of reviewer KSg3 for additional information beyond the heuristic.

We found that the median heuristic is an extremely popular choice for choosing the bandwidth across multiple domains and provides some strong guarantees [1]. Some of it's advantages include:

1. Fully data‑driven and tuning‑free: It requires no extra cross‑validation or held‑out data: you just compute all pairwise distances and take their median.
2. Asymptotic consistency and normality: Under very mild assumptions (e.g. a mix of two distributions), the empirical median of squared distances converges to the true median of a well‑defined “mixture” distance distribution, and indeed satisfies a central‐limit theorem.
3. Separation‑aware behavior: If the two underlying distributions are well separated, there’s even a high‑probability gap between within‑ and between‑group distances. In that case, the median will reliably fall among the “within‑group” distances, yielding a bandwidth tuned to the scale of each cluster rather than to the far‑apart intercluster distances.
4. Keeps the kernel “in the sweet spot”: By construction, it avoids two degenerate extremes; too small a bandwidth or too big a bandwidth. The median sits roughly in the “middle” of the distance spectrum, so you capture meaningful variation without blowing up or collapsing the Gram matrix.

Additionally, we employed the median heuristic for bandwidth selection because it is straightforward, computationally light, and requires no manual tuning, making our method more accessible to users.

A computationally light heuristic is extremely critical, as it has to be used in very iteration of SVGD or SV-CMA-ES. Although several computationally cheap heuristics (like the mean heuristic, Silverman’s rule‑of‑thumb, and Scott’s rule; see the table below) are available they do not in general enjoy non‑asymptotic or distribution‑free theoretical guarantees.

References

[1] Damien Garreau, Wittawat Jitkrittum, and Motonobu Kanagawa. Large sample analysis of the median heuristic. arXiv preprint arXiv:1707.07269, 2018. https://arxiv.org/abs/1707.07269

1. Could you expand the related work to compare with other ergodic planners or diversity-focused methods outside SVES?

We thank the reviewer for pointing out the need for including more related work to better position the contribution.

Classical ergodic planners focus on steering a single trajectory (or a centrally controlled fleet) to match a target spatial distribution (Section 1 and Appendix A). In contrast, our SVES‐diversification methods generate a set of distinct or diverse ergodic trajectories in parallel, leveraging repulsive interactions or signature‐based seeding to ensure both high coverage and trajectory diversity in a unified optimization.

Outside of the ergodic literature, the community has developed a rich variety of quality‐diversity and diversity‐maximization techniques such as MAP-Elites [1], Novelty Search [2] and Determinantal point processes [3]. While these methods successfully assemble static repertoires of diverse solutions, they do not directly optimize for ergodic coverage over time. Our contribution is to bring diversity‐aware techniques into the ergodic planning domain.

At a high level, MAP‑Elites keeps the single best example for each kind of behavior so you end up with a set of good but different options. Novelty Search a evolutionary-computation paradigm, abandons traditional fitness objectives, instead it rewards behavioral divergence; it rewards experiments that lead to brand‑new behaviors, so the search naturally promoting exploration. Determinantal point processes (DPP) give a straightforward way to pick a batch of items that are both high‑quality and as different from one another as possible.

2. Could comparisons be made to other non-SVES baselines?

Prior work suggests that the non-convex form of ergodic search methods are capable of identifying multiple optimal trajectories through variations in initial conditions (see reference [9] from our paper). Hence, the main focus of our work is to produce a set of diverse locally optimal trajectories with effective mode coverage. As long as conditions for optimality are the same across both the SVES based and the non-SVES algorithms both should converge to same or similarly optimal trajectories since SVES guarantees convergence (see [2] from our paper).

3. Is there any hardware or real robot demo?

Thank you for highlighting the importance of real‐world validation. In fact, we have conducted hardware experiments with the Crazyflie drone. Figure 1 in the paper illustrates the paths taken by the Crazyflie drone in a real environment, and the quantitative results are reported in Section 7.2. To further underscore these real‐robot findings, additional images and videos can be added after this rebuttal.

4. Could you provide more details on how sensitive results are to kernel parameters?

We once again appreciate the reviewer suggesting a deeper investigation into bandwidth sensitivity as this is indeed very important and not well presented in the paper. We will add the following augments to the Appendix to strengthen the paper.

We employed the median heuristic for bandwidth selection because it is straightforward, computationally light, and requires no manual tuning, making our method more accessible to users. Additionally, refer to [4] for details on the guarantees provided by the median heuristic.

However, we find that setting $h=0.1$ for all kernels in the experiment from Section 7—the signature kernel still outperforms the other kernels. We did not find the optimal bandwidth for each kernel (say by grid searching) and compare them. This limitation is acknowledged in Section 9.

But in theory, we expect the signature kernel to retain its advantage thanks to its theoretical properties (discussed in lines 154-158 & 768-775 of our paper), most notably, the injectivity of its feature map. For a deeper mathematical treatment, we refer the reviewer to [11,14,15,19,61,62] in our paper. To illustrate one consequence of injectivity, consider three simple paths

1. p_1(t) = (x_1(t),y_1(t)) = (t, 0) \quad t \in [0, 2\pi]\

2. $p_2(t) = (x_2(t),y_2(t)) = (t, sin(t)) \quad t \in [0, 2\pi]$

3. $p_3(t) = (x_3(t),y_3(t)) = (t, sin(t + \pi)) \quad t \in [0, 2\pi]$

It is easy to see that $k^{RBF}(p_1,p_2) = k^{RBF}(p_1,p_3)$. In contrast, because the signature transform is injective on all non–tree‑like paths, no two such trajectories share the same signature, and thus the signature kernel cleanly separates these examples. In general the kernelized DTW's and Global alignment kernel's are not feature maps are not injective as well.

References

[1] Jean-Baptiste Mouret and Jeff Clune, Illuminating search spaces by mapping elites, 2015, arXiv:1504.04909, https://arxiv.org/abs/1504.04909

[2] Joel Lehman and Kenneth O. Stanley. Abandoning objectives: Evolution through the search for novelty alone. Evolutionary Computation, 19(2):189–223, Summer 2011. MIT Press, Cambridge, MA, USA. https://doi.org/10.1162/EVCO_a_00025

[3] Alex Kulesza. Determinantal Point Processes for Machine Learning. Foundations and Trends® in Ma- chine Learning, 5(2–3):123–286, 2012. Now Publishers. http://dx.doi.org/10.1561/2200000044

[4] Damien Garreau, Wittawat Jitkrittum, and Motonobu Kanagawa. Large sample analysis of the median heuristic. arXiv preprint arXiv:1707.07269, 2018. https://arxiv.org/abs/1707.07269