Title: Diversifying Parallel Ergodic Search: A Signature Kernel Evolution Strategy
Venue: NeurIPS 2025
Accepted: yes
Keywords: robotics exploration

trajectory optimization

ergodic search

signature transform
Abstract: Effective robotic exploration in continuous domains requires planning trajectories that maximize coverage over a predefined region. A recent development, Stein Variational Ergodic Search (SVES), proposed parallel ergodic exploration (a key approach within the field of robotic exploration), via Stein variational inference that computes a set of candidate trajectories approximating the posterior distribution over the solution space trajectories. While this approach leverages GPU parallelism well, the trajectories in the set might not be distinct enough, leading to a suboptimal set. In this paper, we propose two key methods to diversify the solution set of this approach.
First, we leverage the signature kernel within the SVES framework, introducing a pathwise, sequence-sensitive interaction that preserves the Markovian structure of the trajectories and naturally spreads paths across distinct regions of the search space. Second, we propose a derivative-free evolution-strategy interpretation of SVES that exploits batched, GPU-friendly fitness evaluations and can be paired with approximate gradients whenever analytic gradients of the kernel are unavailable or computationally intractable. The resulting method both retains SVES’s advantages while diversifying the solution set and extending its reach to black-box objectives. Across planar forest search, 3D quadrotor coverage, and model-predictive control benchmarks, our approach consistently reduces ergodic cost and produces markedly richer trajectory sets than SVES without significant extra tuning effort.

Diversifying Parallel Ergodic Search: A Signature Kernel Evolution Strategy

Abstract

Effective robotic exploration in continuous domains requires planning trajectories that maximize coverage over a predefined region. A recent development, Stein Variational Ergodic Search (SVES), proposed parallel ergodic exploration (a key approach within the field of robotic exploration), via Stein variational inference that computes a set of candidate trajectories approximating the posterior distribution over the solution space trajectories. While this approach leverages GPU parallelism well, the trajectories in the set might not be distinct enough, leading to a suboptimal set. In this paper, we propose two key methods to diversify the solution set of this approach. First, we leverage the signature kernel within the SVES framework, introducing a pathwise, sequence-sensitive interaction that preserves the Markovian structure of the trajectories and naturally spreads paths across distinct regions of the search space. Second, we propose a derivative-free evolution-strategy interpretation of SVES that exploits batched, GPU-friendly fitness evaluations and can be paired with approximate gradients whenever analytic gradients of the kernel are unavailable or computationally intractable. The resulting method both retains SVES's advantages while diversifying the solution set and extending its reach to black-box objectives. Across planar forest search, 3D quadrotor coverage, and model-predictive control benchmarks, our approach consistently reduces ergodic cost and produces markedly richer trajectory sets than SVES without significant extra tuning effort.

1 Introduction

Robotic exploration-the autonomous process by which robots survey unknown or partially mapped environments to collect information, build maps, and accomplish tasks such as search-and-rescue, environmental monitoring, or infrastructure inspection. The success of robotic exploration hinders on the reliability of the robot to adapt its exploratory opteration in dynamic or unstructured settings where pre-programmed paths may fail. A key approach within this field is ergodic search [1, 2], in which trajectories are generated so that the robot's time-averaged visitation frequency matches a desired spatial distribution of information. By ensuring that time spent in each region is proportional to its importance, ergodic search provides systematic, efficient coverage of the domain and avoids both redundantly revisiting well-known areas and neglecting critical regions. In such applications where a robot must systematically explore an area, having multiple, robust, high-quality and diverse set of trajectories is critical to adapt to dynamic environments and quickly switch between search

strategies. Moreover, a set of diverse solutions aid in avoiding "bad" local minima and lead to a better optimization outcome.

Recent progress in curiosity- and information-based exploration has enabled robots to survey large, unstructured environments [3-7]. However, most existing methods rely exclusively on an informationmaximizing strategy, which often leads to myopic behavior: agents greedily seek immediate gains in information without regard for long-term, strategically advantageous states.

Ergodic exploration techniques address this limitation by casting exploration as a coverage problem, optimizing time-averaged trajectory statistics to ensure sustained visitation of highinformation regions [8, 9]. Concretely, these methods optimize the spatial distribution of trajectory dwell time against an expected information measure, allowing robots to plan exploratory paths that effectively handle multi-modal search landscapes [1]. Despite their demonstrated effectiveness, current ergodic algorithms still optimize only a single search policy at a time, constraining a robot's ability to

(a) RBF kernel

adapt when tasks or environments change.

Prior work has observed that the non-convex nature of the ergodic objective can yield multiple locally optimal trajectories under different initializations [9]. Yet explicitly reasoning over-and sampling from-a distribution of such trajectories is computationally prohibitive. Moreover, there is no assurance that distinct initial conditions will avoid collapsing onto the same ergodic solution, a critical shortcoming in online exploration scenarios where mode collapse can incur catastrophic performance failures.

Stein variational inference methods show promise in providing the necessary tools to approximate distributions of trajectories in a computationally tractable manner [10]. Motivated by these strengths, [2] proposes Stein variational ergodic search (SVES), a formulation of ergodic exploration as a Stein variational inference problem: by applying Stein variational gradient descent to the space of robot trajectories. However, SVES neither fully exploits the inherent Markovian structure of trajectories nor avoids a strong dependence on gradient information. Moreover, although SVES alludes to promoting diversity among trajectories, it does not concretely define this notion or provide what 'diversity' entails.

Contributions: We address the challenges mentioned above by "diversifying" Stein Variational Ergodic Search (SVES). Our approach is built on two complementary pillars. First, we leverage the signature kernel similar to [11], enabling us to encode the intrinsic sequential (or Markovian) and geometric properties of trajectories. The signature kernel naturally discriminates between subtly different paths, thus promoting diversity. Second, we incorporate Stein Variational Covariance Matrix Adaptation Evolution Strategy (SV-CMA-ES) [12] and Simultaneous Perturbation Stochastic Approximation (SPSA) [13] into SVES, drawing on recent progress in Stein Variational Evolution Strategies, this approach not only eliminates the need for gradients-often costly to compute-by harnessing GPU parallelism, making it viable for real-time applications, but also enables the use of complex non-differentiable or black-box loss functions and constraints. Later we show empirically that this approach leads to better optimization results and a more diverse set of solutions.

Paper organization: The remainder of the paper is organized as follows. In Section 2 we formalize the ergodic search problem, Section 3 reviews Stein variational ergodic search, Section 4 introduces

the signature transform and signature kernel, Sections 5 and 6 present our diversification strategies using the signature kernel and SV-CMA-ES respectively, Section 7 reports empirical results, Section 8 concludes and discusses the limitations and best practices.

2 Problem Statement

We consider a robot with state space X ⊆ R n and control space U ⊆ R m , whose continuous-time dynamics are

A robot's space trajectory x ( t ) : [0 , T ] →X is defined as the solution of the corresponding initialvalue problem (integrated over time):

We denote the set of all such finite-horizon trajectories by τ = ⋃ T> 0 C 0 ( [0 , T ] , X ) , as the set of all continuous trajectories defined on arbitrary finite time intervals. 1 The robot explores a bounded domain S = [0 , B 0 ] ×··· × [0 , B v -1 ] ⊂ R v , v ≤ n, via a projection g : X → S .

We assign each region of S an importance according to a target probability measure π . An ergodic cost E π ( x ) intuitively measures how well a trajectory x ∈ τ covers S in proportion to π : it penalizes regions that are under- or over-visited relative to their importance. 2 Driving E π ( x ) → 0 therefore ensures the robot spends time in each region proportional to its assigned importance (see Definition 5 in Appendix D a formal introduction on ergodic cost). Our objective is to compute feasible trajectories that minimize this ergodic cost while satisfying the constraints of the system. Over a fixed horizon T , we solve

where initial conditions x (0) and trajectory constraints can be accounted for in the equality and inequality constraints h 1 , h 2 , e.g., ˙ x = f ( x, u ) and u ∈ U .

In what follows, we develop Stein-variational and evolutionary-strategy methods to efficiently generate diverse high-quality ergodic trajectories in parallel.

3 Stein variational ergodic search

SVES reformulates ergodic search (a problem where a robot must explore the domain S according to a target distribution, see Appendix D) as an inference problem. A binary indicator O : τ ( X ) →{ 0 , 1 } , flags trajectories that satisfy the optimality criterion. The likelihood of O = 1 given a trajectory x is defined by p ( O = 1 | x ) = exp ( -µ E π ( x ) ) , where E π ( x ) is the ergodic cost and µ > 0 scales its influence. By combining this likelihood with a prior p ( x ) over the space of trajectories via Bayes' rule, one can obtain the posterior p ( x | O ) . We use Stein Variational Gradient Descent (SVGD) [10] (see Appendix B), an iterative algorithm to approximate this posterior p ( x | O ) ∝ p ( x ) exp ( -µ E π ( x ) ) with a set of particles so that their empirical distribution approximates the target posterior. At each iteration, particles are nudged toward to regions of high posterior density and while being repelled from one another to promote coverage, thus forming a sample-based approximation of the true posterior

Integrating dynamics and constraints: The dynamics and constraints h 1 ( x ) = 0 , h 2 ( x ) ≤ 0 , etc., can be folded into a composite cost

with ρ additional penalties and c i > 0 . The likelihood is then given by p ( x ) = exp ( -µ J π ( x ) ) . Throughout this paper, we set µ = 1 .

3.1 SVES over state trajectories

We discretize a continuous trajectory x ( t ) into x = [ x 0 , . . . , x T -1 ] , and chose a Gaussian prior p ( x ) = N (ˆ x, σ 2 I ) , where ˆ x interpolates boundary states. SVGD maintains N 'particles', in our case N trajectories { x i r } N i =1 and updates them via

where r is an arbitrary SVGD iteration. The first SVGD term pulls particles toward low-cost ergodic trajectories (or modes with high density) and is often referred to as the attractive force; the second, the gradient of the kernel disperses them to preserve coverage and is referred to as the repulsive force. Therefore, the choice of an effective kernel is essential for good mode coverage. A key insight to note is that SVGD allows the N particles to be updated in parallel.

3.2 SVES over control sequences

Similar to before, one can optimize over discrete control sequences u = [ u 0 , . . . , u T -1 ] , with the discretized dynamics x t +1 = F ( x t , u t ) , x 0 given , and a prior p ( u ) using SVGD and each controlparticle u i r is updated by

In an MPC-style loop, usually only the first control action is applied, the state is estimated, and the SVGD optimization is repeated for the shifted horizon.

See Appendix E for a more detailed introduction to SVES.

4 Signature Transform and Signature Kernel

The signature (transform) and its corresponding kernel [14] have proven to be highly effective in diverse applications-ranging from robotics to deep learning-where data is naturally represented as a continuous path [14-19]. Informally, the signature transform of a trajectory can be viewed as analogous to the Fourier series, where the trajectory is represented as a countably infinite sequence. We now introduce the signature (transform) of a path/trajectory and its corresponding kernel:

Definition 1. (Informal) Signature (transform) of a path [20]. Consider a continuous path x : I → R d on an interval I ⊂ R . For any subinterval [ s, t ] ⊂ I , the signature of x over [ s, t ] is defined as the sequence

where ⊗ denotes the classical tensor product. This collection of iterated integrals captures the essential features of the path and is invariant under reparameterization (i.e., φ ( x ) = φ ( x ◦ θ ) , for some reparameterization θ ).

Essentially one can view the path signature is a canonical linear embedding of any multivariate path into an (infinite) series of iterated integrals. It is injective on all non-tree-like paths, meaning no two such paths share the same signature (in practice one ensures non-tree-likeness by appending a monotonic 'time' coordinate). It also satisfies time-reversal duality : φ ( x ) a,b ⊗ φ ( ← -x ) a,b = 1 , where 1 is the unit element and ← -x ( t ) = x ( a + b -t ) .

One can approximate signature kernel reasonably well with the truncated signature transform of level L ∈ N and [21, 22] provide efficient means to compute the truncated signature using Dynamic Programming and approximate it using Random Fourier Features.

Definition 2. (Informal) Signature Kernel Let I = [ u, u ′ ] and J = [ v, v ′ ] be two intervals, and consider paths x ∈ C 1 ( I, R d ) and y ∈ C 1 ( J, R d ) . The signature kernel k sig ( x, y ) : I × J → R is defined by k sig s,t ( x, y ) = ⟨ φ ( x ) s , φ ( y ) t ⟩ , where φ ( x ) s and φ ( y ) t denote the signatures of x and y over the intervals [ u, s ] and [ v, t ] , respectively.

Utilizing the kernel trick introduced in [20], one can efficiently compute the full (untruncated) signature kernel. The kernel trick uses the following result:

Theorem 1. [20] (Informal) Let I = [ u, u ′ ] and J = [ v, v ′ ] be 2 intervals, and let x ∈ C 1 ( I, R d ) and y ∈ C 1 ( J, R d ) . Then the signature kernel k x,y satisfies the following partial differential equation:

subject to the boundary conditions k ( x, y ) = 1 ∀ t ∈ J, k sig ( x, y ) = 1 ∀ s ∈ I.

The resulting PDE formulation of the signature kernel enables its efficient computation using any standard numerical scheme, such as finite difference, finite element, or other suitable methods for hyperbolic equations. See Appendix G.5 for a formal introduction to the signature kernel and the reasons why it makes a good feature map for trajectories.

5 Promoting diversity in SVES using signature kernels

The kernels used in the previous work on SVES [2] do not take full advantage of the Markovian property of the trajectories. Hence, we propose to directly apply k sig in equations 4 and 5. Computing the full ( N × N ) Gram matrix for N trajectories of length T in R d via dynamic programming incurs a time complexity of O ( N 2 T 2 d c ) , where c depends on the level of truncation of the signature transform. By exploiting the kernel trick as stated in Theorem 1, the cost of evaluating the kernel between any two paths can computed in as low as O ( T 2 d ) . Although forming the complete Gram matrix still requires N 2 such evaluations and hence retains an O ( N 2 T 2 d ) overall cost, the underlying PDE formulation admits efficient parallelization. In particular, when implemented with an explicit finite-difference scheme on a suitably provisioned GPU, the per-pair computation can be driven down to linear dependence on the trajectory length (in practice), yielding an effective wallclock complexity of O ( T d ) under the assumption that the GPU can accommodate the required number of threads [20].

6 Promoting Diversity in SVES using SV-CMA-ES

CMA-ES updates have been demonstrated to yield more efficient search steps than standard gradient descent across a variety of benchmark problems [23-25]. Moreover, [26] empirically established that SV-CMA-ES, a variant of SVGD that uses CMA updates, can surpass SVGD in update efficiency, leading to faster and more accurate convergence to the ground truth under certain conditions. In the spirit of these findings, we integrate SV-CMA-ES into the SVES framework in lieu of SVGD. A background in SV-CMA-ES can be found in Appendix C. SVGD's update requires computing the gradient of the log density ∇ log p ( x ) (a process that typically involves multiple sequential backand-forth evaluations of the log-density, in practice) which can be very expensive [10]. However, SV-CMA-ES updates involve the evaluation of a fitness function via many independent samples, which turn is used to compute ∆ j ≈ σ j ∑ m k =1 w j,k y j,k . a gradient-free surrogate of ∇ log p ( x ) (see C Equation 11). Consequently, on massively parallel hardware (e.g., GPUs), SV-CMA-ES may reduce wall-clock time compared to gradient-based SVGD, in cases where computing the gradient of the log density is very expensive or not available. In the case of Ergodic search, the fitness function f ( · ) in the update (Equation 12) is instantiated as the ergodic cost J π ( · ) .

Moreover, SV-CMA-ES's gradient-free exploration-exploitation scheme further enhances trajectory diversity: it simultaneously samples across multiple subpopulations (exploration) and then refines its search by exploiting a select subset of high-fitness (elite) trajectories.

Evolution Strategies for Kernel Gradient Estimation: When the analytical gradient of the kernel is not readily available or is computationally intractable, Simultaneous Perturbation Stochastic Approximation (SPSA) may be employed. SPSA is a gradient-free technique for estimating gradients

of black-box functions via simultaneous random perturbations [13]. SPSA, similar to SV-CMA-ES performs multiple, independent kernel evaluations in parallel, thus can also exploit GPU parallelism to deliver lower overall wall-clock execution times. We found that the noise from SPSA, in some cases aids in breaking out of a "bad" local minima, but in most cases provide no significant performance benefits. By coupling SV-CMA-ES with SPSA, we obtain a fully gradient-free algorithm for generating ergodic trajectories, which is the method adopted in our experimental evaluation (and hence we will refer to SV-CMA-ES coupled with SPSA as just SV-CMA-ES from this point on).

7 Experimental results

In this section, we present results to empirically demonstrate the effectiveness and applicability of the methods discussed above in a variety of simulated and real experiments (and their exact setup can be found in Appendix K). Specifically, we are interested in addressing the following:

How diverse are the trajectories produced using the signature kernel?

How ergodic are the trajectories produced with the signature kernel?

What are the convergence rates of SV-CMA-ES and SVGD?

How ergodic and diverse are the trajectories produced using SV-CMA-ES compared to SVGD?

To address 1) and 2) we will benchmark the signature kernel with other well-known kernels, specifically, the RBF kernel, kernelized DTW, Global Alignement kernel and the Markov RBF kernel. A brief description of these kernels can be found in Appendix G. At each iteration of the SVGD or SV-CMA-ES a new bandwidth for the kernels was chosen using the median heuristic. 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 (refer to [27] for details on the guarantees provided by the median heuristic).

(a) RBF Kernel

(b) Kernelized DTW

(c) Signature kernel

The diversity of N trajectories is measured by det( K f ) , where K f ∈ R N × N is the Gram matrix with entries ( K f ) ij = k Fréchet ( x i , x j ) , where k Fréchet is kernel build using the Fréchet distance. See Appendix H for details.

7.1 Multiscale constrained exploration

7.1.1 Signature kernel performance

To evaluate the performance of the signature kernel within the SVES framework (using SVGD), we instantiate ten distinct priors p k ( · ) for k = 1 , . . . , 10 as detailed in Section 3.1 & E.2, each with variance σ 2 = 0 . 01 . For each prior, we generate N = 6 trajectory samples and compare the kernels according to the following criteria in SVES:

Average trajectory cost, 1 N ∑ N i =1 J π ( x i ) . 1)

Mean ergodic cost, 1 N ∑ N i =1 E π ( x i ) . 3)

Best trajectory cost, min 1 ≤ i ≤ N J π ( x i ) . 2)

Trajectory diversity. -log ( 1 -det( K f ) ) 4)

All Stein variational gradient updates employ a step size of ϵ = 0 . 01 , and iterations are terminated when ∥ ∥ x ( i ) -x ( i -1) ∥ ∥ < 1 . 25 × 10 -3 or maximum of 3000 iterations is reached. The experimental results are summarized in Figure 3 and an elaborate statistical analysis can be found in Appendix J (Table 2).

3.5

3.39e+00

exp(Best trajectory cost):lower is better

2.83e+00

exp(Average trajectory cost):lowerisbetter

3.0

exp(Mean ergodiccost):lowerisbetter

Trajectory diversity:higherisbetter

Average Metric Value over 10 priors

2.21e+00

2.5

1.93e+00

1.77e+00

2.0

1.63e+00

00+369T

1.69e+00

1.71e+00

1.67e+00

1.51e+00

1.53e+00

1.52e+00

1.5

00+061

1.18e+00

1.21e+00

00+61

1.20e+00

1.0

0.5

4.69e-05

0.0

GlobalAlignment

SignatureKernel

MarkovRBF

RBF

Kernel

7.1.2 SV-CMA-ES vs SVGD

To assess the comparative performance of SV-CMA-ES and standard SVGD on the ergodic trajectory optimization, we initialize both methods with N = 5 trajectories drawn from the same Gaussian prior (variance σ 2 = 0 . 01 ). All other hyperparameters are held constant. Here we use the RBF kernel. At each optimization iteration, we record:

Average trajectory cost, 1 N ∑ N i =1 J π ( x i ) . 1)

Trajectory diversity, -log ( 1 -det( K f ) ) 3)

Figure 4 illustrates the per-iteration evolution of these metrics. SV-CMA-ES maintains similar costs compared to SVGD, while promoting trajectory diversity.

7.2 3D Coverage using the Crazyflie drone

7.2.1 Signature kernel performance

We follow the exact method used in section 7.1.1, but in example here set N = 10 , ϵ = 0 . 02 , and iterations are terminated when ∥ ∥ x ( i ) -x ( i -1) ∥ ∥ < 1 . 25 × 10 -3 or a maximum of 1500 iterations is reached.

The experimental results are summarized in Figure 5 and an elaborate statistical analysis can be found in Appendix J (Table 3).

Best trajectory cost min 1 ≤ i ≤ N J π ( x i ) . 2)

Trajectory Cost)

SV-CMA-ES 0.1

10

SV-CMA-ES 0.2

/Cost)

1

SV-CMA-ES 0.5

1

Trajectory Diversity

8

SV-CMA-ES 1

Trajectory

SVGD 0.01

SVGD 0.05

6

0

SVGD 0.1

0

(Average

SVGD 0.2

(Best

4

-1

1

2

800

1600

2400

800

1600

2400

800

1600

2400

Num. Iter.

Num.Iter.

Num.Iter.

exp(Best trajectory cost):lowerisbetter

1.03e+01

10

exp(Averagetrajectory cost):lowerisbetter

exp(Mean ergodic cost): lowerisbetter

8.94e+00

Average Metric Value over 10 priors

exp(Trajectory diversity): higher is better

8

7.30e+00

7.03e+00

5.17e+00

9

4.45e+00

3.75e+00

3.76e+00

3.49e+00

3.29e+00

3.13e+00

00+260

4

2.76e+00

2.74e+00

2.33e+00

2.04e+00

1.91e+00

1.77e+00

1.75e+00

2

0

RBF

Global Alignment

Signature Kernel

Kernel

Kernels

7.2.2 SV-CMA-ES vs SVGD

We once again follow the method in section 7.1.2 with N = 5 trajectories. Figure 6 illustrates the per-iteration evolution of these metrics.

7.3 Stein variational model-predictive control

7.3.1 Signature Kernel Performance

We instantiate ten distinct priors p k ( · ) for k = 1 , . . . , 10 similar to 7.1.1. For each prior, we record the following at each model predictive control (MPC) iteration j = 1 , . . . , 200 .

Average trajectory cost, ¯ J j := 1 N ∑ N i =1 J j π ( u i ) . 1)

Ergodic cost of the best trajectory, E j, ∗ := E j π ( u i ) : i = arg min k ∈{ 1 ,...,N } J j π ( u k ) . 3)

Best trajectory cost, J j, ∗ := min 1 ≤ i ≤ N J j π ( u i ) . 2)

Trajectory diversity, D j := -log ( 1 -K j f ) 4)

Trajectory Cost)

1.5

SV-CMA-ES 0.2

SV-CMA-ES 0.1

Trajectory Cost)

1.5

30

SV-CMA-ES 0.5

Trajectory Diversity

SV-CMA-ES 1

1.0

SVGD 0.01

1.0

SVGD 0.05

20

SVGD 0.1

(Average

0.5

SVGD 0.2

0.5

log (Best 7

10

0.0

0.0

9

800

1600

2400

800

1600

2400

800

1600

2400

Num.Iter.

Num.Iter.

Num.Iter.

The experimental results for one such prior are summarized Table 1. The experimental results for all 10 priors are summarized in Appendix J Table 4.

Markov RBF failed to converge to a meaningful solution in most of the MPC iterations-resulting in essentially random trajectories and thus artificially high diversity values.

7.4 Ablations: kernel bandwidth sensitivity

We provide a small ablation study on the effect of the choice of bandwidth on kernels in the 3D Coverage using the Crazyflie drone experiment from Section 7.2. Figure 7 compares 3 different computationally light heuristics [28] (mean heuristic, Sliverman's rule of thumb and Scott's rule) for selecting the bandwidth and Figure 8 shows the effect of manually tuning the kernel.

Silverman'srule-of-thumb

Meanheuristic

Scott'srule

priors

17.5

exp(Average trajectory cost): lower is better

exp(Best trajectory cost):lower is better

Average Metric Value over10priors

AverageMetric Value over10 priors

15.0

exp(Mean ergodic cost): lower is better

Trajectory diversity:higher is better

12.5

10.0

7.5

5.0

2.5

0.0

RBF

Signature

RBF

Global AlignmentMarkov

Signature

RBF

GlobalAlignmentMarkov

Signature

Kernel

RBF

Kernel

Kernel

RBF

Kernel

Kernel

RBF

Kernel

Kernels

Kernels

Kernels

(a) Bandwitdh vs best trajectory cost

(b) Bandwidth vs average trajectory cost

(c) Bandwidth vs mean ergodic cost

(d) Bandwidth vs trajectory diversity

1.35

1.45

1.30

exp(Best trajectory cost)

1.25

1.40

1.20

1.35

1.15

1.30

1.10

1.25

1.05

Signature Kernel

1.20

Signature Kernel

1.00

+

+

RBF

1.15

RBF

0.95

Kernelized DTW

Kernelized DTW

1.10

0.025

0.05

Bandwidth

0.01

0.025

0.05

Bandwidth

Signature Kernel

Signature Kernel

0.80

RBF

12

RBF

exp(Mean ergodic cost)

0.75

Kernelized DTW

Trajectory diversity

10

Kernelized DTW

0.70

8

0.65

6

0.60

4

0.55

0.50E

.o

05

0.025

Bandwidth

0.025

0.05

Bandwidth

8 Conclusion, Limitations and best practices

In summary, integrating the signature kernel into our ergodic search framework consistently lowers ergodic cost and increases trajectory diversity, compared to standard choices by promoting more effective mode coverage. Moreover, swapping in SV-CMA-ES further enhances exploration diversity, while completely being a gradient-free approach, underscoring the practical benefits of our approach. See Appendix J for the visual representation of the trajectories.

However our approach has several limitations. First, the runtime of the methods discussed in this paper depends heavily on the available CPU/GPU parallelism and the dimensionality of the problem: as the number of particles and dimensions increases, both communication and computation overhead grow, which can degrade performance on hardware with limited cores/threads or when scaling to very high dimensions. Second, the SV-CMA-ES instantiation usually requires performing a singular value decomposition (SVD) or Cholesky factorization at every iteration. Consequently, the scalability (and hence the real time applicability) of this method depends heavily on the problem size, use case, cost of computing the gradients, available hardware etc., and practitioners should be aware of these trade-offs. Future work could explore incorporating recent advances (e.g., diagonal covariance matrices [29]) in vanilla CMA to improve efficiency. Third, while the signature kernel provides a very strong repulsive force that promotes diverse exploration, it can also inadvertently push trajectories into obstacles, outside bounded domains, etc. We noticed this specially in the example in section 7.3, which explains the higher total average trajectory cost. To mitigate these effects, practitioners should consider using more conservative cost designs when employing the signature kernel, opting for alternative kernels (e.g., RBF) that exert milder repulsion or running SVGD for more iterations (see Appendix F for the exact convergence guarantees; we also recommend the reader to refer to [2, 30]). Fourth, SV-CMA-ES does not provide rich theoretical guarantees provided by SVGD and hence SVES. Finally, we did not explore the potential of the Fréchet Distance kernel for producing diverse trajectories due to its computational cost.

References

L. M. Miller, Y. Silverman, M. A. MacIver, and T. D. Murphey, 'Ergodic exploration of distributed information,' IEEE Transactions on Robotics , vol. 32, no. 1, pp. 36-52, 2016.

D. Lee, C. Lerch, F. Ramos, and I. Abraham, 'Stein variational ergodic search,' in Robotics: Science and Systems XX, Delft, The Netherlands, July 15-19, 2024 , D. Kulic, G. Venture, K. E. Bekris, and E. Coronado, Eds., 2024. [Online]. Available: https://doi.org/10.15607/RSS.2024.XX.001

D. Pathak, P. Agrawal, A. A. Efros, and T. Darrell, 'Curiosity-driven exploration by selfsupervised prediction,' in Proceedings of the 34th International Conference on Machine Learning - Volume 70 , ser. ICML'17. JMLR.org, 2017, p. 2778-2787.

Y. Burda, H. Edwards, D. Pathak, A. J. Storkey, T. Darrell, and A. A. Efros, 'Large-scale study of curiosity-driven learning,' in 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019 . OpenReview.net, 2019. [Online]. Available: https://openreview.net/forum?id=rJNwDjAqYX

P. Mazzaglia, O. Catal, T. Verbelen, and B. Dhoedt, 'Curiosity-driven exploration via latent bayesian surprise,' 2022. [Online]. Available: https://arxiv.org/abs/2104.07495

Y. Silverman, L. M. Miller, M. A. MacIver, and T. D. Murphey, 'Optimal planning for information acquisition,' in 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems , 2013, pp. 5974-5980.

W. Chen, R. Khardon, and L. Liu, 'Ak: Attentive kernel for information gathering,' 06 2022.

G. Mathew and I. Mezi´ c, 'Metrics for ergodicity and design of ergodic dynamics for multi-agent systems,' Physica D: Nonlinear Phenomena , vol. 240, no. 4, pp. 432-442, 2011. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S016727891000285X

L. M. Miller and T. D. Murphey, 'Trajectory optimization for continuous ergodic exploration,' in 2013 American Control Conference , 2013, pp. 4196-4201.

Q. Liu and D. Wang, 'Stein variational gradient descent: A general purpose bayesian inference algorithm,' in Advances in Neural Information Processing Systems , D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, and R. Garnett, Eds., vol. 29. Curran Associates, Inc., 2016. [Online]. Available: https://proceedings.neurips.cc/paper_files/paper/2016/file/ b3ba8f1bee1238a2f37603d90b58898d-Paper.pdf

L. Barcelos, T. Lai, R. Oliveira, P. Borges, and F. Ramos, 'Path signatures for diversity in probabilistic trajectory optimisation,' The International Journal of Robotics Research , vol. 43, no. 11, pp. 1693-1710, 2024. [Online]. Available: https://journals.sagepub.com/doi/abs/10.1177/02783649241233300

C. Braun, R. Lange, and M. Toussaint, 'Stein variational evolution strategies,' 10 2024.

J. Spall, 'Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,' IEEE Transactions on Automatic Control , vol. 37, no. 3, pp. 332-341, 1992.

D. Lee and H. Oberhauser, 'The signature kernel,' 2023. [Online]. Available: https://arxiv.org/abs/2305.04625

I. Chevyrev and A. Kormilitzin, 'A primer on the signature method in machine learning,' 03 2016.

P. Bonnier, P. Kidger, I. Perez Arribas, C. Salvi, and T. Lyons, 'Deep signature transforms,' 05 2019.

W. Yang, T. Lyons, H. Ni, C. Schmid, and L. Jin, Developing the Path Signature Methodology and Its Application to LandmarkBased Human Action Recognition . Cham: Springer International Publishing, 2022, pp. 431-464. [Online]. Available: https://doi.org/10.1007/978-3-030-98519-6_18

M. Ohnishi, I. Akinola, J. Xu, A. Mandlekar, and F. Ramos, 'Signatures meet dynamic programming: Generalizing Bellman equations for trajectory following,' in Proceedings of the 6th Annual Learning for Dynamics amp; Control Conference , ser. Proceedings of Machine Learning Research, A. Abate, M. Cannon, K. Margellos, and A. Papachristodoulou, Eds., vol. 242. PMLR, 15-17 Jul 2024, pp. 466-479. [Online]. Available: https://proceedings.mlr.press/v242/ohnishi24a.html

Z. Yin, T. Lai, L. Barcelos, J. Jacob, Y. Li, and F. Ramos, 'Diverse motion planning with stein diffusion trajectory inference,' in 2025 IEEE International Conference on Robotics and Automation (ICRA) , 2025, pp. 15 610-15 616.

C. Salvi, T. Cass, J. Foster, T. Lyons, and W. Yang, 'The signature kernel is the solution of a goursat pde,' SIAM Journal on Mathematics of Data Science , vol. 3, no. 3, pp. 873-899, 2021. [Online]. Available: https://doi.org/10.1137/20M1366794

F. J. Kiraly and H. Oberhauser, 'Kernels for sequentially ordered data,' Journal of Machine Learning Research , vol. 20, no. 31, pp. 1-45, 2019. [Online]. Available: http://jmlr.org/papers/v20/16-314.html

C. Tóth, H. Oberhauser, and Z. Szabó, 'Random fourier signature features,' SIAM Journal on Mathematics of Data Science , vol. 7, no. 1, pp. 329-354, 2025. [Online]. Available: https://doi.org/10.1137/23M1620478

J. Martens, 'New insights and perspectives on the natural gradient method,' Journal of Machine Learning Research , vol. 21, no. 146, pp. 1-76, 2020. [Online]. Available: http://jmlr.org/papers/v21/17-678.html

Y. Akimoto, Y. Nagata, I. Ono, and S. Kobayashi, 'Theoretical foundation for cma-es from information geometry perspective,' Algorithmica , vol. 64, pp. 698-716, 2012. [Online]. Available: https://api.semanticscholar.org/CorpusID:6488573

T. Glasmachers, T. Schaul, S. Yi, D. Wierstra, and J. Schmidhuber, 'Exponential natural evolution strategies,' in Proceedings of the 12th Annual Conference on Genetic and Evolutionary Computation , ser. GECCO '10. New York, NY, USA: Association for Computing Machinery, 2010, p. 393-400. [Online]. Available: https://doi.org/10.1145/1830483.1830557

C. V. Braun, R. T. Lange, and M. Toussaint, 'Stein variational evolution strategies,' 2024. [Online]. Available: https://arxiv.org/abs/2410.10390

D. Garreau, W. Jitkrittum, and M. Kanagawa, 'Large sample analysis of the median heuristic,' 2018. [Online]. Available: https://arxiv.org/abs/1707.07269

S. Sheather, 'Density estimation,' Statistical Science , vol. 19, 11 2004.

R. Ros and N. Hansen, 'A simple modification in cma-es achieving linear time and space complexity,' in Parallel Problem Solving from Nature - PPSN X , G. Rudolph, T. Jansen, N. Beume, S. Lucas, and C. Poloni, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008, pp. 296-305.

A. Korba, A. Salim, M. Arbel, G. Luise, and A. Gretton, 'A non-asymptotic analysis for stein variational gradient descent,' 2021. [Online]. Available: https://arxiv.org/abs/2006.09797

N. Ratliff, M. Zucker, J. A. Bagnell, and S. Srinivasa, 'Chomp: Gradient optimization techniques for efficient motion planning,' in 2009 IEEE International Conference on Robotics and Automation , 2009, pp. 489-494.

M. Zucker, N. Ratliff, A. D. Dragan, M. Pivtoraiko, M. Klingensmith, C. M. Dellin, J. A. Bagnell, and S. S. Srinivasa, 'Chomp: Covariant hamiltonian optimization for motion planning,' The International Journal of Robotics Research , vol. 32, no. 9-10, pp. 1164-1193, 2013. [Online]. Available: https://doi.org/10.1177/0278364913488805

A. Byravan, B. Boots, S. S. Srinivasa, and D. Fox, 'Space-time functional gradient optimization for motion planning,' in 2014 IEEE International Conference on Robotics and Automation (ICRA) , 2014, pp. 6499-6506.

Z. Marinho, A. Dragan, A. Byravan, B. Boots, S. Srinivasa, and G. Gordon, 'Functional gradient motion planning in reproducing kernel hilbert spaces,' 2016. [Online]. Available: https://arxiv.org/abs/1601.03648

M. Kalakrishnan, S. Chitta, E. Theodorou, P. Pastor, and S. Schaal, 'Stomp: Stochastic trajectory optimization for motion planning,' in 2011 IEEE International Conference on Robotics and Automation , 2011, pp. 4569-4574.

N. Hansen, S. D. Müller, and P. Koumoutsakos, 'Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (cma-es),' Evolutionary Computation , vol. 11, no. 1, pp. 1-18, 2003.

P. Hämäläinen, A. Babadi, X. Ma, and J. Lehtinen, 'Ppo-cma: Proximal policy optimization with covariance matrix adaptation,' in 2020 IEEE 30th International Workshop on Machine Learning for Signal Processing (MLSP) , 2020, pp. 1-6.

B. Tjanaka, M. C. Fontaine, D. H. Lee, A. Kalkar, and S. Nikolaidis, 'Training diverse high-dimensional controllers by scaling covariance matrix adaptation map-annealing,' 2023. [Online]. Available: https://arxiv.org/abs/2210.02622

N. Hansen, 'The cma evolution strategy: A tutorial,' 2023. [Online]. Available: https://arxiv.org/abs/1604.00772

J. Schulman, J. Ho, A. Lee, I. Awwal, H. Bradlow, and P. Abbeel, 'Finding locally optimal, collision-free trajectories with sequential convex optimization,' 06 2013.

M. Mukadam, J. Dong, X. Yan, F. Dellaert, and B. Boots, 'Continuous-time gaussian process motion planning via probabilistic inference,' The International Journal of Robotics Research , vol. 37, no. 11, pp. 1319-1340, 2018. [Online]. Available: https://doi.org/10.1177/0278364918790369

J. Dong, M. Mukadam, F. Dellaert, and B. Boots, 'Motion planning as probabilistic inference using gaussian processes and factor graphs,' 06 2016.

M. Mukadam, J. Dong, F. Dellaert, and B. Boots, 'Steap: simultaneous trajectory estimation and planning,' Autonomous Robots , vol. 43, no. 2, p. 415-434, Jul. 2018. [Online]. Available: http://dx.doi.org/10.1007/s10514-018-9770-1

A. Lambert and B. Boots, 'Entropy regularized motion planning via stein variational inference,' 2021. [Online]. Available: https://arxiv.org/abs/2107.05146

H. Yu and Y. Chen, 'A gaussian variational inference approach to motion planning,' IEEE Robotics and Automation Letters , vol. 8, no. 5, p. 2518-2525, May 2023. [Online]. Available: http://dx.doi.org/10.1109/LRA.2023.3256134

A. Lambert and B. Boots, 'Entropy regularized motion planning via stein variational inference,' 2021. [Online]. Available: https://arxiv.org/abs/2107.05146

H. Choset, 'Coverage of known spaces: The boustrophedon cellular decomposition,' Auton. Robots , vol. 9, pp. 247-253, 12 2000.

D. L. Applegate, R. E. Bixby, V. Chvatál, and W. J. Cook, The Traveling Salesman Problem: A Computational Study . Princeton University Press, 2006. [Online]. Available: http://www.jstor.org/stable/j.ctt7s8xg

G. Laporte, H. Mercure, and Y. Nobert, 'Generalized travelling salesman problem through n sets of nodes: the asymmetrical case,' Discrete Applied Mathematics , vol. 18, no. 2, pp. 185-197, 1987. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ 0166218X87900205

G. Chirikjian, Stochastic Models, Information Theory, and Lie Groups, Volume 2: Analytic Methods and Modern Applications , ser. Applied and Numerical Harmonic Analysis. Birkhäuser Boston, 2011. [Online]. Available: https://books.google.com.au/books?id=338COWrZDMUC

D. Dong, H. Berger, and I. Abraham, 'Time optimal ergodic search,' 2023. [Online]. Available: https://arxiv.org/abs/2305.11643

S. E. Scott, T. C. Redd, L. Kuznetsov, I. Mezi´ c, and C. K. Jones, 'Capturing deviation from ergodicity at different scales,' Physica D: Nonlinear Phenomena , vol. 238, no. 16, pp. 1668-1679, 2009. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ S0167278909001523

L. Dressel and M. J. Kochenderfer, 'On the optimality of ergodic trajectories for information gathering tasks,' in 2018 Annual American Control Conference (ACC) , 2018, pp. 1855-1861.

T. Howell, N. Gileadi, S. Tunyasuvunakool, K. Zakka, T. Erez, and Y. Tassa, 'Predictive sampling: Real-time behaviour synthesis with mujoco,' 12 2022.

G. Williams, P. Drews, B. Goldfain, J. M. Rehg, and E. A. Theodorou, 'Aggressive driving with model predictive path integral control,' in 2016 IEEE International Conference on Robotics and Automation (ICRA) , 2016, pp. 1433-1440.

G. Williams, A. Aldrich, and E. A. Theodorou, 'Model predictive path integral control: From theory to parallel computation,' Journal of Guidance, Control, and Dynamics , vol. 40, no. 2, pp. 344-357, 2017. [Online]. Available: https://doi.org/10.2514/1.G001921

M. Sun, F. Baldini, P. Trautman, and T. Murphey, 'Move beyond trajectories: Distribution space coupling for crowd navigation,' 07 2021.

A. Lambert, F. Ramos, B. Boots, D. Fox, and A. Fishman, 'Stein variational model predictive control,' in Proceedings of the 2020 Conference on Robot Learning , ser. Proceedings of Machine Learning Research, J. Kober, F. Ramos, and C. Tomlin, Eds., vol. 155. PMLR, 16-18 Nov 2021, pp. 1278-1297. [Online]. Available: https://proceedings.mlr.press/v155/lambert21a.html

A. Le, G. Chalvatzaki, A. Biess, and J. Peters, 'Accelerating motion planning via optimal transport,' 10 2023.

C. Hughes, H. Warren, D. Lee, F. Ramos, and I. Abraham, 'Ergodic trajectory optimization on generalized domains using maximum mean discrepancy,' CoRR , vol. abs/2410.10599, 2024. [Online]. Available: https://doi.org/10.48550/arXiv.2410.10599

T. Power and D. Berenson, 'Constrained stein variational trajectory optimization,' Trans. Rob. , vol. 40, p. 3602-3619, Jan. 2024. [Online]. Available: https://doi.org/10.1109/TRO.2024. 3428428

B. Ghojogh, A. Ghodsi, F. Karray, and M. Crowley, 'Reproducing kernel hilbert space, mercer's theorem, eigenfunctions, nyström method, and use of kernels in machine learning: Tutorial and survey,' 2021. [Online]. Available: https://arxiv.org/abs/2106.08443

M. Cuturi, J.-P. Vert, O. Birkenes, and T. Matsui, 'A kernel for time series based on global alignments,' 05 2007, pp. II-413.

P.-F. Marteau and S. Gibet, 'On recursive edit distance kernels with application to time series classification,' IEEE Transactions on Neural Networks and Learning Systems , vol. 26, no. 6, p. 1121-1133, Jun. 2015. [Online]. Available: http://dx.doi.org/10.1109/TNNLS.2014.2333876

D. Haussler, 'Convolution kernels on discrete structures ucsc-crl-99-10,' 06 2001.

The Signature of a Path . Berlin, Heidelberg: Springer Berlin Heidelberg, 2007, pp. 25-40. [Online]. Available: https://doi.org/10.1007/978-3-540-71285-5_2

T. Lyons, 'Rough paths, signatures and the modelling of functions on streams,' 05 2014.

M. Fréchet, Sur quelques points de calcul fonctionnel . na, 1906. [Online]. Available: https://books.google.com/books?id=WR6smgEACAAJ

N. Megiddo, 'Applying parallel computation algorithms in the design of serial algorithms,' J. ACM , vol. 30, no. 4, p. 852-865, Oct. 1983. [Online]. Available: https://doi.org/10.1145/2157.322410

R. Cole, 'Parallel merge sort,' SIAM Journal on Computing , vol. 17, no. 4, pp. 770-785, 1988.

Appendices

Appendix Contents

A Additional related work

Trajectory optimization Trajectory optimization encompasses a wide range of techniques that initiate with a preliminary, suboptimal path and iteratively refine it to converge on a (potentially local) optimum by minimizing an associated cost function. Foundational contributions in this domain include Covariant Hamiltonian Optimization for Motion Planning (CHOMP) [31] and its related approaches [32-34], which exploit the covariance of trajectories alongside Hamiltonian Monte Carlo to perform annealed functional gradient descent. A significant drawback of these methods is their reliance on fully differentiable cost functions. Stochastic Trajectory Optimization for Motion Planning (STOMP) [35] overcomes this by estimating gradients from stochastic samples of noisy trajectories, thereby accommodating cost functions that lack differentiability.

Another influential research direction involves quality diversity algorithms, notably the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [36-38], which adopts a derivative-free strategy using multivariate normal distributions to generate and update candidate solutions. Although CMAES is robust and exhibits ergodic behavior capable of managing multimodal problems, it typically requires multiple initializations and a higher number of evaluations compared to gradient-based methods [39]. Other techniques, such as TrajOpt [40], solve sequential quadratic programs with continuous-time collision checking; while these methods are efficient, they often yield only locally optimal solutions and necessitate fine parametrization of waypoints. To mitigate this, [34] limit the optimization to a Reproducing Kernel Hilbert Space (RKHS) with a squared-exponential kernel, although this may overlook the cost contributions between sparsely distributed waypoints. Methods proposed in GPMP [41-43] model trajectories as Gaussian Processes (GP) to obtain a maximum a posteriori (MAP) solution.

More closely aligned with our approach, variational inference methods introduced in [44] and [45] infer the posterior distribution over trajectories using Stein variational gradient descent (SVGD) or

natural gradient updates. In contrast, the RKHS induced in SVMP fails to capture the sequential dynamics of trajectories by treating them as static, high-dimensional vectors-resulting in reduced repulsive forces and less coordination across functional dimensions. Kernel Signature Variational Gradient Descent (SigSVGD) [11], overcomes these shortcomings by integrating path signatures to explicitly encode the sequential structure of trajectories. This approach not only enhances diversity but also increases the probability of discovering superior local optima. Furthermore, leveraging insights from both motion planning and control-as-inference allows SigSVGD to extend naturally to Model Predictive Control (MPC) frameworks (e.g., Stein Variational MPC (SVMPC) [46]), where preserving the sequential structure of control trajectories is crucial.

Robotic exploration Robotic exploration is primarily concerned with steering agents into unknown territories, while the coverage problem entails designing trajectories and strategically placing sensors to ensure that a confined area is thoroughly examined. Early solutions often relied on discrete grid-based strategies-employing methods such as advanced boustrophedon search patterns (the antecedent to the lawnmower pattern [47]) and variations of the traveling salesperson problem [48, 49]-to list all viable exploration routes. Although grid-based methods offer guarantees of completeness, scaling these techniques to continuous domains is inherently difficult because the number of potential trajectories becomes unbounded. Recent strategies in reinforcement learning and information-driven exploration have sought to surmount these obstacles by using 'curiosity' measures [3, 4] derived from information theory [50]. However, many of these approaches tend to be short-sighted, concentrating on immediate rewards and often settling on a single-mode solution, which restricts the system's overall flexibility.

Subsequent developments have brought forth ergodicity-based methods for exploration. By reducing the disparity between the expected spatial information distribution and the time-averaged visitation of a path [8], ergodic exploration techniques yield sophisticated coverage patterns across continuous domains [1, 51]. In fact, the spectral, multi-scale structure of the ergodic metric [52] suggests the existence of numerous exploitable local solutions-an element that is essential for resilient, real-time exploration [53]. Despite these advances, a persistent challenge remains in devising methods that compute sets of 'good,' locally optimal ergodic trajectories which can facilitate efficient strategy switching in dynamic scenarios.

Control as inference The control as inference framework has recently emerged as a powerful perspective by recasting trajectory optimization issues within a probabilistic framework. The inherent non-convexity of many robotic problems-due to complex dynamics and contact interactions-creates significant hurdles for traditional gradient-based optimization techniques. Approaches such as predictive sampling [54] and model-predictive path integral control (MPPI) [55, 56] have been developed to mitigate these difficulties. However, these methods often produce only a single locally optimal solution, thereby overlooking the potential for multiple, equally effective strategies. This limitation is especially problematic in rapidly changing environments-for instance, in crowd-based navigation-where relying on one trajectory can be inadequate [57, 58]. Moreover, the utilization of optimal transport and variational techniques has shown promise by reformulating trajectory optimization as an inference problem [58, 59]. These frameworks enable the approximation of complex solution distributions that encompass multiple local optima, thus providing the necessary flexibility for swift adaptation.

B Stein variational gradient descent (SVGD)

Variational inference often requires approximating an intractable posterior distribution p ( x ) by a tractable distribution q ( x ) from a parameterized family of distributions D , typically through the minimization of the Kullback-Leibler (KL) divergence. Stein Variational Gradient Descent (SVGD) [10] offers a non-parametric approach that iteratively transports a set of particles { x i } n i =1 to match the target distribution.

The core idea of SVGD is to gradually push the particles in the direction that most rapidly decreases the KL divergence. Consider an incremental, smooth transform

where ϕ ( x ) is a smooth perturbation function and ϵ > 0 is a small step size. A key theoretical result shows that the derivative of the KL divergence with respect to ϵ (evaluated at ϵ = 0 ) is

with the Stein operator defined as

This result, based on Stein's identity, motivates the introduction of the kernelized Stein discrepancy (KSD) as a measure of the mismatch between q ( x ) and p ( x ) .

By constraining ϕ to lie within the unit ball of a reproducing kernel Hilbert space (RKHS) associated with a positive definite kernel k ( · , · ) , one can derive a closed-form expression for the optimal perturbation direction:

Here, the first term in the expectation directs particles toward regions of high probability under p ( x ) , while the second term serves as a repulsive force, ensuring that the particles remain diverse and do not collapse into a single mode.

The practical SVGD update for the particles is then given by

where the empirical approximation of the optimal direction is

Notably, when n = 1 the update reduces to the standard gradient ascent used in maximum a posteriori (MAP) estimation, whereas using multiple particles allows SVGD to capture the full posterior distribution.

SVGD's formulation as a functional gradient descent in an RKHS not only avoids the need to compute Jacobians or their inverses but also scales well with large datasets by permitting mini-batch approximations of ∇ log p ( x ) and allows for parallel computation of each particle. This scalability, along with its ability to maintain particle diversity through the repulsive kernel term, makes SVGD an attractive and efficient method for Bayesian inference-and it forms a key component in extensions to more complex tasks.

C Stein variational covariance matrix adaptation evolution strategy

To overcome the reliance on gradient information in standard SVGD while retaining its powerful repulsive interactions, Stein Variational Covariance Matrix Adaptation Evolution Strategy (SV-CMAES) embeds the adaptive, natural-gradient-free CMA-ES into the multi-particle SVGD framework [26]. Conceptually, each SVGD 'particle' is no longer a point but the mean of its own Gaussian search distribution. By interpreting the CMA-ES mean update as an estimate of the score function ∇ log p ( x ) , we obtain a gradient-free yet highly explorative algorithm that tracks both local fitness landscapes and global diversity.

Let n be the number of particles and m the sub-population size per particle. Particle i carries a Gaussian

and we evaluate their fitness f ( ξ i,k ) . By ranking these m samples and selecting the top λ as the elite sub-population, we assign recombination weights w i,k (non-zero only for those λ elites, which may be positive or negative depending on their ranked fitness; see [39]).

In vanilla SVGD each particle x i moves according to

In SV-CMA-ES we replace the intractable score by the CMA-ES mean shift

and approximate the expectation over all n Gaussians. The resulting SV-CMA-ES update is

where α x subsumes the original SVGD step-size ϵ . This blend preserves SVGD's repulsion through ∇ k while guiding particles toward high-fitness regions via CMA-ES's ∆ j .

The self-adaptation of σ i and C i follows CMA-ES's mechanisms:

and

Practical considerations

The driving-force term in (12) averages over all n particles, which can shrink steps when modes are distant and cause premature step-size reduction. To address this, one may employ a hybrid update that uses only the particle's own CMA-ES shift for the first term and retains the full repulsion:

where γ ( t ) is an annealing factor (e.g. γ ( t ) = max(log T t , 1) ) that gradually reduces repulsion to fine-tune convergence near local optima and prevents premature convergence.

Algorithm 1 Stein Variational CMA-ES

Require: kernel k ( · , · ) ; sub-population size m ; elite size λ ; hyper-parameters α x , α σ , α 1 , α λ , α c , d σ ; n particles; T iterations

1: Initialize: ( x i , σ i , C i ) for i = 1 , . . . , n

2: for t = 1 , . . . , T do

3: for i = 1 , . . . , n do Sample sub-population

Evaluate & accumulate statistics

8: compute f ( ξ i,j ) and ranks → weights w i,j

Update mean

Covariance path & matrix

15: compute h σ,i and d ( h σ,i ) (CSA rule)

18: end for

19: end for

All loops other than the outer optimization loop can be parallelized.

D Ergodic search and ergodic trajectory optimization

We denote the state space of a robot as X ⊆ X and the control space as U ⊆ U , where X and U are some Banach spaces. We also denote the the continuous-time dynamics of the robot as f ( x, u ) : X × U → T X , where T X denotes the tangent bundle of the state-space X . Next, a robot's space trajectory x ( t ) : R ≥ 0 →X is defined as the solution to the following initial value problem

with initial condition x (0) ∈ X , a control trajectory u ( t ) : R ≥ 0 → U . In addition, we will denote the bounded (and compact) domain (with bounds B i ) where the robot explores as S = [0 , B 0 ] × . . . × [0 , B v -1 ] , where v ≤ n . Let g ( x ) : X → S be a map that projects state space X to exploration space S .

▷ outer optimisation loop

▷ one search distribution per particle

Definition 3. Time-Averaged Trajectory Statistics [2]. Let L be the Lebesgue measure on R ≥ 0 then for each T ∈ R ≥ 0 , the probability measure Ψ T on S that defines the time-averaged trajectory visitation statistics integrated along time [0 , T ] is defined by

for a A ⊂ S is a Borel set.

Definition 4. Ergodicity [2, 9, 8]. A trajectory x ( t ) is said to be ergodic with respect to a Borel probability measure π on S if Ψ T converges weakly to π as T →∞ , formally,

The trajectory statistics measure can be simplified as an integral of delta functions, where

Intuitively, a trajectory is ergodic with respect to the measure π if it explores A in a manner that is commensurate with π as T →∞ .

Definition 5. Ergodic Cost Function [2]. Given a probability measure π on S . A π -ergodic cost function is defined as E π : τ ( X ) → R such that for any infinite trajectory x ( t ) : R ≥ 0 →X , if

as the set of all continuous trajectories defined on arbitrary finite time intervals. Throughout this article we stick to X = R n and U = R m and, we use the spectral methods and construct a metric in the Fourier space to define the ergodic metric for trajectory optimization [2, 8, 52, 1].

Definition 6. Spectral Ergodic Cost Function. Let π be a probability measure on S . Let K v ⊂ N v be the set of all integer k fundamental frequencies that define the cosine Fourier basis function

where h k is a normalizing factor (see [8]). For a finite trajectory x ( t ) : [0 , T ] →X , let Ψ T be the measure defined in Eq. (24) . The spectral ergodic cost function is defined as

where Ψ k T and µ k are the k th Fourier decomposition modes of Ψ T and µ , respectively (using Eq. (26) ), and Λ k = (1 + ∥ k ∥ 2 ) -v +1 2 is a weight coefficient that places higher importance on lower-frequency modes.

In particular, the spectral ergodic cost function defined for a probability measure π forms a metric [8].

Alternatively, one can use the kernel MMD to construct a ergodic metric for trajectories on any generalized domains [60].

We now can formulate an (ergodic) trajectory optimization using the ergodic metric as

where h 1 are the quality constraints e.g., ˙ x = f ( x, u ) and h 2 are the inequality constraints.

then x ( t ) is ergodic, where

E Stein variational ergodic search

E.1 Likelihood of ergodicity and stein variations

Stein Variational Ergodic Search (SVES) [2] recasts the trajectory optimization problem as a (variational) inference problem. They begin by introducing a binary indicator function

which flags whether a given trajectory satisfies a desired optimality condition. Let π be a probability measure defined on the exploration domain S . The likelihood that a trajectory x exhibits the desired ergodic behavior through an ergodic cost function E π ( x ) is given by

where µ > 0 is a scaling parameter that governs the influence of the ergodic cost.

By combining this likelihood with a prior p ( x ) over the space of trajectories via Bayes' rule, one can obtain the posterior p ( x | O ) . This posterior is approximated using Stein Variational Gradient Descent (SVGD). In this framework, the update direction that minimizes the Kullback-Leibler divergence (from equation 8) is given by

where k ( x, · ) is a positive definite kernel on the space of trajectories. This vector field steers each trajectory sample toward regions of lower ergodic cost while ensuring diversity across samples.

E.2 Ergodic stein variational trajectory optimization

For computational tractability, we discretize trajectories. Let a trajectory be represented as

with x t ∈ X . An empirical distribution over N discrete trajectory samples is given by

Typically one assumes a Gaussian prior of the form

where ˆ x is typically chosen as an interpolation between specified boundary conditions.

Constraints on the trajectory can be integrated into the cost likelihood function in the following form

where c 1 , c 2 are (positive) Lagrange multipliers that form an inner product with equality and inequality functions h 1 , h 2 , and ρ : τ ( X ) → R is any additional penalty terms on the trajectory x . 3 Now, the ergodic Stein variational update step using J π is given by

using the kernel a k ( x , · ) . This procedure exploits the repulsive characteristics of the kernel to prevent sample (or mode) collapse, thereby ensuring that the resulting trajectories remain diverse well-dispersed and effective for exploration. It is crucial to note that maintaining diversity and preventing mode collapse fundamentally depends on selecting an appropriate and effective kernel. The algorithm outlining this procedure is as follows:

Algorithm 2 Stein Variational Ergodic Trajectory Opt.

E.3 Ergodic stein variational control

This approach can easily be extended to the control domain by optimizing over sequences of control inputs rather than trajectories directly. Let the control sequence be represented as

and let the the system evolves according to the discrete-time dynamics

We define an empirical distribution over control sequences by

and assume a prior p ( u ) on the control sequences. With a kernel k ( u, · ) defined on the control space, the SVGD update for controls is expressed as

where J π ( x i r | u i r , x 0 ) represents the ergodic (with constraints) cost incurred by the trajectory generated under the control sequence u i starting from the initial state x 0 . This control formulation is well-suited for online applications, where the first control action is executed and the sequence is re-optimized at each time step to adapt to new information.

The algorithm summarizes the approach for control optimization within a model-predictive control framework is as follows:

Algorithm 3 Stein Variational Ergodic Control

F Convergence of SVGD in the Ergodic Search Setting

The work in [2] shows that the general convergence results for SVGD from [30] remain valid in the ergodic search setting when using an RBF kernel. We now generalize this convergence result to any differentiable kernel.

Consider a discretized ergodic search on a bounded state space X ⊂ R n and a normalized workspace W = [0 , 1] v with T time points. Without loss of generality we set X = [0 , 1] n , and thus SVGD acts on the space X T = [0 , 1] Tn . We fix a mapping g : X → X . Moreover, we work in the population limit of infinitely many initial samples from the prior.

Let E π be the spectral ergodic cost function with respect to a measure π on W , and let p be the prior on X T . Then the posterior

can be written in the form

Define the potential function as is standard in SVGD.

Furthermore, let k : X T ×X T → R be a positive definite kernel with associated RKHS H . Convergence of SVGD is then characterized by the kernel Stein discrepancy (KSD). In particular, for any measure q , its discrepancy to η is

where ϕ ∗ q ∈ H T (noting X T ⊂ R Tn ) is given by

In particular, KSD η ( q ) coincides with the norm of the SVGD update gradient ϕ ∗ r in (32). Here we define SVGD in the population limit in the same way as the finite-sample case. The population gradient at iteration r is ϕ ∗ r := ϕ ∗ ( q r ) , where q r is the pushforward of q r -1 under the map U r : X T → X T ,

with step size ϵ > 0 and initial distribution q 0 = p .

Moreover, the convergence results in [30] rest on three assumptions on the kernel k , the potential V , and the updates U r . Specifically, there exist constants B,C,M > 0 such that:

(A2) The Hessian H V of V in (28) is well-defined with operator norm ∥ H V ∥ op ≤ M .

(A3) For all r , KSD η ( q r ) ≤ C .

Now we can formally state the generalized convergence guarantee from [2] as follows:

Theorem 2. Let X = [0 , 1] n , let η = p ( x | O ) be the target with potential V from (28) , and assume p is smooth on X T . Let k be a positive kernel on X T with RKHS H . Then there exists a step size ϵ < S , where S depends on B,C,M , such that

where c ϵ is a constant depending on ϵ , M , and B .

Proof. This result is a special case of [30, Corollary 6], and in order to prove this result, we must show that assumptions ( A 1) , ( A 2) , ( A 3) above hold. First, ( A 1) holds as long as the kernel k is differentiable, and X T is a bounded domain. Next, we note that for a discrete path x = [ x 0 , . . . , x T -1 ] , the spectral ergodic cost fucntion has the form

which is smooth since F k from Eq. (29) is smooth. Because the prior p is also smooth, the potential V is smooth, and the Hessian is well-defined. Furthermore, the Hessian is bounded since X T is a bounded domain, so ( A 2) is satisfied. Finally, since ( A 1) and ( A 2) is satisfied, it suffices to show that

to show ( A 3) , from the discussion in [30, Section 5]. However, this is satisfied since q r is a probability measure and W T is a bounded domain, so ( A 3) is satisfied.

G Kernels

G.1 Radial basis function (RBF) kernel

The Radial Basis Function (RBF) kernel [62], also known as the Gaussian kernel, is one of the most widely used kernels in machine learning and other applications due to its strong theoretical properties and practical performance. Given two inputs x, y ∈ R d (In the case of our trajectories, x = [ x 1 , . . . , x T ] where x i ∈ R n , we flatten them into a vector in R Tn ), the RBF kernel is defined as

where h > 0 is the kernel bandwidth, and is chosen using the median heuristic. This choice of kernel corresponds to an inner product in an infinite-dimensional reproducing kernel Hilbert space (RKHS), making it a universal approximator under mild conditions on h .

G.2 Markov RBF kernel

Given two state trajectories

we define a Markov kernel by decomposing similarity along the Markov chain:

where each k is a some (positive definite) kernel and G is a graph of connected points separated by time i.e., G chosen to encode the Markov-structure along time (typically the nearest-neighbour edges t ↔ t +1 ).

Choice of the temporal graph G . For a sparse Markov kernel one usually takes G nn = { ( t, t + 1) } T -1 t =1 , so that only nearest-neighbour time indices interact. This nearest-neighbour choice was the one employed in the experiment described in section 7.2.

In the complete-graph variant used in the experiments in sections 7.1 & 7.3.

̸

which connects every ordered pair of distinct time indices. Substituting this into the definition gives

̸

and dividing the double sum by T 2 simply rescales the kernel without affecting positive-definiteness or SVGD updates.

A typical choice for the base kernel is the radial basis function (RBF) with bandwidth h set by, the median heuristic in our experiments. This construction yields an expressive yet efficient measure of trajectory similarity that honors the one-step dependence characteristic of Markov processes.

G.3 Global alignment kernel

Let x = [ x 1 , . . . , x n ] and y = [ y 1 , . . . , y m ] be two trajectories (of possibly different lengths) in R d . Traditional vector kernels (e.g., Gaussian or polynomial) cannot directly handle variable-length sequences nor account for temporal distortions. Dynamic alignment methods such as Dynamic Time Warping (DTW) (see Appendix G.4) overcome this by finding an optimal alignment, but do not yield positive-definite kernels in general.

The Global Alignment (GA) kernel [63] addresses these issues by summing over all possible alignments between x and y . Formally, let

be the set of all monotonic alignments between the indices of x and y . Given a local positive-definite kernel (often called static kernel)

Intuitively, K ( x, y ) aggregates the similarity contributions of every alignment, acting as a 'soft-max' over DTW-like scores. In our experiments we chose the RBF kernel as the static kernel.

Positive definiteness. Under mild conditions on k (for instance, if both k and k/ (1+ k ) are positive definite), K is itself a Mercer kernel.

Efficient computation. Although | A ( x , y ) | grows exponentially, k ga ( x , y ) admits a simple dynamic programming algorithm in O ( nmd ) time. Define

Then the GA kernel is defined as

This quadratic-time computation matches the complexity of standard DTW while incorporating richer alignment information. And this can be further reduced to a wall-clock time complexity of O ( md ) (without loss of generality on n and m ) on a GPU that can accommodate the required number of threads.

G.4 Kernelized dynamic time warping

Let x = [ x 1 , . . . , x n ] and y = [ y 1 , . . . , y m ] be two trajectories (of possibly different lengths) in R d . Traditional vector kernels cannot handle temporal misalignments and variable lengths inherent in time-series comparison. Dynamic Time Warping (DTW) addresses this by computing an optimal alignment path π ∗ that minimizes the cumulative local distance:

Kernelization of DTW (KDTW). [64] To turn DTW into a Mercer kernel many but very similar ideas have been used, we will state 2 such ideas. One approach is to exponentiate the optimal cost, yielding the dynamic time-alignment kernel :

An alternative, inspired by convolution kernels [65], is to exponentiate local distances first and then maximize over alignments:

Neither construction guarantees differentiability.

We use k DTW1 in our experiments and h is chosen using the median heuristic at every iteration of SVGD or SV-CMA-ES. We approximated the gradient of the k DTW1 kernel using a centered finitedifference scheme in our SVGD experiments. Although the KDTW kernels are not differentiable in the classical sense, we observed no adverse effects in our experiments. Nevertheless, these theoretical issues warrant careful attention in future work.

Computational complexity. Both k DTW1 and k DTW2 admit O ( nmd ) dynamic-programming algorithms, replacing the usual min operations of DTW with exponentiation and max (or further min ) steps, while preserving quadratic time complexity. And this can be further reduced to a wall-clock time complexity of O ( md ) (without loss of generality on n and m ) on a GPU that can accommodate the required number of threads.

G.5 Signature kernel

Definition 7. Signature (transform) of a path (or trajectory) [20]. Let V be a Banach space and define the tensor algebra of formal power series

which is equipped with the natural operations induced by the tensor product and has the unit element (1 , 0 , 0 , . . . ) .

Now, consider a continuous path x : I → V on a compact interval I ⊂ R with finite p -variation (for some p < 2 ). For any subinterval [ s, t ] ⊂ I , the signature of x over [ s, t ] is defined as the sequence

This collection of iterated integrals captures the essential features of the path and is invariant under reparameterization.

Recall, the signature of a trajectory is injective (for all non-tree-like paths) and, remains unchanged under any reparametrization. This invariance effectively filters out the complex, infinite-dimensional group of symmetries. Moreover, the collection of linear functionals defined on the signature not only forms an algebra under pointwise multiplication but also has the power to separate points [66]. Consequently, by the Stone-Weierstrass theorem, for any compact set C of continuous paths with bounded variation, the set of such linear functionals is dense in the space of continuous real-valued functions on C [67]. These properties together establish the signature as an ideal feature map for representing our trajectories in ergodic search.

For a path x of finite p -variation with p > 1 the signature terms decay factorially [67]

where ∥ x ∥ p, [ s,t ] is the p -variation of the path on [ s, t ] . Therefore, one can approximate signature kernel reasonably well with the truncated signature transform of level L ∈ N ,

where ⊕ denotes the direct sum, and [21] and [22] provide efficient means to compute the truncated signature using Dynamic Programming and approximate it using Random Fourier Features, respectively.

Definition 8. Signature Kernel Let I = [ u, u ′ ] and J = [ v, v ′ ] be two compact intervals, and consider paths x ∈ C 1 ( I, V ) and y ∈ C 1 ( J, V ) , where V is a Banach space. The signature kernel k sig ( x, y ) : I × J → R is defined by

where φ ( x ) s and φ ( y ) t denote the signatures of x and y over the intervals [ u, s ] and [ v, t ] , respectively.

Utilizing the kernel trick introduced in [20], one can efficiently compute the full (untruncated) signature kernel.

Theorem 3. Let I = [ u, u ′ ] and J = [ v, v ′ ] be compact intervals, and let x ∈ C 1 ( I, V ) and y ∈ C 1 ( J, V ) . Then the signature kernel k x,y satisfies the following linear, second-order hyperbolic partial differential equation:

subject to the boundary conditions

The resulting PDE formulation of the signature kernel enables its efficient computation using any standard numerical scheme, such as finite difference, finite element, or other suitable methods. An important point to note is the assumptions on Theorem 3, will be satisfied for any reasonable discretizations of the trajectories and the domain over which the PDE is solved.

One must take extra cation if they wish to use the signature kernel in SV-CMA-ES, since the trajectories produced in initial few iterations of SV-CMA-ES may not be very well behaved, can produce very rough trajectories, breaking the assumptions of Theorem 3.

G.6 Median heuristic

The median heuristic [27] sets the bandwidth

using the empirical median of all pairwise distances among particles (divided by log N ) to provide a robust, data-driven scale. We adopt

which yields slightly improved performance (in terms of computational speed) in our experiments. 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. We recommend referring to [27] for more (theoretical) guarantees, advantages and details on the median heuristic.

H Measuring trajectory diversity

̸

One can interpret the kernel k ( x , y ) as a similarity measure between two trajectories x and y . In particular, for a normalized kernel function (i.e., k ( x, x ) = 1 ), k ( x , y ) → 1 as x → y , k ( x , y ) → 0 as ∥ x -y ∥ → ∞ . Given a collection of trajectories { x i } N i =1 , one may quantify their diversity by forming the Gram matrix K ∈ R N × N , K ij = k ( x i , x j ) , K ii = 1 , and computing its determinant. Notably, det( K ) → 0 if x i → x j for any i = j, det( K ) → 1 as ∥ x i -x j ∥ → ∞ for all i = j, thereby providing a principled measure of particle diversity.

Throughout the experiments we shall use the following kernel for measuring diversity,

̸

where d continuous Frechet ( · , · ) is the (Continuous) Frechet distance (see Appendix I) and h = 0 . 1 . We denote the Gram matrix of this kernel with K f .

One may choose any finite-valued kernel to measure diversity, we chose k Frechet simply due to the fact that it has not been used in experiments to benchmark the signature kernel.

I Fréchet distance

The Fréchet distance [68] is a classical similarity measure between trajectories that respects both spatial geometry and the ordering of points along each trajectory. Intuitively, it is the minimum leash-length necessary for a person and a dog to walk along two continuous paths, each at its own varying speed, without backtracking.

Definition 9 (Continuous Fréchet Distance) . Let f, g : [0 , 1] → ( X , d ) be continuous curves in a metric space. A reparameterization is a continuous, nondecreasing, surjection α : [0 , 1] → [0 , 1] satisfying

(and similarly for β ). The continuous Fréchet distance is

In our case with discretized trajectories in R d ,

we use the discrete Fréchet distance.

Definition 10 (Discrete Fréchet Distance) . A coupling of x and y is a pair of index sequences σ = ( σ (1) , . . . , σ ( K )) and τ = ( τ (1) , . . . , τ ( K )) such that σ (1) = 1 , σ ( K ) = n, τ (1) = 1 , τ ( K ) = m , and each of σ, τ is non-decreasing with increments at most 1. The discrete Fréchet distance is

Computational complexity The continuous Fréchet distance can be computed in

time [69, 70]. Here-and unlike most of the literature that treats d as a hidden constantd varies in our setting and so is made explicit in the O ( · ) notation.

Properties

Metric Properties. The continuous Fréchet distance is only a pseudometric on the space of parameterized curves-distinct parameterizations of the same geometric trace satisfy d Continuous Frechet ( f, g ) = 0 -but it induces a true metric on the quotient of curves under reparameterization. In contrast, the discrete Fréchet distance d Frechet on point-sequences satisfies all four metric axioms (non-negativity, symmetry, identity of indiscernibles, and the triangle inequality) and therefore is a bona fide metric on the set of discrete trajectories

Stability. If two curves f, f ′ and g, g ′ satisfy ∥ f -f ′ ∥ ∞ ≤ ε and ∥ g -g ′ ∥ ∞ ≤ ε , then ∣ ∣ d ( f, g ) -d ( f ′ , g ′ ) ∣ ∣ ≤ ε.

Order-sensitivity. Unlike the Hausdorff distance, the Fréchet distance respects the natural parameterization of trajectories, making it especially powerful for comparing time-indexed data.

Because of these advantages, the Fréchet distance has become a cornerstone in shape matching, motion analysis, and trajectory mining.

J Additional results

J.1 Multiscale constrained exploration

(a) RBF Kernel

(b) Global Alignment Kernel

(c) Signature kernel

(a) SVGD (

ϵ

= 0

.

05

)

(b) SV-CMA-ES (

α

x

= 0

.

2

)

J.2 3D Coverage using the Crazyflie drone

(a) SVGD (

ϵ

= 0

.

01

)

(b) SV-CMA-ES (

α

x

= 0

.

1

)

J.3 Stein variational model-predictive control

K Implementational and experimental deatils

In this section we provide the full Implementational and experimental details.

K.1 Implementational details

Elite sample selection and log-rank weights for SV-CMA-ES In our experiments we rank the m samples by selecting the top

as the elite sub-population, we assign positive log-rank recombination weights

and set w i,k = 0 for k > λ . ('Active' negative weights [39] are disabled in the present implementation.)

Enhancing the signature kernel's performance A common enhancement is to first embed each trajectory into a (possibly infinite-dimensional) feature space via a static feature map φ static : R d -→ H , associated, for example, with a radial basis or Matérn kernel. Concretely, one defines

Empirical studies demonstrate that this pre-lifting step substantially improves the performance of the signature kernel. Therefore, the ergodic Stein variational update step in the state space using the

signature kernel and a static kernel is given by

An analogous update applies in the control space. Throughout this paper, we emploied the feature map corresponding to the radial basis kernel (RBF) (see Appendix G).

Hardware used All experiments were run on the a computer with:

Operating System:

Ubuntu 20.04.6 LTS

CPU:

AMDRyzen 9 5900HS

RAM:

16 GB

GPU:

NVIDIA GeForce RTX 3060

CUDA Version: 12.2

Assets used The codebases provided by the following works [2, 22, 20] and the GitHub repository https://github.com/khdlr/softdtw_jax were either directly incorporated into our experiments or proved essential as implementation references.

K.2 Experimental details

K.2.1 Multiscale constrained exploration

We consider the two-dimensional spatial domain S = [0 , 100] × [0 , 100] m , equipped with the uniform measure π . To improve numerical conditioning, we introduce the affine mapping g : S → [0 , 1] × [0 , 1] , g ( x ) = x 100 . 4 Trajectories of dimension n = 2 and length T = 100 with a time step of ∆ t = 1 s are generated via Algorithm described in sections 3.1 & E.2. A maximum k max = 8 of basis functions per dimension is employed for constructing the ergodic metric (see Appendix D Definition 6). The cost functional is specified as

where the obstacle penalty is defined by c obs ( x ) = max ( 0 , ∥ x -x c ∥ 2 -r ) , with x c and r denoting the obstacle center and radius, respectively. The initial and final states, x init and x final , are prescribed on the boundary of S .

K.2.2 3D Coverage using the Crazyflie drone

We consider a three-dimensional domain S = [0 , 3] × [0 , 3] × [0 . 5 , 1 . 5] , equipped with the uniform measure π . Trajectories are computed via Algorithm described in section 3.1 & E.2, using a cost functional analogous to that of the multiscale forest example:

The trajectory length is set to T = 150 with a time step of ∆ t = 0 . 1 s. A maximum k max = 8 of basis functions per dimension is employed for constructing the ergodic metric (see Appendix D Definition 6).

K.2.3 Stein variational model-predictive control

We consider coverage over a quad-modal target distribution

comprising four Gaussian components

on the unit-square domain S = [0 , 1] × [0 , 1] . The agent obeys single-integrator dynamics x t +1 = x t +∆ t u t , with ∆ t = 0 . 1 . Trajectory planning is carried out over a horizon of T = 20 time steps, using N = 20 control-sequence samples per iteration. Each planning loop is executed for up to 200 iterations.

A maximum k max = 10 of basis functions per dimension is employed for constructing the ergodic metric (see Appendix D Definition 6). Ten obstacles are placed uniformly at random in S ; each obstacle moves at the same rate ∆ t , with velocity directions sampled from N (0 , σ 2 ) where σ = 0 . 01 .

The cost functional for a control sequence u = ( u 0 , . . . , u T -1 ) is defined as

with scaling parameter µ = 10 . The initial control prior is taken as p ( u ) = N ( 0 , 0 . 01 I ) .

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: The claim made in the abstract and introduction are supported in section 7

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: Limitations are discussed in section 8.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: Look at Appendix F.

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

All assumptions should be clearly stated or referenced in the statement of any theorems.

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

Theorems and Lemmas that the proof relies upon should be properly referenced.

4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes] .

Justification: The exact experimental setup is are provides in sections 7 & Appendix K.

Guidelines:

The answer NA means that the paper does not include experiments.

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [No]

Justification: We extended the code used from the paper [2]. The work in this paper is closed source and therefore we did not open our code.

Guidelines:

The answer NA means that paper does not include experiments requiring code.

Please see the NeurIPS code and data submission guidelines ( https://nips.cc/ public/guides/CodeSubmissionPolicy ) for more details.

While we encourage the release of code and data, we understand that this might not be possible, so 'No' is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines ( https: //nips.cc/public/guides/CodeSubmissionPolicy ) for more details.

The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

6. Experimental setting/details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [NA] .

Justification: This paper does not introduce a machine learning based algorithm.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: We do provide statistical data in section J. Although no p-test is performed.

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments compute resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: This is provided in section K.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

9. Code of ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes] .

Justification: We acknowledge the code of ethics and can guarantee that none of them were violated.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [NA] .

Justification: The social impact of this work is not quantifiable at the moment.

Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA] .

Justification: The paper poses no such risk. The paper builds on a trajectory optimization algorithm.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: They have been acknowledged in section K.

Guidelines:

The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

The authors should state which version of the asset is used and, if possible, include a URL.

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset's creators.

13. New assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [NA]

Justification: The paper does not release new assets, but will provide assets soon, and will be documented.

Guidelines:

The answer NA means that the paper does not release new assets.

Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

The paper should discuss whether and how consent was obtained from people whose asset is used.

At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

14. Crowdsourcing and research with human subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA] .

Justification: The paper does not involve crowdsourcing nor research with human subjects

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

15. Institutional review board (IRB) approvals or equivalent for research with human subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA] .

Justification: The paper does not involve crowdsourcing nor research with human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

16. Declaration of LLM usage

Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.

Answer: [NA] .

Justification: LLMs were used only for improving the english used in the paper and, writing and debugging snippets of code. They were not used for any core method development.

Guidelines:

The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.

Please refer to our LLM policy ( https://neurips.cc/Conferences/2025/LLM ) for what should or should not be described.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new method of ergodic search for robots based on the SVES framework, and it aims to improve the diversity of the generated trajectories. The key innovation is to replace the traditional RBF kernel with the Signature Kernel, which improves the kernel's sensitivity to differences in paths. The paper also introduces a PDE formulation of computing the signature kernel that allows parallel computation. The experiments show that with the signature kernel, SVES has better state coverage and convergence results.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n1. The paper contains both qualitative comparisons and quantitative results of trajectories generated using the RBF kernel and the Signature kernel, illustrating that the Signature kernel helps cover the state space. \n2. Using signature kernels for ergodic exploration is an innovative idea, whereas previously signature kernels were applied to trajectory optimization in robotics, as in “Path Signatures for Diversity in Probabilistic Trajectory Optimisation.”\n3. The paper extends the theoretical framework in the prior work Stein Variational Ergodic Search (SVES). \n\n\nWeakness:\n1. The reason why the signature kernel can improve the state coverage is still unclear. In Line 78, the author claims \"The signature kernel naturally discriminates between **subtly different paths**, thus promoting diversity.\" Can you visualize how the signature kernel helps discriminate the local differences between two similar paths, or provide a detailed explanation about why the RBF kernel fails to do so while the signature kernel can?\n2. The experiments only compare SVES with the signature kernel with SVES based on other kernels. However, there have been plenty of kernel-free methods for the robot exploration problem. For example, Burda et al. (2018) “Exploration by Random Network Distillation” introduces **intrinsic rewards** to improve the trajectories' diversity. Also, the **curiosity-driven** approaches in \"Aggressive Quadrotor Flight Using Curiosity-Driven Reinforcement Learning\" and \"Curiosity-Driven Reinforcement Learning based Low-Level Flight Control\" also help improve the state coverage during training. I am confused about whether the SVES framework is necessary for the ergodic search problem.\n3. The experiments are only conducted in the 3D Coverage Drone Experiment, which follows a similar setting in Lee et al. (2024) \"Stein Variational Ergodic Search\". I am concerned about the applicability of the signature kernel in other robotic tasks.\n\nQUESTIONS:\nTypo:\nYou assume $h_2(x) \\leq 0$ in Line 122, but the last term in Eq. 3 is $\\max(0, h_2(x))$, which equals to 0.\n\n\nQuestion:\n1. 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.\n2. Could you explain in more detail how the signature kernel differs from the RBF kernel when it comes to comparing two similar trajectories?\n\nLIMITATIONS:\nThe authors addressed the limitations and potential negative societal impact of their work.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nEfficiently exploring a bounded domain within a finite amount of time is a challenging and central problem in robotics. Ergodic exploration converts exploration into a coverage problem, optimizing the time-average spatial distribution of the robot against an expected information measure. While this can handle multi-modal landscapes, in practice, the non-convex nature of the objective makes this difficult. Stein Variational Ergodic Search (SVES) was proposed to mitigate this problem, leveraging Stein Variational Gradient Descent (SVGD), which approximates the distributions of trajectories in a particle-based fashion. However, the choice of kernel in SVGD can have a huge impact on the diversity of the trajectories considered. In this paper, the authors propose to use the signature kernel, which can better discriminate between trajectories and respects its Markovian structure. In addition, the authors propose to use Stein Variational Covariance Matrix Adaptation Evolution Strategy (SV-CMA-ES) instead of SVGD, which is gradient-free and better take advantage of GPU parallelism. This is motivated by both the fact that gradients can be expensive to compute and that search-based strategies have shown empirical benefits over gradient-based methods. To estimate the gradient of the signature kernel, they propose to use Simultaneous Perturbation Stochastic Approximation (SPSA), which is also a gradient-free method that evaluates a black-box function via random perturbations. They evaluate SV-CMA-ES across two benchmark systems and five choices of kernels (including the signature kernel). Their results show that trajectories produced via the signature kernel are indeed more diverse, lower cost, and slightly more ergodic than other choices of kernels. They demonstrate improved convergence of SV-CMA-ES over SVGD on both benchmark systems. Finally, they use the signature kernel in SV-MPC, a model predictive control algorithm which leverages SVGD to perform the updates. They show that the signature kernel improves yields trajectories with more diversity while remaining low cost. They also found that the Markov RBF kernel failed to converge to meaningful solutions.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- Efficient exploration of a predefined region in a timely manner is an important and core problem in robotics. This paper combines existing tools in a novel way to solve this problem, with clear benefits in terms of trajectory cost and diversity on two simulated platforms. They also show their choice of kernel can be used in an MPC loop, with clear benefits over other choices of kernels.\n- They perform sufficient ablations that show the concrete benefits of each of their design choices. In particular, they evaluate four different choices of baseline kernels compared with their choice of signature kernel. They also explore the convergence of SV-CMA-ES and compare it to SVGD, showing the benefits of this choice of optimizer.\n- Figure 1 is nicely placed and immediately introduces the reader to the empirical benefits of their approach, with a clear improvement in the diversity of trajectories. Figure 2 also makes it clear which kernel works better for this application.\n- The appendix is contains most of the additional information necessary to understand the paper, making it fairly self-contained.\n- In terms of writing, I enjoyed how the main paper focused on the high-level points, relegating the details to the appendix for the interested reader. I also liked how the core questions of the evaluations were outlined in the introduction of Section 7.\n\nWeaknesses:\n- Using only two benchmark systems on similar tasks is a great starting point, but the evaluations could be more extensive. Exploring systems with different choices of dimensionality, types of dynamics, and obstacle types and configurations would strengthen this paper.\n- Figure 3 and 5 make it difficult to clearly see how each method compares. I know there's a table in the appendix, but the visualization of the data could be much clearer. For instance, pairing up by metric rather than kernel could help make the differences distinguishable. Or you could show the distributions with box or violin plots, instead of averages, dividing each metric into a separate sub-panel. While not critical, it would make it much easier to understand the results.\n- In general, there isn't enough discussion of the results or conclusions drawn. In particular, Figures 4 and 6 provide the comparison in convergence of SV-CMA-ES versus SVGD without any discussion or interpretation of the results in the text. There are also a lot of curves in the plot, with substantial overlap, making it hard to interpret. The paper would be stronger if the discussion of these results was elaborated.\n- It would be great to have a little exposition on the benchmark problems used to evaluate the method. While I like relegating a lot of the details to the appendix, the considered problems do need to be introduced more in the main paper.\n- Model predictive control (MPC) is discussed and used in this paper, without really being defined, at least in the main paper. Since it is an important part of the evaluation, I would suggest providing at least a couple sentences describing how it works and adding citations to SV-MPC in the main paper.\n- Similarly, it would be great if the median heuristic would be defined, or at least given a citation. I know it's a fairly standard practice, but for completeness, it would be good to include.\n\nQUESTIONS:\n- 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.\n\nLIMITATIONS:\n- The authors provide a number of limitations, mostly centered around computational cost. They also talk about how the repulsive force with the signature kernel can be almost too strong, encouraging poor choices, such a collisions into obstacles. They also provide some remedies for these issues and possible future directions.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nEfficient robotic exploration requires information-aware coverage of the domain, avoiding well-understood areas, and focuses more attention on areas that are not well-understood. The paper proposes two methods to diversify Stein Variational Ergodic Search (SVES), which reduce ergodic cost and produce richer trajectory sets than SVES.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper does a good job of explaining the problem statement, as well as explaining the underlying SVES and Signature Transform and Signature Kernel. Qualitatively, the signature kernel seems to produce richer and diverse trajectories than the other methods in Figure 2. This is further reinforced in Figure 3, where the trajectory diversity is significantly higher than the other methods, while the trajectory and ergodic costs are marginally lower. The signature kernel really stands out from methods in real-world drone tests. SV-CMA-ES shows similar costs compared to SVGD, while having higher average trajectory diversity. \n\nThe authors are honest about the limitations of their method, including dependence on hardware, and scalability of the approach. The one critique of the paper is that it claims SV-CMA-ES is a real-time method (in Contributions subsection), due to elimination of gradient computation, but no analysis is provided on this. I would have liked to see a runtime analysis of SV-CMA-ES versus other methods. The time complexity gives an idea of runtime for the signature kernel, but more quantitative evaluation is required. For the most part however, this is a complete work with supplementary results as well.\n\nSubmission is clearly written and well-organized. It seems reproducible as experimental and implementation details are provided. Results seem impactful for more efficient robotic exploration, particularly for active perception, where the robot is trying to map its environment in an unknown setting. Due to real-world tests on a drone, it seems like the method is viable for robotics practitioners, although little analysis of runtime is provided. This method improves prior works, and the SV-CMA-ES method is of considerable interest due to its ability to possibly run real-time on a drone.\nThe work is original as it tackles the problem of diversifying SVES as well as incorporating gradient-free approaches. It is also clear how the method advances previous works, showcasing both qualitative and quantitative comparisons. The work also provides new insights into the meaning of “diversity” for trajectories and improves upon different metrics, such as ergodic cost, trajectory cost and diversity. However, the motivation of integrating SV-CMA-ES is not well-formulated as the paper says “more efficient search steps”, but no quantitative evaluation is provided to back up this claim.\n\nQUESTIONS:\n1.\tPlease provide runtime analysis on a common hardware platform. Ideally benchmark with different number of particles, and problem dimensions.\n2.\tAre there other efficient popular heuristics other than the median heuristic for selecting bandwidth of kernels? Might be interesting to benchmark these.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-03"}

comments: SUMMARY:
This paper improves Stein Variational Ergodic Search (SVES) for robotic exploration by making trajectories more diverse and lowering ergodic cost. It does this by using a signature kernel and a gradient-free evolution strategy (SV-CMA-ES). Experiments on various tasks demonstrate these improvements.

STRENGTHS AND WEAKNESSES:
Strengths:
1. Clear technical improvement on SVES by combining signature kernels and SV-CMA-ES.
2. Experiments support claims with consistent gains in trajectory diversity and lower ergodic cost.
3. Well-structured and clearly presented paper.

Weaknesses:
1. Could include more related work to better position the contribution.
2. The main novelty seems to be the combination of known components, not new core methods.
3. Missing baselines from recent literature for broader comparison.

QUESTIONS:
1. Could you expand the related work to compare with other ergodic planners or diversity-focused methods outside SVES? 
2. Could comparisons be made to other non-SVES baselines?
3. Is there any hardware or real robot demo?
4. Could you provide more details on how sensitive results are to kernel parameters?

LIMITATIONS:
Yes, the authors acknowledge key technical limitations, including hardware scalability, computational cost for large particle sets, and practical issues like the signature kernel pushing trajectories into obstacles.
TITLE: Official Review
CLARITY: 2
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-03