GeoGP: Geometry-Invariant One-Shot GP

• The paper introduces novel geometry-invariant kernels that enable Gaussian processes to learn robust representations from minimal data in complex, non-Euclidean spaces.
• GeoGP employs specialized coordinate transformations and spectral methods to enforce invariance under translations, rotations, scaling, and other symmetry operations.
• The framework achieves efficient one-shot learning through sparse approximations, random feature expansions, and manifold-aware covariance constructions, demonstrating practical success in robotics and geospatial modeling.

Geometry-Invariant One-Shot Gaussian Process (GeoGP) designates a class of probabilistic models that combine the flexibility of Gaussian processes (GPs) with explicit invariance to geometric transformations and the ability to learn from minimal data—often a single example or sparse observations. GeoGP models address complex scenarios where intrinsic geometry, symmetry groups, or non-Euclidean constraints must be respected, such as data defined on manifolds, robotic learning-from-demonstration, geospatial inference, shape reconstruction, and physical system modeling. These frameworks fundamentally differ from conventional GP approaches by employing specialized kernels, coordinate representations, random feature expansions, and/or equivariant neural or spectral architectures that guarantee invariant predictions under translation, rotation, scaling, reflection, or transformation governed by a symmetry group.

1. Geometric Kernels and Covariance Construction

Conventional GPs assume Euclidean geometry and typically operate with stationary kernels (e.g., RBF, Matérn) relying on Euclidean distances. GeoGP approaches modify this paradigm to encode geometric invariance using several strategies:

• Heat Kernel Construction via Manifold Brownian Motion. As illustrated in “Intrinsic Gaussian processes on complex constrained domains” (Niu et al., 2018), the kernel is built from the transition density of Brownian motion or the solution to the heat equation on a Riemannian manifold. For $\mathcal{M}$, the kernel

$$K_{heat}(x, y, t) = (2\pi t)^{-d/2} \exp\left(-\frac{\|x-y\|^2}{2t}\right)$$

on $\mathbb{R}^d$ generalizes to curved or constrained domains via simulation of manifold Brownian paths and local SDEs respecting the metric tensor $g$. The resulting GP respects boundaries, gaps, and curvature.

• Graph Laplacian Kernels. In “Graph Based Gaussian Processes on Restricted Domains” (Dunson et al., 2020), the covariance structure is defined spectrally via the eigenpairs of a kernel-normalized graph Laplacian $L = (A - I)/\varepsilon^2$, approximating the intrinsic heat kernel on the data manifold. The covariance matrix

$$\mathcal{H}_{\varepsilon,K,t} = (m+n) \sum_{i=0}^{K-1} e^{-\mu_i t} \tilde{v}_i \tilde{v}_i^T$$

uses spectral decay to control smoothness and adaptation to local geometry.

• Group-Invariant Spectral Construction on Symmetric Spaces. “Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces II” (Azangulov et al., 2023) develops kernels expressed via zonal spherical functions and group-invariant spectral measures:

$$k(x,x') = \int_{\mathfrak{a}^*} \pi^{(\lambda)}(g_2^{-1} g_1) d\mu_k(\lambda)$$

thus encoding stationarity and invariance under the group action, with random Fourier feature approximations supporting one-shot sample generation.

• Diffusion and Curvature-Aware Convolutional Kernels. “Geometry-Aware Hierarchical Bayesian Learning on Manifolds” (Fan et al., 2021) introduces periodic-potential kernels capturing mean curvature flow:

$$K(\|v\|,\lambda,\phi) = \frac{1}{4\pi}\sum_{n=1}^N \exp(-\lambda_n\|v\|)\cos(\lambda_n\|v\|+\phi_n)$$

enforcing geometry-awareness for mesh or point cloud data.

2. Coordinate Representations and Invariance Strategies

Geometry invariance is further achieved by transforming data into representations that explicitly neutralize translation, rotation, scaling, or more complex symmetries:

• Polar and Relative Coordinates for Trajectory Learning. “Prompt2Auto: From Motion Prompt to Automated Control via Geometry-Invariant One-Shot Gaussian Process Learning” (Yang et al., 17 Sep 2025) employs polar coordinates, recentering at the initial demonstration point:

$$r(t_k) = \sqrt{[p_1(t_k) - p_1(t_0)]^2 + [p_2(t_k) - p_2(t_0)]^2}, \quad \theta(t_k) = \arctan2(p_2(t_k) - p_2(t_0), p_1(t_k) - p_1(t_0))$$

with subsequent embeddings $[\cos \theta, \sin \theta]$ and normalization, ensuring invariance to translation, rotation, and scaling.

• Kernel Features for Non-Euclidean Spaces. In (Azangulov et al., 2023), random feature expansions are built on non-Euclidean group actions, sampling from the Haar measure and spectral measure, resulting in feature maps

$\phi_{\lambda,h}(g) = e^{(i\lambda+\rho)^T a(hg)}$

where the constructed GP is invariant across the full group structure.

• Cluster-Based Subpart Abstraction for One-Shot Concept Learning. “Abstracted Gaussian Prototypes for One-Shot Concept Learning” (Zou et al., 30 Aug 2024) models images by clustering pixel subparts with Gaussian Mixture Models (GMMs), producing prototypes invariant up to feature permutation and spatial arrangement, facilitating breadth in generative and classification tasks.

3. Efficient Learning: One-Shot and Sparse Approximations

GeoGP frameworks are designed to operate effectively with minimal data, often a single demonstration:

• Integral Gaussian Processes and RKHS Confinement. “Learning Integral Representations of Gaussian Processes” (Tan et al., 2018) introduces IGPs, where sample paths are guaranteed to live in the RKHS of the kernel, supporting geometry invariance and fast computation. Sufficient dimension reduction (SDR) selects optimally informative subspaces, leveraging generalized eigenvalue decompositions and the representer theorem, leading to “one-shot” dimensionality reduction:

$$f_n(\cdot) \approx \sum_{j=1}^m \beta_j\left[\sum_{i=1}^n W_{ij}\kappa(\cdot, x_i)\right]$$

and EM-based updates using the Woodbury identity.

• Sparse and Inducing Point Strategies. (Niu et al., 2018) demonstrates that using a restricted set of inducing locations and simulating BM only from these points reduces computational cost ($\mathcal{O}(n \cdot m^2)$ scaling).
• Scalable Random Feature and Neural Approximations. (Azangulov et al., 2023, Mathieu et al., 2023) show that random Fourier feature expansions built from group-theoretic bases permit fixed-budget, scalable GP inference compatible with automatic differentiation and modern software stacks.

4. Practical Applications and Empirical Outcomes

GeoGP methodologies are validated across domains where geometric structure is fundamental:

• Trajectory Learning and Robotic Control. (Yang et al., 17 Sep 2025) demonstrates robust adaptation to human demonstration prompts under arbitrary translation, rotation, and scaling, supporting passive/active skill autonomy with multi-skill libraries. Quantitative results show exact GP models trained on Cartesian coordinates fail to generalize under geometric transformations; GeoGP maintains low error under all tested transformations.
• Surface Reconstruction and Object Modeling. (Holalkere et al., 24 Mar 2025) introduces a one-shot stochastic Poisson surface reconstruction pipeline with geometric GPs on the torus $\mathbb{T}^d$, supporting local queries (collision, ray casting, next-view planning) without dependency on global volumetric grids—output-sensitive runtime and sharper, correlated uncertainty estimates than grid-based approaches.
• Geospatial and Environmental Modeling. (Dunson et al., 2020, Niu et al., 2018) apply heat kernel and Laplacian-based GPs on complex spatial domains (e.g., "Swiss Roll", Aral Sea chlorophyll), with simulation studies showing reduced RMSE and realistic pattern reconstruction relative to traditional GPs or smoothers.
• Manifold-Valued Computer Vision Tasks. (Fan et al., 2021) achieves superior accuracy on non-rigid shape retrieval and point cloud classification tasks by aggregating curvature-aware features via geometry-aware convolutional kernels.
• Localization and Sensor Fusion. (Yuan et al., 22 Dec 2024) feathers UWB ranging with continuous-time LiDAR-inertial odometry via GP modeling, delivering geometry-invariant anchor calibration and one-shot global localization robust to NLoS/environmental ambiguity.

5. Mathematical Formulation and Implementation Details

GeoGP models are underpinned by specialized mathematical structures, most notably:

• Heat Kernel/SDE Connection:

$$\frac{\partial K_{heat}(x_0, x, t)}{\partial t} = \frac{1}{2} \Delta_x K_{heat}(x_0, x, t)$$

with $K_{heat}(x_0, x, 0) = \delta(x_0, x)$.

• Graph Laplacian Spectrum:

$$\mathcal{H}_{\varepsilon,K,t} = (m+n) \sum_{i=0}^{K-1} e^{-\mu_i t} \tilde{v}_i \tilde{v}_i^T$$

• Group-Invariant Kernel Expansion:

$$k(x,x') = \int_{\mathfrak{a}^*} \pi^{(\lambda)}(g_2^{-1} g_1) d\mu_k(\lambda)$$

• Polar Coordinate Transformation:

$$r_k, \theta_k \quad \rightarrow \quad [r_k/r_{max}, (\cos \theta_k + 1)/2, (\sin \theta_k + 1)/2]$$

• One-shot GP Conditioning:

$$(f | v)(\cdot) = f(\cdot) + K_{f,v}(K_{v,v} + \Sigma)^{-1}(v - v(x) - \epsilon)$$

Implementation typically relies on scalable random feature approximations, efficient spectral decomposition (for GL-GPs and Fourier/toroidal methods), sparse variational inference, and equivariant neural architectures.

6. Comparative Analysis and Limitations

GeoGP frameworks surpass conventional GPs where the latter confound disconnected regions, fail in spaces with complex boundaries, or require embedding or heavy data augmentation. Heat kernel or graph Laplacian-based approaches capture intrinsic distances, classical approaches only Euclidean. One-shot and output-sensitive designs are favorable over two-stage reconstruction pipelines or methods tied to mesh resolution.

Limitations include:

• Fourier truncation and approximation errors for high-frequency detail (see (Holalkere et al., 24 Mar 2025));
• Dense kernel operations possibly limiting scalability for extreme data sizes;
• Some methods rely on periodic boundary conditions or specific manifold topologies;
• Group-invariant feature sampling requires sophisticated spectral and group measure machinery.

7. Future Directions

Prospects span:

• Full rigid body invariance in SE(3) for 3D tasks;
• Extension to volumetric and medical imaging data;
• End-to-end integration with deep learning for scalable, uncertainty-aware manifold representations;
• Efficient online learning (streaming GP and sparsification);
• Deployment in logistics, robotics, large-scale sensor fusion, and physical system identification.

The synthesis and development of geometry-invariant kernel structures, coordinate transformations, and efficient learning rules make GeoGP models an attractive path for probabilistic modeling under complex geometric constraints, with practical impact across computational science, robotics, graphics, and data-driven physical modeling.