Low-Dimensional Embeddings for Gaussian Kernels on Manifolds

We propose a family of random feature maps for scalable kernel machines on low-dimensional subspaces, ie on the Grassmannian manifold. Such representations are useful when data classes or clusters are well described by the span of a few samples. Classical Grassmannian kernels, including the projection and Binet-Cauchy kernels, require full Gram matrices, which leads to prohibitive computational and memory costs for large high-dimensional subspace datasets. We address this limitation using random features based on rank-one projections of subspace projection matrices followed by bounded non-linear transforms, either periodic or binary, to control the resulting distributions. We show that inner products in the random feature space approximate well-defined rotation-invariant Grassmannian kernels that depend only on the principal angles between subspaces. When the number of features is sufficiently large relative to the intrinsic subspace dimension, the approximation holds uniformly over all fixed-dimensional subspaces with high probability. For periodic transforms, the approximated kernel has a closed-form expression with tunable behaviour between inverse Binet-Cauchy and Gaussian-type regimes. Binary transforms yield compact one-bit subspace features, although no closed-form kernel is known. Structured rank-one projections based on randomised fast Fourier transforms further reduce computation without sacrificing practical accuracy. Experiments on synthetic data and ETH-80 classification tasks show that these features accurately preserve Grassmannian geometry while reducing computation, memory, and storage. Rank-one embeddings therefore provide a practical and scalable alternative to classical Grassmannian kernels.

We show that, up to isotropic scaling, the Gaussian RBF reproducing kernel Hilbert space (RKHS) is asymptotically isometric to Euclidean space in the large bandwidth limit. This strongly suggests that kernel-based constructions reliant on metric properties of the RKHS will yield results for Gaussian RBF kernels that similarly approach those of linear kernels for large bandwidths. The asymptotic behavior of Gaussian CKA can be understood in this light. We further consider kernel PCA, showing that Gaussian RBF eigenvalues, eigenprojections, and principal components all converge to those of classical (linear) PCA as bandwidth $σ\rightarrow \infty$. For a given data representation, both the RKHS feature embeddings and the orthogonal PCA eigenframes of the two kernel types differ asymptotically by a geometric similarity transformation, up to a residual of size $O \left (\fracρσ \right )^2$, where $ρ$ is a measure of geometric eccentricity of the representation, equal to the ratio of maximum to median pairwise distance between data examples. Experiments over a diverse collection of data sets demonstrate that $ρ$ provides a simple and reliable predictor of dataset-specific convergence behavior in the top principal directions.

Kernel methods, and Gaussian Processes (GPs) in particular, require a Hilbertian distance measure---one whose square is conditionally negative definite (CND)---to guarantee positive semi-definiteness (PSD) of the kernel matrix; a condition that fails for many natural input spaces, including smooth manifolds and spaces of probability distributions. We propose the Sparse Landmark Embedding (SLE) kernel, which eliminates this requirement entirely. Each input is embedded into a sparse feature vector via compactly supported bump functions centered at all |D| training points; applying any standard PSD kernel in this embedding space yields a kernel that is provably PSD for arbitrary distance measures. The compact support automatically controls embedding sparsity, keeping kernel matrices well-conditioned and computationally tractable despite the high ambient dimension. We provide theoretical guarantees on PSD, sparsity, stability, and universal approximation, and demonstrate, using geodesic and Wasserstein distances, that the SLE kernel matches or substantially exceeds domain-specific baselines in both predictive accuracy and uncertainty quantification.