Multilevel kernel approximation of log-Euclidean manifold-valued functions

Abstract

We study kernel approximation and multilevel residual correction for functions with values in a manifold that admits a global logarithm onto a finite-dimensional Hilbert space, such as positive definite matrices, exponential Lie groups, and Hadamard manifolds. Approximating in logarithmic coordinates keeps the approximant on the manifold and transfers Hilbert-space error estimates verbatim. To express these errors in the natural geometry of the target, we prove metric comparisons. In particular, we obtain a sharp, scale-invariant bound between the log-Euclidean and affine-invariant distances on positive definite matrices, with a constant that depends only on condition numbers. For multilevel approximation, compactly supported kernels scaled to the fill distance give algebraic convergence with sparse, uniformly well-conditioned systems. These guarantees require appropriate sample locations and approximation scales. Compact groups are treated through a principal logarithm, where a bi-invariant metric gives error bounds with constant one. Finally, the log-Euclidean and Karcher means differ by $\mathcal O(r^3)$ near the identity and by $\mathcal O(r^2)$ around a general point, where $r$ is the logarithmic data radius and the second bound is sharp.