GARFLN: Geodesic Adaptive Riemannian Functional Link Network

Abstract

The random vector functional link (RVFL) network is a lightweight single-hidden-layer neural architecture that augments input features through random nonlinear transformations and direct input–output connections, enabling closed-form training and universal approximation in Euclidean spaces. Despite its computational efficiency and strong generalization on vectorized data, RVFL fundamentally assumes flat geometry, which limits its applicability to non-Euclidean domains such as symmetric positive definite (SPD) manifolds, hyperbolic embeddings, and spherical representations. In such settings, the Euclidean random projections in RVFL fail to preserve intrinsic curvature and geodesic structure, leading to distorted feature embeddings and degraded performance. To overcome these limitations, we introduce the geodesic adaptive Riemannian functional link network (GARFLN), a novel curvature-aware extension of RVFL designed for manifold-valued data. GARFLN integrates curvature-adaptive geodesic kernels and tangent space projections within a dual-branch architecture: the intrinsic branch operates directly on the manifold to capture local geometric relationships, while the extrinsic branch leverages Euclidean embeddings to retain global contextual information. A random vector field displacement mechanism perturbs samples along geodesics, enhancing representational diversity while maintaining geometric consistency. This design preserves RVFL's closed-form learning efficiency while extending its expressivity to curved spaces. Furthermore, we provide a rigorous theoretical analysis establishing the geometric consistency and Universal Approximation Theorem for GARFLN, proving its capacity to approximate any continuous function on compact Riemannian manifolds. Comprehensive experiments conducted on UCI and KEEL benchmark datasets, supported by detailed statistical analyses, demonstrate that GARFLN consistently achieves superior generalization performance and robustness compared to state-of-the-art baseline models across diverse learning tasks. The source code of the proposed GARFLN is available at https://github.com/mtanveer1/GARFLN. © 2026 Elsevier Ltd