Learning Reconstructive Embeddings in Reproducing Kernel Hilbert Spaces via the Representer Theorem

doi:10.1109/ojcs.2026.3682462

Learning Reconstructive Embeddings in Reproducing Kernel Hilbert Spaces via the Representer Theorem

2026-04-09

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E

Enrique Feito-Casares

F

Francisco-Manuel Melgarejo-Meseguer

J

José Luis Rojo‐Álvarez

DOI：10.1109/ojcs.2026.3682462

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Motivated by the growing interest in representation learning approaches that uncover the latent structure of high-dimensional data, this work proposes new algorithms for reconstruction-based manifold learning within Reproducing-Kernel Hilbert Spaces (RKHS). Each observation is first reconstructed as a linear combination of the other samples in the RKHS, by optimizing a vector form of the Representer Theorem for their autorepresentation property. A separable operator-valued kernel extends the formulation to vector-valued data while retaining the simplicity of a single scalar similarity function. A subsequent kernel-alignment task projects the data into a lower-dimensional latent space whose Gram matrix aims to match the high-dimensional reconstruction kernel, thus transferring the auto-reconstruction geometry of the RKHS to the embedding. Therefore, the proposed algorithms represent an extended approach to the autorepresentation property, exhibited by many natural data, by using and adapting well-known results of kernel learning theory. Numerical experiments on both simulated (concentric circles and swiss-roll) and real (cancer molecular activity and IoT network intrusions) datasets provide empirical evidence of the practical effectiveness of the proposed approach.

Keywords:

Reproducing kernel Hilbert space

representer theorem

manifold learning

kernel alignment

dimensionality reduction

data embedding

Learning Reconstructive Embeddings in Reproducing Kernel Hilbert Spaces via the Representer Theorem

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