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

• URL: http://arxiv.org/abs/2601.05811v1
• Date: Fri, 09 Jan 2026 14:35:19 GMT
• Title: Learning Reconstructive Embeddings in Reproducing Kernel Hilbert Spaces via the Representer Theorem
• Authors: Enrique Feito-Casares, Francisco M. Melgarejo-Meseguer, José-Luis Rojo-Álvarez,
• Abstract summary: This work proposes new algorithms for reconstruction-based manifold learning within Reproducing Kernel Hilbert Spaces (RKHS)<n>A separable operator-valued kernel extends the formulation to vector-valued data while retaining the simplicity of a single scalar similarity function.<n>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.
• Score: 2.0573301822495553
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: 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.

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