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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.05811 |
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| _version_ | 1866910193577623552 |
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| author | Feito-Casares, Enrique Melgarejo-Meseguer, Francisco M. Rojo-Álvarez, José-Luis |
| author_facet | Feito-Casares, Enrique Melgarejo-Meseguer, Francisco M. Rojo-Álvarez, José-Luis |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05811 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Learning Reconstructive Embeddings in Reproducing Kernel Hilbert Spaces via the Representer Theorem Feito-Casares, Enrique Melgarejo-Meseguer, Francisco M. Rojo-Álvarez, José-Luis Machine Learning Artificial Intelligence 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. |
| title | Learning Reconstructive Embeddings in Reproducing Kernel Hilbert Spaces via the Representer Theorem |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2601.05811 |