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

Enrique Feito-Casares, Francisco M. Melgarejo-Meseguer, José-Luis Rojo-Álvarez

TL;DR

The paper tackles learning low-dimensional embeddings that preserve the autoreconstructive geometry of data in a Reproducing-Kernel Hilbert Space ($RKHS$). It introduces Autoreconstructive Kernel Embedding (KE), a two-step method: (i) learn reconstruction weights $\boldsymbol{\beta}$ in the high-dimensional RKHS via a closed-form solution to a quadratic reconstruction loss, and (ii) learn a latent embedding by kernel alignment so the latent kernel $\boldsymbol{G}_H$ mirrors the original reconstruction relationships, using a separable operator-valued kernel. The approach yields a Nyström-based out-of-sample extension and a detailed computational analysis, with empirical validation on concentric circles, Swiss roll, cancer biomolecule fingerprints, and IoT intrusion data, where KE often outperforms KPCA and UMAP on synthetic tasks and demonstrates competitive discriminative performance on real-world tasks. Overall, KE provides a principled, geometry-preserving dimension-reduction framework that can be adapted to vector-valued data and domain-specific kernels, with potential applications in visualization, clustering, and anomaly detection.

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.

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

TL;DR

The paper tackles learning low-dimensional embeddings that preserve the autoreconstructive geometry of data in a Reproducing-Kernel Hilbert Space (). It introduces Autoreconstructive Kernel Embedding (KE), a two-step method: (i) learn reconstruction weights in the high-dimensional RKHS via a closed-form solution to a quadratic reconstruction loss, and (ii) learn a latent embedding by kernel alignment so the latent kernel mirrors the original reconstruction relationships, using a separable operator-valued kernel. The approach yields a Nyström-based out-of-sample extension and a detailed computational analysis, with empirical validation on concentric circles, Swiss roll, cancer biomolecule fingerprints, and IoT intrusion data, where KE often outperforms KPCA and UMAP on synthetic tasks and demonstrates competitive discriminative performance on real-world tasks. Overall, KE provides a principled, geometry-preserving dimension-reduction framework that can be adapted to vector-valued data and domain-specific kernels, with potential applications in visualization, clustering, and anomaly detection.

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.
Paper Structure (15 sections, 6 theorems, 35 equations, 3 figures, 3 tables)

This paper contains 15 sections, 6 theorems, 35 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background on Dimensionality Reduction and Kernel Methods
  4. Autoreconstructive Kernel Embedding
  5. Reconstruction Weight Learning
  6. Embedding via Kernel Alignment
  7. Learning Algorithm
  8. Out–of–Sample Embedding
  9. Computational Complexity Analysis
  10. Numerical Experiments
  11. Concentric Circles
  12. Swiss Roll
  13. Cancer Biomolecules
  14. IoT Network Intrusions
  15. Discussion and Conclussion

Key Result

Theorem 1

Let $\mathcal{H}_k$ be a scalar-valued RKHS with kernel $k: \mathcal{X} \times \mathcal{X} \to \mathbb{R}$. Given training data $(\boldsymbol{x}_i, y_i) \in \mathcal{X} \times \mathbb{R}$ for $i=1, \dots, n$, the solution to the regularized empirical risk minimization problem admits a representation of the form

Figures (3)

  • Figure 1: Embeddings of the concentric circles dataset. From left to right: original 2D data, Kernel PCA with 1D output, UMAP with 1D output, and the proposed KE with 1D output. Each point is colored by its cluster label. The horizontal axis for the 1D methods corresponds to the sample index.
  • Figure 2: Embeddings of the Swiss roll dataset. From left to right: original 3D data, Kernel PCA with 2D output, UMAP with 2D output, and the KE with 2D output. Points are colored based on their position along the manifold. The 3D plot shows the original data geometry; the remaining plots show the corresponding 2D projections.
  • Figure 3: Embeddings on the CIC-CIoT2023 dataset. Both visualizations display benign training samples (light blue) and test samples from four traffic categories: Benign, DoS, Mirai, and Web-based. Panel (a) embeddings produced by the UMAP algorithm, while panel (b) embeddings obtained using the proposed KE method.

Theorems & Definitions (20)

  • Definition 1: Manifold Hypothesis Fefferman2016
  • Definition 2: Kernel Function
  • Definition 3: Positive Semi-Definite Kernel
  • Definition 4: Feature Map
  • Definition 5: Operator-Valued Kernel Function
  • Definition 6: Operator-Valued Kernel Matrix
  • Definition 7: Scalar-Valued RKHS
  • Definition 8: Vector-Valued RKHS
  • Definition 9: RKHS Norms
  • Theorem 1: Representer Theorem - Scalar-Valued Case
  • ...and 10 more