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

TL;DR

This paper introduces new algorithms for reconstructive manifold learning in RKHS, leveraging the Representer Theorem to embed high-dimensional data into lower-dimensional spaces effectively.

Contribution

It extends autorepresentation in RKHS to vector-valued data using operator-valued kernels and aligns kernels for improved low-dimensional embeddings.

Findings

Algorithms effectively reconstruct high-dimensional data in low-dimensional space.

Numerical experiments demonstrate practical effectiveness on diverse datasets.

The approach generalizes autorepresentation properties to vector-valued data.

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.