🤖 AI Summary
This work addresses the problem of learning low-dimensional embeddings from high-dimensional data while preserving the intrinsic geometric structure inherent in self-reconstruction. To this end, the authors propose a novel approach grounded in reproducing kernel Hilbert spaces (RKHS), which explicitly models the self-expressive property of data by integrating the representer theorem with separable operator-valued kernels. Geometric structure preservation is achieved through kernel alignment, enabling effective transfer of intrinsic data geometry into the embedding space. The proposed method unifies manifold learning and kernel methods within a coherent framework. Empirical evaluations on both synthetic benchmarks—such as concentric circles and the Swiss roll—and real-world datasets—including molecular activity prediction in cancer research and intrusion detection in IoT systems—demonstrate its superior performance in maintaining reconstruction-based geometric fidelity.
📝 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.