# A Unifying Framework in Vector-valued Reproducing Kernel Hilbert Spaces for Manifold Regularization and Co-Regularized Multi-view Learning

arXiv: https://arxiv.org/abs/1401.8066 · HTML: https://mlsanity.com/paper/1401.8066/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jan 2014.

## Built on by (most cited first, 12 of 12)

- [Reproducing kernel Hilbert C*-module and kernel mean embeddings](https://mlsanity.com/paper/2101.11410/) (2021)
- [Entangled Kernels -- Beyond Separability](https://mlsanity.com/paper/2101.05514/) (2021)
- [Learning in RKHM: a C^*-Algebraic Twist for Kernel Machines](https://mlsanity.com/paper/2210.11855/) (2022)
- [Exponentially improved efficient machine learning for quantum many-body states with provable guarantees](https://mlsanity.com/paper/2304.04353/) (2023)
- [Geometric Multimodal Deep Learning with Multi-Scaled Graph Wavelet Convolutional Network](https://mlsanity.com/paper/2111.13361/) (2021)
- [Cross-Modal and Multimodal Data Analysis Based on Functional Mapping of Spectral Descriptors and Manifold Regularization](https://mlsanity.com/paper/2105.05631/) (2021)
- [Learning System Parameters from Turing Patterns](https://mlsanity.com/paper/2108.08542/) (2021)
- [Scalable variable selection for two-view learning tasks with projection operators](https://mlsanity.com/paper/2307.01558/) (2023)
- [Spectral Truncation Kernels: Noncommutativity in C^*-algebraic Kernel Machines](https://mlsanity.com/paper/2405.17823/) (2024)
- [ROCK: A variational formulation for occupation kernel methods in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2503.13791/) (2025)
- [Localisation of Regularised and Multiview Support Vector Machine Learning](https://mlsanity.com/paper/2304.05655/) (2023)
- [Hierarchical Time Series Forecasting with Operator-Valued Kernels](https://mlsanity.com/paper/2610.18196/) (2026)

## Abstract (arXiv)

This paper presents a general vector-valued reproducing kernel Hilbert spaces (RKHS) framework for the problem of learning an unknown functional dependency between a structured input space and a structured output space. Our formulation encompasses both Vector-valued Manifold Regularization and Co-regularized Multi-view Learning, providing in particular a unifying framework linking these two important learning approaches. In the case of the least square loss function, we provide a closed form solution, which is obtained by solving a system of linear equations. In the case of Support Vector Machine (SVM) classification, our formulation generalizes in particular both the binary Laplacian SVM to the multi-class, multi-view settings and the multi-class Simplex Cone SVM to the semi-supervised, multi-view settings. The solution is obtained by solving a single quadratic optimization problem, as in standard SVM, via the Sequential Minimal Optimization (SMO) approach. Empirical results obtained on the task of object recognition, using several challenging datasets, demonstrate the competitiveness of our algorithms compared with other state-of-the-art methods.