# Entangled Kernels -- Beyond Separability

arXiv: https://arxiv.org/abs/2101.05514 · HTML: https://mlsanity.com/paper/2101.05514/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 58 references that are in the map.
- First posted on arXiv 14 Jan 2021.

## Builds on (most cited first, 13 of 13)

- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [A vector-contraction inequality for Rademacher complexities](https://mlsanity.com/paper/1605.00251/) (2016)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [Beyond Shared Hierarchies: Deep Multitask Learning through Soft Layer Ordering](https://mlsanity.com/paper/1711.00108/) (2017)
- [Random Fourier Features for Operator-Valued Kernels](https://mlsanity.com/paper/1605.02536/) (2016)
- [Operator-Valued Bochner Theorem, Fourier Feature Maps for Operator-Valued Kernels, and Vector-Valued Learning](https://mlsanity.com/paper/1608.05639/) (2016)
- [Scalable Matrix-valued Kernel Learning for High-dimensional Nonlinear Multivariate Regression and Granger Causality](https://mlsanity.com/paper/1408.2066/) (2014)
- [Convex Learning of Multiple Tasks and their Structure](https://mlsanity.com/paper/1504.03101/) (2015)
- [A Unifying Framework in Vector-valued Reproducing Kernel Hilbert Spaces for Manifold Regularization and Co-Regularized Multi-view Learning](https://mlsanity.com/paper/1401.8066/) (2014)
- [Efficient Output Kernel Learning for Multiple Tasks](https://mlsanity.com/paper/1511.05706/) (2015)
- [Multi-view Metric Learning in Vector-valued Kernel Spaces](https://mlsanity.com/paper/1803.07821/) (2018)
- [Forecasting and Granger Modelling with Non-linear Dynamical Dependencies](https://mlsanity.com/paper/1706.08811/) (2017)

## Built on by (most cited first, 7 of 7)

- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Learning in RKHM: a C^*-Algebraic Twist for Kernel Machines](https://mlsanity.com/paper/2210.11855/) (2022)
- [Deep Learning with Kernels through RKHM and the Perron-Frobenius Operator](https://mlsanity.com/paper/2305.13588/) (2023)
- [Spectral Truncation Kernels: Noncommutativity in C^*-algebraic Kernel Machines](https://mlsanity.com/paper/2405.17823/) (2024)
- [Position: Quantum Kernel Machines Should Move Beyond Scalar-Valued Kernels to Realize Their Potential](https://mlsanity.com/paper/2506.03779/) (2025)
- [Operator-Based Generalization Bound for Deep Learning: Insights on Multi-Task Learning](https://mlsanity.com/paper/2512.19184/) (2025)
- [Free Random Projection for In-Context Reinforcement Learning](https://mlsanity.com/paper/2504.06983/) (2025)

## Abstract (arXiv)

We consider the problem of operator-valued kernel learning and investigate the possibility of going beyond the well-known separable kernels. Borrowing tools and concepts from the field of quantum computing, such as partial trace and entanglement, we propose a new view on operator-valued kernels and define a general family of kernels that encompasses previously known operator-valued kernels, including separable and transformable kernels. Within this framework, we introduce another novel class of operator-valued kernels called entangled kernels that are not separable. We propose an efficient two-step algorithm for this framework, where the entangled kernel is learned based on a novel extension of kernel alignment to operator-valued kernels. We illustrate our algorithm with an application to supervised dimensionality reduction, and demonstrate its effectiveness with both artificial and real data for multi-output regression.