# Kernels over Sets of Finite Sets using RKHS Embeddings, with Application to Bayesian (Combinatorial) Optimization

arXiv: https://arxiv.org/abs/1910.04086 · HTML: https://mlsanity.com/paper/1910.04086/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2019.

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

- [Recent Advances in Bayesian Optimization](https://mlsanity.com/paper/2206.03301/) (2022)
- [Combining Latent Space and Structured Kernels for Bayesian Optimization over Combinatorial Spaces](https://mlsanity.com/paper/2111.01186/) (2021)
- [Bayesian Optimization over Permutation Spaces](https://mlsanity.com/paper/2112.01049/) (2021)
- [AntBO: Towards Real-World Automated Antibody Design with Combinatorial Bayesian Optimisation](https://mlsanity.com/paper/2201.12570/) (2022)
- [Tree ensemble kernels for Bayesian optimization with known constraints over mixed-feature spaces](https://mlsanity.com/paper/2207.00879/) (2022)
- [Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures](https://mlsanity.com/paper/2206.07588/) (2022)
- [Lossy compression of matrices by black-box optimisation of mixed integer nonlinear programming](https://mlsanity.com/paper/2204.10579/) (2022)
- [Reproducing kernel Hilbert spaces in the mean field limit](https://mlsanity.com/paper/2302.14446/) (2023)
- [Advances in Set Function Learning: A Survey of Techniques and Applications](https://mlsanity.com/paper/2501.14991/) (2025)
- [BARK: A Fully Bayesian Tree Kernel for Black-box Optimization](https://mlsanity.com/paper/2503.05574/) (2025)
- [A Bayesian Approach to Low-Discrepancy Subset Selection](https://mlsanity.com/paper/2602.14607/) (2026)
- [Inducing Permutation Invariant Priors in Bayesian Optimization for Carbon Capture and Storage Applications](https://mlsanity.com/paper/2605.02409/) (2026)

## Abstract (arXiv)

We focus on kernel methods for set-valued inputs and their application to Bayesian set optimization, notably combinatorial optimization. We investigate two classes of set kernels that both rely on Reproducing Kernel Hilbert Space embeddings, namely the ``Double Sum'' (DS) kernels recently considered in Bayesian set optimization, and a class introduced here called ``Deep Embedding'' (DE) kernels that essentially consists in applying a radial kernel on Hilbert space on top of the canonical distance induced by another kernel such as a DS kernel. We establish in particular that while DS kernels typically suffer from a lack of strict positive definiteness, vast subclasses of DE kernels built upon DS kernels do possess this property, enabling in turn combinatorial optimization without requiring to introduce a jitter parameter. Proofs of theoretical results about considered kernels are complemented by a few practicalities regarding hyperparameter fitting. We furthermore demonstrate the applicability of our approach in prediction and optimization tasks, relying both on toy examples and on two test cases from mechanical engineering and hydrogeology, respectively. Experimental results highlight the applicability and compared merits of the considered approaches while opening new perspectives in prediction and sequential design with set inputs.