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From Sorting Algorithms to Scalable Kernels: Bayesian Optimization in High-Dimensional Permutation Spaces

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Bayesian OptimizationCombinatorial OptimizationPermutation Space
Abstract:

Bayesian Optimization (BO) is a powerful tool for black-box optimization, but its application to high-dimensional permutation spaces is severely limited by the challenge of defining scalable representations. The current state-of-the-art BO approach for permutation spaces relies on an exhaustive Ω(n2)\Omega(n^2) pairwise comparison, inducing a dense representation that is impractical for large-scale permutations. To break this barrier, we introduce a novel framework for generating efficient permutation representations via kernel functions derived from sorting algorithms. Within this framework, the Mallows kernel can be viewed as a special instance derived from enumeration sort. Further, we introduce the \textbf{Merge Kernel} , which leverages the divide-and-conquer structure of merge sort to produce a compact, Θ(nlog⁡n)\Theta(n\log n) to achieve the lowest possible complexity with no information loss and effectively capture permutation structure. Our central thesis is that the Merge Kernel performs competitively with the Mallows kernel in low-dimensional settings, but significantly outperforms it in both optimization performance and computational efficiency as the dimension nn grows. Extensive evaluations on various permutation optimization benchmarks confirm our hypothesis, demonstrating that the Merge Kernel provides a scalable and more effective solution for Bayesian optimization in high-dimensional permutation spaces, thereby unlocking the potential for tackling previously intractable problems such as large-scale feature ordering and combinatorial neural architecture search.

Disclaimer
This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper proposes a sorting-algorithm-driven framework for designing permutation kernels, introducing the Merge Kernel with O(n log n) complexity as an alternative to the quadratic Mallows kernel. It resides in the 'Permutation-Specific Kernel Functions and Representations' leaf, which contains only three papers total, including this work and two siblings. This indicates a relatively sparse research direction within the broader taxonomy of 42 papers across the field, suggesting that permutation-specific kernel design remains an underexplored area compared to general combinatorial embeddings or application domains.

The taxonomy reveals that neighboring leaves address related but distinct challenges: 'Embeddings and Kernels for General Combinatorial and Categorical Spaces' (four papers) develops dictionary-based and graph-based methods not tailored to permutations, while 'Set-Input Kernels and Permutation-Invariant Models' (two papers) handles variable-length inputs rather than fixed-length permutations. The scope notes clarify that this work's focus on sorting-based permutation kernels distinguishes it from these sibling categories. The broader parent branch on 'Kernel Design and Surrogate Modeling' contains seven papers total, indicating that kernel innovation for combinatorial spaces is an active but not overcrowded research frontier.

Among 16 candidates examined across three contributions, only one refutable pair emerged, specifically for the evaluation contribution (9 candidates examined, 1 refutable). The sorting-algorithm framework (6 candidates, 0 refutable) and the Merge Kernel itself (1 candidate, 0 refutable) show no clear prior overlap within this limited search scope. The evaluation contribution's single refutable candidate suggests some overlap in benchmark selection or experimental methodology, but the core algorithmic contributions appear more distinctive. The small candidate pool (16 total) reflects the specialized nature of permutation kernel design rather than exhaustive coverage.

Based on top-16 semantic matches and citation expansion, the framework and Merge Kernel appear novel within the examined scope, though the limited search scale leaves open the possibility of relevant work outside this candidate set. The sparse taxonomy leaf (three papers) and low refutation rate (1 of 16 candidates) suggest the work addresses a genuine gap, but definitive claims require broader literature coverage beyond the current analysis.

Taxonomy

Core-task Taxonomy Papers
42
3
Claimed Contributions
16
Contribution Candidate Papers Compared
1
Refutable Paper

Research Landscape Overview

Core task: Bayesian optimization in high-dimensional permutation spaces. The field addresses the challenge of optimizing expensive black-box functions over discrete combinatorial structures, particularly permutations, where standard continuous optimization techniques fail. The taxonomy reveals several complementary research directions: kernel design and surrogate modeling efforts focus on constructing expressive representations for permutations and combinatorial objects (e.g., Permutation Spaces[3], Combinatorial Structures[5]); acquisition function optimization and batch methods tackle the discrete search problem inherent in selecting next evaluation points; dimensionality reduction approaches (High Dimensional Subspaces[4]) aim to make optimization tractable in large spaces; estimation of distribution algorithms provide probabilistic model-based alternatives; application domains demonstrate practical deployment; and methodological surveys contextualize non-conventional search spaces (Non Conventional Spaces[23]). These branches collectively address the dual challenge of representing structured discrete spaces in ways amenable to Gaussian process modeling while efficiently searching over exponentially large candidate sets. A particularly active line of work centers on designing kernels that respect permutation structure and scale to high dimensions. Scalable Kernels Permutations[0] sits squarely within this effort, developing computationally efficient kernel functions for permutation spaces that maintain expressiveness while avoiding prohibitive computational costs. This contrasts with approaches like Relational Permutation[11], which emphasizes relational structure in permutation representations, and Dictionary Embeddings[2], which leverages embedding techniques to handle combinatorial objects. The tension between representational richness and computational tractability remains central: while expressive kernels like those in Combinatorial Assignment[6] can capture complex structure, they often struggle with scalability, whereas simpler embeddings sacrifice some modeling fidelity for efficiency. Scalable Kernels Permutations[0] navigates this trade-off by focusing explicitly on scalability without abandoning the permutation-specific structure that makes surrogate modeling effective in these discrete domains.

Claimed Contributions

Sorting-algorithm-driven kernel-design framework for permutations

The authors introduce a general framework that generates permutation kernels by deriving feature vectors from the comparison sequences of sorting algorithms. Within this framework, the Mallows kernel is shown to be a special case corresponding to enumeration sort.

6 retrieved papers
Merge Kernel with O(n log n) complexity

The authors instantiate their framework using merge sort to create the Merge Kernel, which achieves a compact feature dimension of O(n log n) that matches the theoretical lower bound for lossless permutation encoding, significantly reducing the quadratic complexity of the Mallows kernel.

1 retrieved paper
Comprehensive evaluation on synthetic and real-world benchmarks

The authors conduct extensive experiments demonstrating that the Merge kernel performs competitively with the Mallows kernel in low-dimensional settings but significantly outperforms it in both optimization performance and computational efficiency as dimensionality increases, validating the scalability of their approach.

9 retrieved papers
Can Refute

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Sorting-algorithm-driven kernel-design framework for permutations

The authors introduce a general framework that generates permutation kernels by deriving feature vectors from the comparison sequences of sorting algorithms. Within this framework, the Mallows kernel is shown to be a special case corresponding to enumeration sort.

Contribution

Merge Kernel with O(n log n) complexity

The authors instantiate their framework using merge sort to create the Merge Kernel, which achieves a compact feature dimension of O(n log n) that matches the theoretical lower bound for lossless permutation encoding, significantly reducing the quadratic complexity of the Mallows kernel.

Contribution

Comprehensive evaluation on synthetic and real-world benchmarks

The authors conduct extensive experiments demonstrating that the Merge kernel performs competitively with the Mallows kernel in low-dimensional settings but significantly outperforms it in both optimization performance and computational efficiency as dimensionality increases, validating the scalability of their approach.

Table of Contents

From Sorting Algorithms to Scalable Kernels: Bayesian Optimization in High-Dimensional Permutation Spaces | Novelty Validation