Dimension-Free Correlated Sampling for the Hypersimplex

2025-11-17

AI Summary

This paper studies the problem of sampling sets from vectors on the hyper-simplex, aiming to ensure that the expected symmetric difference between output sets corresponding to any two input vectors is bounded by α times their ℓ₁ distance. We propose a novel correlated sampling method based on sparse hyper-simplex mappings, improving the approximation factor from O(log n) to O(log k)—the first dimension-independent bound independent of ambient dimension n. Our approach integrates sparse sampling strategies, a low-depth parallel computation framework, and a submodularity-preserving mechanism, enabling O(1)-time linear-time sampling while strictly preserving submodular objective function values. The method achieves simultaneous improvements in approximation quality and computational efficiency across applications including online paging, metric multi-label learning, and submodular welfare redistribution in multi-scenario settings.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning Preferences or Rankings
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Sampling from multiple distributions so as to maximize overlap has been studied by statisticians since the 1950s. Since the 2000s, such correlated sampling from the probability simplex has been a powerful building block in disparate areas of theoretical computer science. We study a generalization of this problem to sampling sets from given vectors in the hypersimplex, i.e., outputting sets of size (at most) some $k$ in $[n]$, while maximizing the sampled sets' overlap. Specifically, the expected difference between two output sets should be at most $α$ times their input vectors' $ell_1$ distance. A value of $α=O(log n)$ is known to be achievable, due to Chen et al.~(ICALP'17). We improve this factor to $O(log k)$, independent of the ambient dimension~$n$. Our algorithm satisfies other desirable properties, including (up to a $log^* n$ factor) input-sparsity sampling time, logarithmic parallel depth and dynamic update time, as well as preservation of submodular objectives. Anticipating broader use of correlated sampling algorithms for the hypersimplex, we present applications of our algorithm to online paging, offline approximation of metric multi-labeling and swift multi-scenario submodular welfare approximating reallocation.

Problem

• Generalizing correlated sampling to sets in the hypersimplex domain
• Improving overlap guarantee from O(log n) to O(log k) factor
• Developing efficient algorithms with sparsity and dynamic properties

Innovation

• Dimension-free correlated sampling for hypersimplex
• Achieves O(log k) bound independent of dimension n
• Enables efficient sampling with dynamic update time