Permutation Learning with Only N Parameters: From SoftSort to Self-Organizing Gaussians

2025-03-17

AI Summary

This paper addresses the computational inefficiency of differentiable permutation learning for large-scale, high-dimensional data. We propose a lightweight permutation modeling approach requiring only $N$ parameters—significantly fewer than Gumbel-Sinkhorn ($O(N^2)$) or low-rank approximations ($O(MN)$). Built upon the SoftSort continuous relaxation framework, our method decouples index reordering from differentiable sorting via iterative design, enabling compact and fully differentiable parameterization of the full permutation matrix. On tasks such as Self-Organizing Gaussians, memory consumption drops to $sim 1/N$ of conventional methods, while sorting accuracy substantially surpasses that of the original SoftSort. The approach exhibits strong scalability and practical deployability, bridging the gap between theoretical expressiveness and real-world efficiency in differentiable sorting and permutation learning.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Sorting and permutation learning are key concepts in optimization and machine learning, especially when organizing high-dimensional data into meaningful spatial layouts. The Gumbel-Sinkhorn method, while effective, requires N*N parameters to determine a full permutation matrix, making it computationally expensive for large datasets. Low-rank matrix factorization approximations reduce memory requirements to 2MN (with M<<N), but they still struggle with very large problems. SoftSort, by providing a continuous relaxation of the argsort operator, allows differentiable 1D sorting, but it faces challenges with multidimensional data and complex permutations. In this paper, we present a novel method for learning permutations using only N parameters, which dramatically reduces storage costs. Our approach builds on SoftSort, but extends it by iteratively shuffling the N indices of the elements to be sorted through a separable learning process. This modification significantly improves sorting quality, especially for multidimensional data and complex optimization criteria, and outperforms pure SoftSort. Our method offers improved memory efficiency and scalability compared to existing approaches, while maintaining high-quality permutation learning. Its dramatically reduced memory requirements make it particularly well-suited for large-scale optimization tasks, such as"Self-Organizing Gaussians", where efficient and scalable permutation learning is critical.

Problem

• Reduces permutation learning parameters from N*N to N
• Improves sorting quality for multidimensional data
• Enhances memory efficiency for large-scale optimization tasks

Innovation

• Uses only N parameters for permutation learning
• Extends SoftSort with iterative index shuffling
• Improves memory efficiency and scalability significantly