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

arXiv: https://arxiv.org/abs/2503.13051 · HTML: https://mlsanity.com/paper/2503.13051/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 20 references that are in the map.
- First posted on arXiv 17 Mar 2025.

## Builds on (most cited first, 10 of 10)

- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Compact 3D Scene Representation via Self-Organizing Gaussian Grids](https://mlsanity.com/paper/2312.13299/) (2023)
- [3DGS.zip: A survey on 3D Gaussian Splatting Compression Methods](https://mlsanity.com/paper/2407.09510/) (2024)
- [SoftSort: A Continuous Relaxation for the argsort Operator](https://mlsanity.com/paper/2006.16038/) (2020)
- [Monotonic Differentiable Sorting Networks](https://mlsanity.com/paper/2203.09630/) (2022)
- [Kissing to Find a Match: Efficient Low-Rank Permutation Representation](https://mlsanity.com/paper/2308.13252/) (2023)
- [Improved Evaluation and Generation of Grid Layouts using Distance Preservation Quality and Linear Assignment Sorting](https://mlsanity.com/paper/2205.04255/) (2022)
- [A Grid-based Method for Removing Overlaps of Dimensionality Reduction Scatterplot Layouts](https://mlsanity.com/paper/1903.06262/) (2019)
- [Creating Sorted Grid Layouts with Gradient-based Optimization](https://mlsanity.com/paper/2503.02730/) (2025)

## Abstract (arXiv)

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 2NM (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 method extends SoftSort by iteratively shuffling the N indices of the elements and applying a few SoftSort optimization steps per iteration. 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.