# Creating Sorted Grid Layouts with Gradient-based Optimization

arXiv: https://arxiv.org/abs/2503.02730 · HTML: https://mlsanity.com/paper/2503.02730/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 33 references that are in the map.
- First posted on arXiv 4 Mar 2025.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Fine-tuning CNN Image Retrieval with No Human Annotation](https://mlsanity.com/paper/1711.02512/) (2017)
- [SoftSort: A Continuous Relaxation for the argsort Operator](https://mlsanity.com/paper/2006.16038/) (2020)
- [AutoShuffleNet: Learning Permutation Matrices via an Exact Lipschitz Continuous Penalty in Deep Convolutional Neural Networks](https://mlsanity.com/paper/1901.08624/) (2019)
- [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)
- [Tree-based Visualization and Optimization for Image Collection](https://mlsanity.com/paper/1507.04913/) (2015)

## Built on by (most cited first, 1 of 1)

- [Permutation Learning with Only N Parameters: From SoftSort to Self-Organizing Gaussians](https://mlsanity.com/paper/2503.13051/) (2025)

## Abstract (arXiv)

Visually sorted grid layouts provide an efficient method for organizing high-dimensional vectors in two-dimensional space by aligning spatial proximity with similarity relationships. This approach facilitates the effective sorting of diverse elements ranging from data points to images, and enables the simultaneous visualization of a significant number of elements. However, sorting data on two-dimensional grids is a challenge due to its high complexity. Even for a small 8-by-8 grid with 64 elements, the number of possible arrangements exceeds 1.3 · 1089 - more than the number of atoms in the universe - making brute-force solutions impractical. Although various methods have been proposed to address the challenge of determining sorted grid layouts, none have investigated the potential of gradient-based optimization. In this paper, we present a novel method for grid-based sorting that exploits gradient optimization for the first time. We introduce a novel loss function that balances two opposing goals: ensuring the generation of a "valid" permutation matrix, and optimizing the arrangement on the grid to reflect the similarity between vectors, inspired by metrics that assess the quality of sorted grids. While learning-based approaches are inherently computationally complex, our method shows promising results in generating sorted grid layouts with superior sorting quality compared to existing techniques.