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Creating Sorted Grid Layouts with Gradient-based Optimization

Kai Uwe Barthel, Florian Tim Barthel, Peter Eisert, Nico Hezel, Konstantin Schall

TL;DR

The paper tackles the challenge of visually sorting large sets of high-dimensional vectors onto a 2D grid so that spatial proximity mirrors similarity, a problem with combinatorial explosion beyond brute force. It introduces a gradient-based framework that learns a differentiable permutation matrix $P_{soft}$ to reorder inputs and maps them to grid positions, guided by a DPQ-inspired objective that balances permutation validity with neighborhood preservation via a neighborhood loss $L_{nbr}$ and regularizers $L_s$ and $L_p$ controlled by a scheduling parameter $\alpha(t)$. The authors explore four implementations to generate $P_{soft}$ (Gumbel-Sinkhorn, Low-Rank, SoftSort, and a Transformer-based variant) and demonstrate, on multiple image and color datasets, that GradSort achieves state-of-the-art $DPQ_{16}$ scores, albeit with higher computational cost. The work provides a scalable, differentiable approach to grid-based visual sorting for high-dimensional data and outlines concrete avenues for improving efficiency, such as faster convergence, distance-matrix-based objectives, and scalable approximations for very large datasets.

Abstract

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 $1.3 \cdot 10^{89}$ - 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.

Creating Sorted Grid Layouts with Gradient-based Optimization

TL;DR

The paper tackles the challenge of visually sorting large sets of high-dimensional vectors onto a 2D grid so that spatial proximity mirrors similarity, a problem with combinatorial explosion beyond brute force. It introduces a gradient-based framework that learns a differentiable permutation matrix to reorder inputs and maps them to grid positions, guided by a DPQ-inspired objective that balances permutation validity with neighborhood preservation via a neighborhood loss and regularizers and controlled by a scheduling parameter . The authors explore four implementations to generate (Gumbel-Sinkhorn, Low-Rank, SoftSort, and a Transformer-based variant) and demonstrate, on multiple image and color datasets, that GradSort achieves state-of-the-art scores, albeit with higher computational cost. The work provides a scalable, differentiable approach to grid-based visual sorting for high-dimensional data and outlines concrete avenues for improving efficiency, such as faster convergence, distance-matrix-based objectives, and scalable approximations for very large datasets.

Abstract

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 - 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.

Paper Structure

This paper contains 9 sections, 14 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Distance Preserving Grid Layout Algorithms
  4. Learning Latent Permutations
  5. Creating Sorted Grid Layouts with Gradient-based Optimization
  6. Learning Objectives
  7. Evaluated Learnable Permutations
  8. Experiments
  9. Conclusion

Figures (4)

  • Figure 1: The principle of permutation learning involves training a network with weights W to learn the differentiable permutation matrix Psoft based on a specified loss function. The input vectors X are rearranged into Xsort using the permutation matrix Phard, which is a binarized version of Psoft.
  • Figure 2: An example with 256 RGB colors. On the left, the colors are randomly arranged. Sorted examples using different methods to generate the permutation matrix: A) Full rank + Gumbel-Sinkhorn, B) Low-rank + Softmax, C) SoftSort with n weights (achieving only 1D sorting), D) Full rank, Transformer + Gumbel-Sinkhorn.
  • Figure 3: Comparison of the runtime and the achieved sorting quality of different sorting algorithms with the kitchenware image set. GradSort T uses the additional transformer.
  • Figure 4: A comparison of different sorting methods for the color dataset and three image sets. The top row shows the IsoMatch sorting, followed by the Self Sorting Map (SSM). The third row shows the result of a t-SNE projection, mapped onto the grid positions using a linear assignment solver. The fourth row displays the results of Linear Assignment Sorting (LAS). The final row exhibits sortings from our new approach, GradSort. Achieved DPQ16 values are indicated below the respective arrangements.