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Locally orderless networks

Jon Sporring, Peidi Xu, Jiahao Lu, François Lauze, Sune Darkner

TL;DR

This work introduces Locally Orderless Networks (LON), a neural-layer framework that transforms inputs into local histograms by applying scale-space–informed, bell-shaped activations to local convolutions. By linking LOI theory, measure theory, and scale-space with neural networks, LON provides a continuum with Convolutional Neural Networks (CNN) and enables computation of nonlinear functionals such as the gradient magnitude squared. The authors demonstrate theoretically and empirically that LON can emulate CNNs, while also excelling at boundary- and isophote-driven tasks (e.g., perimeter estimation, saliency explainability) and offering more interpretable saliency maps. The results suggest LON as a complementary approach that enhances nonlinear capabilities and explainability, with potential applications in segmentation and other tasks where boundary information is crucial.

Abstract

We present Locally Orderless Networks (LON) and its theoretic foundation which links it to Convolutional Neural Networks (CNN), to Scale-space histograms, and measurement theory. The key elements are a regular sampling of the bias and the derivative of the activation function. We compare LON, CNN, and Scale-space histograms on prototypical single-layer networks. We show how LON and CNN can emulate each other, how LON expands the set of functionals computable to non-linear functions such as squaring. We demonstrate simple networks which illustrate the improved performance of LON over CNN on simple tasks for estimating the gradient magnitude squared, for regressing shape area and perimeter lengths, and for explainability of individual pixels' influence on the result.

Locally orderless networks

TL;DR

This work introduces Locally Orderless Networks (LON), a neural-layer framework that transforms inputs into local histograms by applying scale-space–informed, bell-shaped activations to local convolutions. By linking LOI theory, measure theory, and scale-space with neural networks, LON provides a continuum with Convolutional Neural Networks (CNN) and enables computation of nonlinear functionals such as the gradient magnitude squared. The authors demonstrate theoretically and empirically that LON can emulate CNNs, while also excelling at boundary- and isophote-driven tasks (e.g., perimeter estimation, saliency explainability) and offering more interpretable saliency maps. The results suggest LON as a complementary approach that enhances nonlinear capabilities and explainability, with potential applications in segmentation and other tasks where boundary information is crucial.

Abstract

We present Locally Orderless Networks (LON) and its theoretic foundation which links it to Convolutional Neural Networks (CNN), to Scale-space histograms, and measurement theory. The key elements are a regular sampling of the bias and the derivative of the activation function. We compare LON, CNN, and Scale-space histograms on prototypical single-layer networks. We show how LON and CNN can emulate each other, how LON expands the set of functionals computable to non-linear functions such as squaring. We demonstrate simple networks which illustrate the improved performance of LON over CNN on simple tasks for estimating the gradient magnitude squared, for regressing shape area and perimeter lengths, and for explainability of individual pixels' influence on the result.
Paper Structure (10 sections, 10 equations, 6 figures)

This paper contains 10 sections, 10 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Locally orderless histograms as convolutional networks
  4. Boundaries versus areas
  5. LON can implement nonlinear measures directly
  6. Experiments
  7. Estimating the gradient magnitude squared
  8. Area versus perimeter regression and classification
  9. Explainability by salience maps
  10. Discussion and Conclusion

Figures (6)

  • Figure 1: Comparing LON with the gradient magnitude. The original image (\ref{['fig:original']}) and its local histogram of the vertical and horizontal derivative (\ref{['fig:hIr']}) and (\ref{['fig:hIc']}) at the red mark and with a smoothing kernel of width indicated by the red circle. (\ref{['fig:G']}) and (\ref{['fig:Gh']}), the gradient magnitude squared $\left|I(\mathbf{x})\right|^2$ and the locally orderless network \ref{['eq:grad3']} with $A=\text{id}$.
  • Figure 2: CNN achieves similar behavior as LON, but while LON succeeds in all directions, CNN does not.
  • Figure 3: Process for random shape generation: An image of iid. normal noise smoothed with a Gaussian kernel \ref{['fig:smoothedNoise']}, its threshold and similar components \ref{['fig:objectClipping']}, and examplar objects with added iid. noise \ref{['fig:noisyClippings']}.
  • Figure 4: The mean square error by the logarithm of the number of parameters for CNN with 2 kernels and the ReLU, LON with 2 kernels and 2 bins and 8 bins on the regression task on the perimeter and area of random shapes without iid. noise (\ref{['fig:shape_generation']}). Lower is better.
  • Figure 5: The accuracy by the logarithm of the number of parameters for CNN with 2 kernels and the ReLU, LON with 2 kernels and 2 bins and 8 bins on the the regression task on the perimeter and area of random shapes without noise but trained on many ($\approx$4000) or few ($\approx$1500) examples (\ref{['fig:shape_generation']}). Higher is better.
  • ...and 1 more figures