# Investigating Convolutional Neural Networks using Spatial Orderness

## TL;DR

This paper introduces a new metric called spatial orderness to quantify how well 2D data obeys spatial relationships, revealing insights into CNN training dynamics and kernel properties.

## Contribution

The paper proposes the spatial orderness metric, analyzes its behavior during CNN training, and explores how kernel size affects the spatial orderness of network weights.

## Key findings

- Adding convolutional layers can be counterproductive without spatial order in data.
- Spatial orderness of feature maps increases early in training and decreases before validation improves.
- Smaller kernels produce kernels with higher spatial orderness.

## Abstract

Convolutional Neural Networks (CNN) have been pivotal to the success of many state-of-the-art classification problems, in a wide variety of domains (for e.g. vision, speech, graphs and medical imaging). A commonality within those domains is the presence of hierarchical, spatially agglomerative local-to-global interactions within the data. For two-dimensional images, such interactions may induce an a priori relationship between the pixel data and the underlying spatial ordering of the pixels. For instance in natural images, neighboring pixels are more likely contain similar values than non-neighboring pixels which are further apart. To that end, we propose a statistical metric called spatial orderness, which quantifies the extent to which the input data (2D) obeys the underlying spatial ordering at various scales. In our experiments, we mainly find that adding convolutional layers to a CNN could be counterproductive for data bereft of spatial order at higher scales. We also observe, quite counter-intuitively, that the spatial orderness of CNN feature maps show a synchronized increase during the intial stages of training, and validation performance only improves after spatial orderness of feature maps start decreasing. Lastly, we present a theoretical analysis (and empirical validation) of the spatial orderness of network weights, where we find that using smaller kernel sizes leads to kernels of greater spatial orderness and vice-versa.