Improving image classification of one-dimensional convolutional neural networks using Hilbert space-filling curves

Abstract

Convolutional neural networks (CNNs) have significantly contributed to recent advances in machine learning and computer vision. Although initially designed for image classification, the application of CNNs has stretched far beyond the context of images alone. Some exciting applications, e.g., in natural language processing and image segmentation, implement one-dimensional CNNs, often after a pre-processing step that transforms higher-dimensional input into a suitable data format for the networks. However, local correlations within data can diminish or vanish when one converts higher-dimensional data into a one-dimensional string. The Hilbert space-filling curve can minimize this loss of locality. Here, we study this claim rigorously by comparing an analytical model that quantifies locality preservation with the performance of several neural networks trained with and without Hilbert mappings. We find that Hilbert mappings offer a consistent advantage over the traditional flatten transformation in test accuracy and training speed. The results also depend on the chosen kernel size, agreeing with our analytical model. Our findings quantify the importance of locality preservation when transforming data before training a one-dimensional CNN and show that the Hilbert space-filling curve is a preferential transformation to achieve this goal.

Appendix A: Empirical results for different resolutions

Below we report the results of the empirical study as presented in Section 3 for different resolutions of the input images but with a similar methodology. One change to the example shown in the paper is the choice of radii over which we expand our search. In the general report, we use each distance at which a point can be found for any selection of anchor point. The below results only use the distances along the diagonal of the original two-dimensional input image, but the result remains the same.

Fig. 16

We show the progression of the density plots as introduced in Fig. 5 for increasing resolutions of the input image. The resolutions increase from left to right and top to bottom, spanning ($8\times 8$,$16\times 16$,$32\times 32$,$64\times 64$ pixels). The density plot shows the logarithm of the distance ratio for each combination of an anchor point labeled according to the flatten operation and a radius ($r_{0}$). We find that the Hilbert mapping preserves the relative distance better for most anchor points than the flattened string

Fig. 17

Results show the data distribution for the different resolutions of the input data. Figures on the left show the distribution of the ratio’s $\frac{d_{H}}{d_{F}}(r_{0},\left( i,j\right) )$ for each choice of radius $r_{0}$. The distribution of the logarithm of these ratios is shown in the right figures. Most of these ratios favor the Hilbert SFC mapping