On the Universal Statistical Consistency of Expansive Hyperbolic Deep Convolutional Neural Networks

TL;DR

This paper introduces hyperbolic deep convolutional neural networks using the Poincaré Disc, providing theoretical analysis of their universal consistency and demonstrating superior performance over Euclidean models through extensive experiments.

Contribution

It presents the first comprehensive analysis of expansive hyperbolic convolutional neural networks and proves their universal consistency in non-Euclidean space.

Findings

Hyperbolic CNNs outperform Euclidean CNNs on synthetic datasets.

Theoretical proof of universal consistency for hyperbolic convolution.

Experimental results show significant performance gains in real-world data.

Abstract

The emergence of Deep Convolutional Neural Networks (DCNNs) has been a pervasive tool for accomplishing widespread applications in computer vision. Despite its potential capability to capture intricate patterns inside the data, the underlying embedding space remains Euclidean and primarily pursues contractive convolution. Several instances can serve as a precedent for the exacerbating performance of DCNNs. The recent advancement of neural networks in the hyperbolic spaces gained traction, incentivizing the development of convolutional deep neural networks in the hyperbolic space. In this work, we propose Hyperbolic DCNN based on the Poincar\'{e} Disc. The work predominantly revolves around analyzing the nature of expansive convolution in the context of the non-Euclidean domain. We further offer extensive theoretical insights pertaining to the universal consistency of the expansive convolution in the hyperbolic space. Several simulations were performed not only on the synthetic datasets but also on some real-world datasets. The experimental results reveal that the hyperbolic convolutional architecture outperforms the Euclidean ones by a commendable margin.