Shift-Equivariant Complex-Valued Convolutional Neural Networks

TL;DR

This paper introduces a novel complex-valued neural network architecture that guarantees shift equivariance and invariance through specialized downsampling and upsampling layers, improving performance on vision tasks with polarimetric SAR images.

Contribution

It extends learnable polyphase sampling to complex-valued neural networks and proposes a new projection layer from complex to real for enhanced shift-equivariant properties.

Findings

Improved shift invariance in classification tasks.

Enhanced shift equivariance in reconstruction and segmentation.

Effective application to polarimetric SAR image analysis.

Abstract

Convolutional neural networks have shown remarkable performance in recent years on various computer vision problems. However, the traditional convolutional neural network architecture lacks a critical property: shift equivariance and invariance, broken by downsampling and upsampling operations. Although data augmentation techniques can help the model learn the latter property empirically, a consistent and systematic way to achieve this goal is by designing downsampling and upsampling layers that theoretically guarantee these properties by construction. Adaptive Polyphase Sampling (APS) introduced the cornerstone for shift invariance, later extended to shift equivariance with Learnable Polyphase up/downsampling (LPS) applied to real-valued neural networks. In this paper, we extend the work on LPS to complex-valued neural networks both from a theoretical perspective and with a novel building block of a projection layer from $\mathbb{C}$ to $\mathbb{R}$ before the Gumbel Softmax. We finally evaluate this extension on several computer vision problems, specifically for either the invariance property in classification tasks or the equivariance property in both reconstruction and semantic segmentation problems, using polarimetric Synthetic Aperture Radar images.