Understanding Geometry of Encoder-Decoder CNNs

TL;DR

This paper develops a unified theoretical framework to understand the geometric properties of encoder-decoder CNNs, explaining their expressibility and the role of skip connections in deep learning applications.

Contribution

It introduces a mathematical framework linking encoder-decoder CNNs to nonlinear basis representations via convolutional framelets, highlighting the exponential growth of expressibility with depth.

Findings

Encoder-decoder CNNs relate to nonlinear basis representations.

Skip connections enhance expressibility and optimization.

Expressibility increases exponentially with network depth.

Abstract

Encoder-decoder networks using convolutional neural network (CNN) architecture have been extensively used in deep learning literatures thanks to its excellent performance for various inverse problems. However, it is still difficult to obtain coherent geometric view why such an architecture gives the desired performance. Inspired by recent theoretical understanding on generalizability, expressivity and optimization landscape of neural networks, as well as the theory of convolutional framelets, here we provide a unified theoretical framework that leads to a better understanding of geometry of encoder-decoder CNNs. Our unified mathematical framework shows that encoder-decoder CNN architecture is closely related to nonlinear basis representation using combinatorial convolution frames, whose expressibility increases exponentially with the network depth. We also demonstrate the importance of skipped connection in terms of expressibility, and optimization landscape.