# Towards Understanding the Invertibility of Convolutional Neural Networks

## TL;DR

This paper investigates the approximate invertibility of CNNs by developing a mathematical model based on sparse signal recovery, connecting it to compressive sensing, and validating it with empirical results on real images.

## Contribution

It introduces a theoretical framework linking CNN invertibility to sparse signal recovery and compressive sensing, providing insights into CNN reconstruction capabilities.

## Key findings

- Empirical networks align with the mathematical model
- Reasonable image reconstructions are achievable using the model
- Discussion of model limitations in practical CNNs

## Abstract

Several recent works have empirically observed that Convolutional Neural Nets (CNNs) are (approximately) invertible. To understand this approximate invertibility phenomenon and how to leverage it more effectively, we focus on a theoretical explanation and develop a mathematical model of sparse signal recovery that is consistent with CNNs with random weights. We give an exact connection to a particular model of model-based compressive sensing (and its recovery algorithms) and random-weight CNNs. We show empirically that several learned networks are consistent with our mathematical analysis and then demonstrate that with such a simple theoretical framework, we can obtain reasonable re- construction results on real images. We also discuss gaps between our model assumptions and the CNN trained for classification in practical scenarios.