Rate-Optimal Denoising with Deep Neural Networks

TL;DR

This paper provides theoretical guarantees for deep neural network-based image denoising, showing that both autoencoder and generative model approaches can effectively reduce noise energy proportionally to the ratio of latent to image dimensions.

Contribution

It offers the first theoretical analysis demonstrating that deep neural network denoising methods reduce noise energy by a factor of O(k/n), validating their effectiveness.

Findings

Autoencoder reduces noise energy by O(k/n)

Gradient-based generative model denoising achieves O(k/n) noise reduction

Numerical experiments confirm theoretical predictions

Abstract

Deep neural networks provide state-of-the-art performance for image denoising, where the goal is to recover a near noise-free image from a noisy observation. The underlying principle is that neural networks trained on large datasets have empirically been shown to be able to generate natural images well from a low-dimensional latent representation of the image. Given such a generator network, a noisy image can be denoised by i) finding the closest image in the range of the generator or by ii) passing it through an encoder-generator architecture (known as an autoencoder). However, there is little theory to justify this success, let alone to predict the denoising performance as a function of the network parameters. In this paper we consider the problem of denoising an image from additive Gaussian noise using the two generator based approaches. In both cases, we assume the image is well described by a deep neural network with ReLU activations functions, mapping a $k$-dimensional code to an $n$-dimensional image. In the case of the autoencoder, we show that the feedforward network reduces noise energy by a factor of $O(k/n)$. In the case of optimizing over the range of a generative model, we state and analyze a simple gradient algorithm that minimizes a non-convex loss function, and provably reduces noise energy by a factor of $O(k/n)$. We also demonstrate in numerical experiments that this denoising performance is, indeed, achieved by generative priors learned from data.