Noise2Noise: Learning Image Restoration without Clean Data **Abstract:** This paper explores the application of basic statistical reasoning to signal reconstruction using machine learning, specifically focusing on learning to map corrupted observations to clean signals. The key finding is that it is possible to restore images effectively using only corrupted examples, achieving performance comparable to or even surpassing that of methods using clean data, without explicit image priors or likelihood models of the corruption. The authors demonstrate this approach in various practical scenarios, including photographic noise removal, denoising synthetic Monte Carlo images, and reconstructing undersampled MRI scans. **Introduction:** Signal reconstruction from corrupted or incomplete measurements is a significant area of statistical data analysis. Recent advancements in deep neural networks have led to interest in avoiding traditional, explicit statistical modeling of signal corruptions and instead learning to map corrupted observations to clean versions. This involves training regression models, such as convolutional neural networks (CNNs), with pairs of corrupted inputs and clean targets, minimizing the empirical risk. However, obtaining clean training targets can be challenging or tedious. **Theoretical Background:** The paper discusses the theoretical underpinnings of this approach, including the minimization of loss functions and the properties of neural network training. It highlights that the optimal network parameters remain unchanged when input-conditioned target distributions are replaced with arbitrary distributions that have the same conditional expected values. This allows for the use of corrupted targets in training, even without explicit likelihood models of the corruption. **Practical Experiments:** The authors experimentally validate the effectiveness of noisy-target training in various scenarios. They demonstrate that corrupted targets can lead to similar or better performance compared to clean targets in tasks such as denoising, text removal, and MRI reconstruction. The experiments show that using more realizations of corruption and more latent clean images can improve performance, even with finite data and a fixed capture budget. **Discussion:** The paper concludes that simple statistical arguments can lead to new capabilities in learned signal recovery using deep neural networks. It shows that it is possible to recover signals under complex corruptions without observing clean signals, achieving performance levels equal or close to using clean target data. This approach has significant benefits in many applications by removing the need for potentially strenuous collection of clean data. **Acknowledgments:** The authors acknowledge contributions from various individuals and organizations, including NVIDIA Research staff and Runa Lober and Gunter Sprenger for synthetic offline training data.Noise2Noise: Learning Image Restoration without Clean Data **Abstract:** This paper explores the application of basic statistical reasoning to signal reconstruction using machine learning, specifically focusing on learning to map corrupted observations to clean signals. The key finding is that it is possible to restore images effectively using only corrupted examples, achieving performance comparable to or even surpassing that of methods using clean data, without explicit image priors or likelihood models of the corruption. The authors demonstrate this approach in various practical scenarios, including photographic noise removal, denoising synthetic Monte Carlo images, and reconstructing undersampled MRI scans. **Introduction:** Signal reconstruction from corrupted or incomplete measurements is a significant area of statistical data analysis. Recent advancements in deep neural networks have led to interest in avoiding traditional, explicit statistical modeling of signal corruptions and instead learning to map corrupted observations to clean versions. This involves training regression models, such as convolutional neural networks (CNNs), with pairs of corrupted inputs and clean targets, minimizing the empirical risk. However, obtaining clean training targets can be challenging or tedious. **Theoretical Background:** The paper discusses the theoretical underpinnings of this approach, including the minimization of loss functions and the properties of neural network training. It highlights that the optimal network parameters remain unchanged when input-conditioned target distributions are replaced with arbitrary distributions that have the same conditional expected values. This allows for the use of corrupted targets in training, even without explicit likelihood models of the corruption. **Practical Experiments:** The authors experimentally validate the effectiveness of noisy-target training in various scenarios. They demonstrate that corrupted targets can lead to similar or better performance compared to clean targets in tasks such as denoising, text removal, and MRI reconstruction. The experiments show that using more realizations of corruption and more latent clean images can improve performance, even with finite data and a fixed capture budget. **Discussion:** The paper concludes that simple statistical arguments can lead to new capabilities in learned signal recovery using deep neural networks. It shows that it is possible to recover signals under complex corruptions without observing clean signals, achieving performance levels equal or close to using clean target data. This approach has significant benefits in many applications by removing the need for potentially strenuous collection of clean data. **Acknowledgments:** The authors acknowledge contributions from various individuals and organizations, including NVIDIA Research staff and Runa Lober and Gunter Sprenger for synthetic offline training data.