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1. Seismic data denoising by deep-residual networks

   https://doi.org/10.1190/segam2018-2998619.1  

   Jin, Yuchen; Wu, Xuqing; Chen, Jiefu; Han, Zhu; Hu, Wenyi (August 2018, SEG Technical Program Expanded Abstracts 2018)

   Deep learning leverages multi-layer neural networks architecture and demonstrates superb power in many machine learning applications. The deep denoising autoencoder technique extracts better coherent features from the seismic data. The technique allows us to automatically extract low-dimensional features from high dimensional feature space in a non-linear, data-driven, and unsupervised way. A properly trained denoising autoencoder takes a partially corrupted input and recovers the original undistorted input. In this paper, a novel autoencoder built upon the deep residual network is proposed to perform noise attenuation on the seismic data. We evaluate the proposed method with synthetic datasets and the result confirms the effective denoising performance of the proposed approach. 

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2. Denoising and Regularization via Exploiting the Structural Bias of Convolutional Generators

   Heckel, Reinhard; Soltanolkotabi, Mahdi (May 2020, International Conference on Learning Representations)

   Convolutional Neural Networks (CNNs) have emerged as highly successful tools for image generation, recovery, and restoration. A major contributing factor to this success is that convolutional networks impose strong prior assumptions about natural images. A surprising experiment that highlights this architectural bias towards natural images is that one can remove noise and corruptions from a natural image without using any training data, by simply fitting (via gradient descent) a randomly initialized, over-parameterized convolutional generator to the corrupted image. While this over-parameterized network can fit the corrupted image perfectly, surprisingly after a few iterations of gradient descent it generates an almost uncorrupted image. This intriguing phenomenon enables state-of-the-art CNN-based denoising and regularization of other inverse problems. In this paper, we attribute this effect to a particular architectural choice of convolutional networks, namely convolutions with fixed interpolating filters. We then formally characterize the dynamics of fitting a two-layer convolutional generator to a noisy signal and prove that early-stopped gradient descent denoises/regularizes. Our proof relies on showing that convolutional generators fit the structured part of an image significantly faster than the corrupted portion. 

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3. Denoising and Regularization via Exploiting the Structural Bias of Convolutional Generators. R. Heckel and M. Soltanolkotabi

   Heckel, Reinhard; Soltanolkotabi, Mahdi (May 2020, International Conference on Learning Representations)

   Convolutional Neural Networks (CNNs) have emerged as highly successful tools for image generation, recovery, and restoration. This success is often attributed to large amounts of training data. However, recent experimental findings challenge this view and instead suggest that a major contributing factor to this success is that convolutional networks impose strong prior assumptions about natural images. A surprising experiment that highlights this architectural bias towards natural images is that one can remove noise and corruptions from a natural image without using any training data, by simply fitting (via gradient descent) a randomly initialized, over-parameterized convolutional generator to the single corrupted image. While this over-parameterized network can fit the corrupted image perfectly, surprisingly after a few iterations of gradient descent one obtains the uncorrupted image. This intriguing phenomena enables state-of-the-art CNN-based denoising and regularization of linear inverse problems such as compressive sensing. In this paper we take a step towards demystifying this experimental phenomena by attributing this effect to particular architectural choices of convolutional networks, namely convolutions with fixed interpolating filters. We then formally characterize the dynamics of fitting a two layer convolutional generator to a noisy signal and prove that earlystopped gradient descent denoises/regularizes. This results relies on showing that convolutional generators fit the structured part of an image significantly faster than the corrupted portion 

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4. Learning Provably Robust Estimators for Inverse Problems via Jittering

   Krainovic, Anselm; Soltanolkotabi, Mahdi; Heckel, Reinhard (December 2023, Proceedings of Neural Information Processing Systems)

   Deep neural networks provide excellent performance for inverse problems such as denoising. However, neural networks can be sensitive to adversarial or worst-case perturbations. This raises the question of whether such networks can be trained efficiently to be worst-case robust. In this paper, we investigate whether jittering, a simple regularization technique that adds isotropic Gaussian noise during training, is effective for learning worst-case robust estimators for inverse problems. While well studied for prediction in classification tasks, the effectiveness of jittering for inverse problems has not been systematically investigated. In this paper, we present a novel analytical characterization of the optimal -worst-case robust estimator for linear denoising and show that jittering yields optimal robust denoisers. Furthermore, we examine jittering empirically via training deep neural networks (U-nets) for natural image denoising, deconvolution, and accelerated magnetic resonance imaging (MRI). The results show that jittering significantly enhances the worst-case robustness, but can be suboptimal for inverse problems beyond denoising. Moreover, our results imply that training on real data which often contains slight noise is somewhat robustness enhancing. 

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5. Deep Decoder: Concise Image Representations from Untrained Non-convolutional Networks

   Heckel, R.; Hand., P. (April 2019, International Conference on Learning Representations)

   Deep neural networks, in particular convolutional neural networks, have become highly effective tools for compressing images and solving inverse problems including denoising, inpainting, and reconstruction from few and noisy measurements. This success can be attributed in part to their ability to represent and generate natural images well. Contrary to classical tools such as wavelets, image-generating deep neural networks have a large number of parameters---typically a multiple of their output dimension---and need to be trained on large datasets. In this paper, we propose an untrained simple image model, called the deep decoder, which is a deep neural network that can generate natural images from very few weight parameters. The deep decoder has a simple architecture with no convolutions and fewer weight parameters than the output dimensionality. This underparameterization enables the deep decoder to compress images into a concise set of network weights, which we show is on par with wavelet-based thresholding. Further, underparameterization provides a barrier to overfitting, allowing the deep decoder to have state-of-the-art performance for denoising. The deep decoder is simple in the sense that each layer has an identical structure that consists of only one upsampling unit, pixel-wise linear combination of channels, ReLU activation, and channelwise normalization. This simplicity makes the network amenable to theoretical analysis, and it sheds light on the aspects of neural networks that enable them to form effective signal representations. 

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