Denoising and Regularization via Exploiting the Structural Bias of Convolutional Generators
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. On the contrary, a number of recent experimental results 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 structural bias towards simple, natural images is that one can remove various kinds of noise and corruptions from a corrupted natural image by simply fitting (via gradient descent) a randomly initialized, over-parameterized convolutional generator to this single image. While this over-parameterized model can eventually fit the corrupted image perfectly, surprisingly after a few iterations of gradient descent one obtains the uncorrupted image, without using any training data. This intriguing phenomena has enabled state-of-the-art CNN-based denoising as well as regularization in 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 fixed convolutional operations. 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. This results relies on showing that convolutional generators fit the structured part of an image significantly faster than the corrupted portion.
The paper provides a theoretical study of regularization capabilities of over-parameterized convolutional generators trained via gradient descent, in the context of denoising with an approach similar to the *deep image prior*. The authors show that when using appropriate upsampling operators, gradient descent biases the reconstructed images towards low-frequency components, while the noise components, which typically consist of high-frequency patterns, take longer to fit, so that early stopping can provide a useful bias for denoising. The proofs are based on recent results for *lazy* training of over-parameterized networks, namely neural tangent kernels. I find the paper interesting and novel, as it is the first to my knowledge to study the implicit bias of optimization for such generator-based denoising approaches, and provides interesting theoretical and empirical denoising results. While the current results are for a simple architecture with one hidden layer, they may pave the way for a formal study of more complex architectures. Nevertheless, the paper presents some limitations which should be addressed further in the paper: * in terms of denoising capabilities, it seems that the results for the non-linear, over-parameterized network are essentially similar to the linear case (with an appropriately designed operator J in Section 3). Are there any benefits to the non-linear case that I am missing? Whether or not this is a limitation of the study, it should be discussed further in the paper. * Some comments on the empirical results reported in Figure 2: (i) how are hyper-parameters chosen? this seems to be crucial for the effective use of such denoising strategies, and it is unclear how robust these methods are, e.g., to the early stopping time; (ii) there are other methods that only learn on the given image (thus without the need for a training set) which outperform BM3D [e.g. 1,2], it is unclear how the proposed methods compare to those. other minor comments: - Figure 2 *maintained on a large test set of images* more details are welcome, including on hyperparameter selection - Section 1.1: * *demystifying* -> understanding? * different notations are used for the (same?) iterates, C^t and C_tau - Section 2.3 * *it follow that optimizing...*: please clarify. I agree that the optimum is the same, but is the implicit bias from initialization the same? - end of Section 3: early stoped -> stopped - Section 4 * after Definition 2, *this in turn implies that the jacobian spectrum ...rapidly*: clarify, is it a consequence of the analysis? * after Thm 1: maybe give some insight about the proof and the use of NTK? also, discuss similarities or differences with the linear case [1] Dabov et al. (2009) BM3D Image Denoising with Shape-Adaptive Principal Component Analysis [2] Mairal et al. (2009) Non-local Sparse Models for Image Restoration
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Denoising and Regularization via Exploiting the Structural Bias of Convolutional Generators
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. On the contrary, a number of recent experimental results 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 structural bias towards simple, natural images is that one can remove various kinds of noise and corruptions from a corrupted natural image by simply fitting (via gradient descent) a randomly initialized, over-parameterized convolutional generator to this single image. While this over-parameterized model can eventually fit the corrupted image perfectly, surprisingly after a few iterations of gradient descent one obtains the uncorrupted image, without using any training data. This intriguing phenomena has enabled state-of-the-art CNN-based denoising as well as regularization in 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 fixed convolutional operations. 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. This results relies on showing that convolutional generators fit the structured part of an image significantly faster than the corrupted portion.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



The paper provides a theoretical study of regularization capabilities of over-parameterized convolutional generators trained via gradient descent, in the context of denoising with an approach similar to the *deep image prior*. The authors show that when using appropriate upsampling operators, gradient descent biases the reconstructed images towards low-frequency components, while the noise components, which typically consist of high-frequency patterns, take longer to fit, so that early stopping can provide a useful bias for denoising. The proofs are based on recent results for *lazy* training of over-parameterized networks, namely neural tangent kernels. I find the paper interesting and novel, as it is the first to my knowledge to study the implicit bias of optimization for such generator-based denoising approaches, and provides interesting theoretical and empirical denoising results. While the current results are for a simple architecture with one hidden layer, they may pave the way for a formal study of more complex architectures. Nevertheless, the paper presents some limitations which should be addressed further in the paper: * in terms of denoising capabilities, it seems that the results for the non-linear, over-parameterized network are essentially similar to the linear case (with an appropriately designed operator J in Section 3). Are there any benefits to the non-linear case that I am missing? Whether or not this is a limitation of the study, it should be discussed further in the paper. * Some comments on the empirical results reported in Figure 2: (i) how are hyper-parameters chosen? this seems to be crucial for the effective use of such denoising strategies, and it is unclear how robust these methods are, e.g., to the early stopping time; (ii) there are other methods that only learn on the given image (thus without the need for a training set) which outperform BM3D [e.g. 1,2], it is unclear how the proposed methods compare to those. other minor comments: - Figure 2 *maintained on a large test set of images* more details are welcome, including on hyperparameter selection - Section 1.1: * *demystifying* -> understanding? * different notations are used for the (same?) iterates, C^t and C_tau - Section 2.3 * *it follow that optimizing...*: please clarify. I agree that the optimum is the same, but is the implicit bias from initialization the same? - end of Section 3: early stoped -> stopped - Section 4 * after Definition 2, *this in turn implies that the jacobian spectrum ...rapidly*: clarify, is it a consequence of the analysis? * after Thm 1: maybe give some insight about the proof and the use of NTK? also, discuss similarities or differences with the linear case [1] Dabov et al. (2009) BM3D Image Denoising with Shape-Adaptive Principal Component Analysis [2] Mairal et al. (2009) Non-local Sparse Models for Image Restoration
assistant
The paper provides a theoretical study of regularization capabilities of over-parameterized convolutional generators trained via gradient descent, in the context of denoising with an approach similar to the *deep image prior*. The authors show that when using appropriate upsampling operators, gradient descent biases the reconstructed images towards low-frequency components, while the noise components, which typically consist of high-frequency patterns, take longer to fit, so that early stopping can provide a useful bias for denoising. The proofs are based on recent results for *lazy* training of over-parameterized networks, namely neural tangent kernels. I find the paper interesting and novel, as it is the first to my knowledge to study the implicit bias of optimization for such generator-based denoising approaches, and provides interesting theoretical and empirical denoising results. While the current results are for a simple architecture with one hidden layer, they may pave the way for a formal study of more complex architectures. Nevertheless, the paper presents some limitations which should be addressed further in the paper: * in terms of denoising capabilities, it seems that the results for the non-linear, over-parameterized network are essentially similar to the linear case (with an appropriately designed operator J in Section 3). Are there any benefits to the non-linear case that I am missing? Whether or not this is a limitation of the study, it should be discussed further in the paper. * Some comments on the empirical results reported in Figure 2: (i) how are hyper-parameters chosen? this seems to be crucial for the effective use of such denoising strategies, and it is unclear how robust these methods are, e.g., to the early stopping time; (ii) there are other methods that only learn on the given image (thus without the need for a training set) which outperform BM3D [e.g. 1,2], it is unclear how the proposed methods compare to those. other minor comments: - Figure 2 *maintained on a large test set of images* more details are welcome, including on hyperparameter selection - Section 1.1: * *demystifying* -> understanding? * different notations are used for the (same?) iterates, C^t and C_tau - Section 2.3 * *it follow that optimizing...*: please clarify. I agree that the optimum is the same, but is the implicit bias from initialization the same? - end of Section 3: early stoped -> stopped - Section 4 * after Definition 2, *this in turn implies that the jacobian spectrum ...rapidly*: clarify, is it a consequence of the analysis? * after Thm 1: maybe give some insight about the proof and the use of NTK? also, discuss similarities or differences with the linear case [1] Dabov et al. (2009) BM3D Image Denoising with Shape-Adaptive Principal Component Analysis [2] Mairal et al. (2009) Non-local Sparse Models for Image Restoration