arXiv subjects

Reinhard Heckel

Publications and source records attributed to Reinhard Heckel.

At least 55 records · Page 3Linked to original sources

arXiv2022

Test-Time Training Can Close the Natural Distribution Shift Performance Gap in Deep Learning Based Compressed Sensing

Deep learning based image reconstruction methods outperform traditional methods. However, neural networks suffer from a performance drop when applied to images from a different distribution than the training images. For example, a model trained for reconstructing knees in accelerated magnetic resonance imaging (MRI) does not reconstruct brains well, even though the same network trained on brains reconstructs brains perfectly well. Thus there is a distribution shift performance gap for a given neural network, defined as the difference in performance when training on a distribution $P$ and training on another distribution $Q$, and evaluating both models on $Q$. In this work, we propose a domain adaptation method for deep learning based compressive sensing that relies on self-supervision during training paired with test-time training at inference. We show that for four natural distribution shifts, this method essentially closes the distribution shift performance gap for state-of-the-art architectures for accelerated MRI.

eess.IV↗

arXiv2022

Regularization-wise double descent: Why it occurs and how to eliminate it

The risk of overparameterized models, in particular deep neural networks, is often double-descent shaped as a function of the model size. Recently, it was shown that the risk as a function of the early-stopping time can also be double-descent shaped, and this behavior can be explained as a super-position of bias-variance tradeoffs. In this paper, we show that the risk of explicit L2-regularized models can exhibit double descent behavior as a function of the regularization strength, both in theory and practice. We find that for linear regression, a double descent shaped risk is caused by a superposition of bias-variance tradeoffs corresponding to different parts of the model and can be mitigated by scaling the regularization strength of each part appropriately. Motivated by this result, we study a two-layer neural network and show that double descent can be eliminated by adjusting the regularization strengths for the first and second layer. Lastly, we study a 5-layer CNN and ResNet-18 trained on CIFAR-10 with label noise, and CIFAR-100 without label noise, and demonstrate that all exhibit double descent behavior as a function of the regularization strength.

cs.LG↗

arXiv2022

Image-to-Image MLP-mixer for Image Reconstruction

Neural networks are highly effective tools for image reconstruction problems such as denoising and compressive sensing. To date, neural networks for image reconstruction are almost exclusively convolutional. The most popular architecture is the U-Net, a convolutional network with a multi-resolution architecture. In this work, we show that a simple network based on the multi-layer perceptron (MLP)-mixer enables state-of-the art image reconstruction performance without convolutions and without a multi-resolution architecture, provided that the training set and the size of the network are moderately large. Similar to the original MLP-mixer, the image-to-image MLP-mixer is based exclusively on MLPs operating on linearly-transformed image patches. Contrary to the original MLP-mixer, we incorporate structure by retaining the relative positions of the image patches. This imposes an inductive bias towards natural images which enables the image-to-image MLP-mixer to learn to denoise images based on fewer examples than the original MLP-mixer. Moreover, the image-to-image MLP-mixer requires fewer parameters to achieve the same denoising performance than the U-Net and its parameters scale linearly in the image resolution instead of quadratically as for the original MLP-mixer. If trained on a moderate amount of examples for denoising, the image-to-image MLP-mixer outperforms the U-Net by a slight margin. It also outperforms the vision transformer tailored for image reconstruction and classical un-trained methods such as BM3D, making it a very effective tool for image reconstruction problems.

cs.CV↗

arXiv2021

Interpolation can hurt robust generalization even when there is no noise

Numerous recent works show that overparameterization implicitly reduces variance for min-norm interpolators and max-margin classifiers. These findings suggest that ridge regularization has vanishing benefits in high dimensions. We challenge this narrative by showing that, even in the absence of noise, avoiding interpolation through ridge regularization can significantly improve generalization. We prove this phenomenon for the robust risk of both linear regression and classification and hence provide the first theoretical result on robust overfitting.

stat.ML↗

arXiv2021

Provable Continual Learning via Sketched Jacobian Approximations

An important problem in machine learning is the ability to learn tasks in a sequential manner. If trained with standard first-order methods most models forget previously learned tasks when trained on a new task, which is often referred to as catastrophic forgetting. A popular approach to overcome forgetting is to regularize the loss function by penalizing models that perform poorly on previous tasks. For example, elastic weight consolidation (EWC) regularizes with a quadratic form involving a diagonal matrix build based on past data. While EWC works very well for some setups, we show that, even under otherwise ideal conditions, it can provably suffer catastrophic forgetting if the diagonal matrix is a poor approximation of the Hessian matrix of previous tasks. We propose a simple approach to overcome this: Regularizing training of a new task with sketches of the Jacobian matrix of past data. This provably enables overcoming catastrophic forgetting for linear models and for wide neural networks, at the cost of memory. The overarching goal of this paper is to provided insights on when regularization-based continual learning algorithms work and under what memory costs.

stat.ML↗

arXiv2021

Achieving the Capacity of a DNA Storage Channel with Linear Coding Schemes

Due to the redundant nature of DNA synthesis and sequencing technologies, a basic model for a DNA storage system is a multi-draw "shuffling-sampling" channel. In this model, a random number of noisy copies of each sequence is observed at the channel output. Recent works have characterized the capacity of such a DNA storage channel under different noise and sequencing models, relying on sophisticated typicality-based approaches for the achievability. Here, we consider a multi-draw DNA storage channel in the setting of noise corruption by a binary erasure channel. We show that, in this setting, the capacity is achieved by linear coding schemes. This leads to a considerably simpler derivation of the capacity expression of a multi-draw DNA storage channel than existing results in the literature.

cs.IT↗

arXiv2021

Data augmentation for deep learning based accelerated MRI reconstruction with limited data

Deep neural networks have emerged as very successful tools for image restoration and reconstruction tasks. These networks are often trained end-to-end to directly reconstruct an image from a noisy or corrupted measurement of that image. To achieve state-of-the-art performance, training on large and diverse sets of images is considered critical. However, it is often difficult and/or expensive to collect large amounts of training images. Inspired by the success of Data Augmentation (DA) for classification problems, in this paper, we propose a pipeline for data augmentation for accelerated MRI reconstruction and study its effectiveness at reducing the required training data in a variety of settings. Our DA pipeline, MRAugment, is specifically designed to utilize the invariances present in medical imaging measurements as naive DA strategies that neglect the physics of the problem fail. Through extensive studies on multiple datasets we demonstrate that in the low-data regime DA prevents overfitting and can match or even surpass the state of the art while using significantly fewer training data, whereas in the high-data regime it has diminishing returns. Furthermore, our findings show that DA can improve the robustness of the model against various shifts in the test distribution.

eess.IV↗

arXiv2021

Active Sampling Count Sketch (ASCS) for Online Sparse Estimation of a Trillion Scale Covariance Matrix

Estimating and storing the covariance (or correlation) matrix of high-dimensional data is computationally challenging because both memory and computational requirements scale quadratically with the dimension. Fortunately, high-dimensional covariance matrices as observed in text, click-through, meta-genomics datasets, etc are often sparse. In this paper, we consider the problem of efficient sparse estimation of covariance matrices with possibly trillions of entries. The size of the datasets we target requires the algorithm to be online, as more than one pass over the data is prohibitive. In this paper, we propose Active Sampling Count Sketch (ASCS), an online and one-pass sketching algorithm, that recovers the large entries of the covariance matrix accurately. Count Sketch (CS), and other sub-linear compressed sensing algorithms, offer a natural solution to the problem in theory. However, vanilla CS does not work well in practice due to a low signal-to-noise ratio (SNR). At the heart of our approach is a novel active sampling strategy that increases the SNR of classical CS. We demonstrate the practicality of our algorithm with synthetic data and real-world high dimensional datasets. ASCS significantly improves over vanilla CS, demonstrating the merit of our active sampling strategy.

cs.DS↗

arXiv2021

Measuring Robustness in Deep Learning Based Compressive Sensing

Deep neural networks give state-of-the-art accuracy for reconstructing images from few and noisy measurements, a problem arising for example in accelerated magnetic resonance imaging (MRI). However, recent works have raised concerns that deep-learning-based image reconstruction methods are sensitive to perturbations and are less robust than traditional methods: Neural networks (i) may be sensitive to small, yet adversarially-selected perturbations, (ii) may perform poorly under distribution shifts, and (iii) may fail to recover small but important features in an image. In order to understand the sensitivity to such perturbations, in this work, we measure the robustness of different approaches for image reconstruction including trained and un-trained neural networks as well as traditional sparsity-based methods. We find, contrary to prior works, that both trained and un-trained methods are vulnerable to adversarial perturbations. Moreover, both trained and un-trained methods tuned for a particular dataset suffer very similarly from distribution shifts. Finally, we demonstrate that an image reconstruction method that achieves higher reconstruction quality, also performs better in terms of accurately recovering fine details. Our results indicate that the state-of-the-art deep-learning-based image reconstruction methods provide improved performance than traditional methods without compromising robustness.

eess.IV↗

arXiv2021

Accelerated MRI with Un-trained Neural Networks

Convolutional Neural Networks (CNNs) are highly effective for image reconstruction problems. Typically, CNNs are trained on large amounts of training images. Recently, however, un-trained CNNs such as the Deep Image Prior and Deep Decoder have achieved excellent performance for image reconstruction problems such as denoising and inpainting, \emph{without using any training data}. Motivated by this development, we address the reconstruction problem arising in accelerated MRI with un-trained neural networks. We propose a highly optimized un-trained recovery approach based on a variation of the Deep Decoder and show that it significantly outperforms other un-trained methods, in particular sparsity-based classical compressed sensing methods and naive applications of un-trained neural networks. We also compare performance (both in terms of reconstruction accuracy and computational cost) in an ideal setup for trained methods, specifically on the fastMRI dataset, where the training and test data come from the same distribution. We find that our un-trained algorithm achieves similar performance to a baseline trained neural network, but a state-of-the-art trained network outperforms the un-trained one. Finally, we perform a comparison on a non-ideal setup where the train and test distributions are slightly different, and find that our un-trained method achieves similar performance to a state-of-the-art accelerated MRI reconstruction method.

eess.IV↗

arXiv2021

Super-Resolution Radar

In this paper we study the identification of a time-varying linear system from its response to a known input signal. More specifically, we consider systems whose response to the input signal is given by a weighted superposition of delayed and Doppler shifted versions of the input. This problem arises in a multitude of applications such as wireless communications and radar imaging. Due to practical constraints, the input signal has finite bandwidth B, and the received signal is observed over a finite time interval of length T only. This gives rise to a delay and Doppler resolution of 1/B and 1/T. We show that this resolution limit can be overcome, i.e., we can exactly recover the continuous delay-Doppler pairs and the corresponding attenuation factors, by solving a convex optimization problem. This result holds provided that the distance between the delay-Doppler pairs is at least 2.37/B in time or 2.37/T in frequency. Furthermore, this result allows the total number of delay-Doppler pairs to be linear up to a log-factor in BT, the dimensionality of the response of the system, and thereby the limit for identifiability. Stated differently, we show that we can estimate the time-frequency components of a signal that is S-sparse in the continuous dictionary of time-frequency shifts of a random window function, from a number of measurements, that is linear up to a log-factor in S.

cs.IT↗

arXiv2020

Early Stopping in Deep Networks: Double Descent and How to Eliminate it

Over-parameterized models, such as large deep networks, often exhibit a double descent phenomenon, whereas a function of model size, error first decreases, increases, and decreases at last. This intriguing double descent behavior also occurs as a function of training epochs and has been conjectured to arise because training epochs control the model complexity. In this paper, we show that such epoch-wise double descent arises for a different reason: It is caused by a superposition of two or more bias-variance tradeoffs that arise because different parts of the network are learned at different epochs, and eliminating this by proper scaling of stepsizes can significantly improve the early stopping performance. We show this analytically for i) linear regression, where differently scaled features give rise to a superposition of bias-variance tradeoffs, and for ii) a two-layer neural network, where the first and second layer each govern a bias-variance tradeoff. Inspired by this theory, we study two standard convolutional networks empirically and show that eliminating epoch-wise double descent through adjusting stepsizes of different layers improves the early stopping performance significantly.

cs.LG↗

arXiv2020

Unsupervised Learning with Stein's Unbiased Risk Estimator

Learning from unlabeled and noisy data is one of the grand challenges of machine learning. As such, it has seen a flurry of research with new ideas proposed continuously. In this work, we revisit a classical idea: Stein's Unbiased Risk Estimator (SURE). We show that, in the context of image recovery, SURE and its generalizations can be used to train convolutional neural networks (CNNs) for a range of image denoising and recovery problems without any ground truth data. Specifically, our goal is to reconstruct an image $x$ from a noisy linear transformation (measurement) of the image. We consider two scenarios: one where no additional data is available and one where we have measurements of other images that are drawn from the same noisy distribution as $x$, but have no access to the clean images. Such is the case, for instance, in the context of medical imaging, microscopy, and astronomy, where noise-less ground truth data is rarely available. We show that in this situation, SURE can be used to estimate the mean-squared-error loss associated with an estimate of $x$. Using this estimate of the loss, we train networks to perform denoising and compressed sensing recovery. In addition, we also use the SURE framework to partially explain and improve upon an intriguing results presented by Ulyanov et al. in "Deep Image Prior": that a network initialized with random weights and fit to a single noisy image can effectively denoise that image. Public implementations of the networks and methods described in this paper can be found at https://github.com/ricedsp/D-AMP_Toolbox.

stat.ML↗

arXiv2020

Compressive sensing with un-trained neural networks: Gradient descent finds the smoothest approximation

Un-trained convolutional neural networks have emerged as highly successful tools for image recovery and restoration. They are capable of solving standard inverse problems such as denoising and compressive sensing with excellent results by simply fitting a neural network model to measurements from a single image or signal without the need for any additional training data. For some applications, this critically requires additional regularization in the form of early stopping the optimization. For signal recovery from a few measurements, however, un-trained convolutional networks have an intriguing self-regularizing property: Even though the network can perfectly fit any image, the network recovers a natural image from few measurements when trained with gradient descent until convergence. In this paper, we provide numerical evidence for this property and study it theoretically. We show that---without any further regularization---an un-trained convolutional neural network can approximately reconstruct signals and images that are sufficiently structured, from a near minimal number of random measurements.

cs.LG↗

arXiv2020

Image recognition from raw labels collected without annotators

Image classification problems are typically addressed by first collecting examples with candidate labels, second cleaning the candidate labels manually, and third training a deep neural network on the clean examples. The manual labeling step is often the most expensive one as it requires workers to label millions of images. In this paper we propose to work without any explicitly labeled data by i) directly training the deep neural network on the noisy candidate labels, and ii) early stopping the training to avoid overfitting. With this procedure we exploit an intriguing property of standard overparameterized convolutional neural networks trained with (stochastic) gradient descent: Clean labels are fitted faster than noisy ones. We consider two classification problems, a subset of ImageNet and CIFAR-10. For both, we construct large candidate datasets without any explicit human annotations, that only contain 10%-50% correctly labeled examples per class. We show that training on the candidate examples and regularizing through early stopping gives higher test performance for both problems than when training on the original, clean data. This is possible because the candidate datasets contain a huge number of clean examples, and, as we show in this paper, the noise generated through the label collection process is not nearly as adversarial for learning as the noise generated by randomly flipping labels.

cs.LG↗

arXiv2020

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. 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.

cs.LG↗

arXiv2020

Deep Phase Decoder: Self-calibrating phase microscopy with an untrained deep neural network

Deep neural networks have emerged as effective tools for computational imaging including quantitative phase microscopy of transparent samples. To reconstruct phase from intensity, current approaches rely on supervised learning with training examples; consequently, their performance is sensitive to a match of training and imaging settings. Here we propose a new approach to phase microscopy by using an untrained deep neural network for measurement formation, encapsulating the image prior and imaging physics. Our approach does not require any training data and simultaneously reconstructs the sought phase and pupil-plane aberrations by fitting the weights of the network to the captured images. To demonstrate experimentally, we reconstruct quantitative phase from through-focus images blindly (i.e. no explicit knowledge of the aberrations).

eess.IV↗

arXiv2020

DNA-Based Storage: Models and Fundamental Limits

Due to its longevity and enormous information density, DNA is an attractive medium for archival storage. In this work, we study the fundamental limits and trade-offs of DNA-based storage systems by introducing a new channel model, which we call the noisy shuffling-sampling channel. Motivated by current technological constraints on DNA synthesis and sequencing, this model captures three key distinctive aspects of DNA storage systems: (1) the data is written onto many short DNA molecules; (2) the molecules are corrupted by noise during synthesis and sequencing and (3) the data is read by randomly sampling from the DNA pool. We provide capacity results for this channel under specific noise and sampling assumptions and show that, in many scenarios, a simple index-based coding scheme is optimal.

cs.IT↗