Various iterative reconstruction algorithms for inverse problems can be unfolded as neural networks. Empirically, this approach has often led to improved results, but theoretical guarantees are still scarce. While some progress on generalization properties of neural networks have been made, great challenges remain. In this chapter, we discuss and combine these topics to present a generalization error analysis for a class of neural networks suitable for sparse reconstruction from few linear measurements. The hypothesis class considered is inspired by the classical iterative soft-thresholding algorithm (ISTA). The neural networks in this class are obtained by unfolding iterations of ISTA and learning some of the weights. Based on training samples, we aim at learning the optimal network parameters via empirical risk minimization and thereby the optimal network that reconstructs signals from their compressive linear measurements. In particular, we may learn a sparsity basis that is shared by all of the iterations/layers and thereby obtain a new approach for dictionary learning. For this class of networks, we present a generalization bound, which is based on bounding the Rademacher complexity of hypothesis classes consisting of such deep networks via Dudley's integral. Remarkably, under realistic conditions, the generalization error scales only logarithmically in the number of layers, and at most linear in number of measurements.
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Aiming at high-dimensional (HD) data acquisition and analysis, snapshot compressive imaging (SCI) obtains the 2D compressed measurement of HD data with optical imaging systems and reconstructs HD data using compressive sensing algorithms. While the Plug-and-Play (PnP) framework offers an emerging solution to SCI reconstruction, its intrinsic denoising process is still a challenging problem. Unfortunately, existing denoisers in the PnP framework either suffer limited performance or require extensive training data. In this paper, we propose an efficient and effective shallow-learning-based algorithm for video SCI reconstruction. Revisiting dictionary learning methods, we empower the PnP framework with a new denoiser, the kernel singular value decomposition (KSVD). Benefited from the advent of KSVD, our algorithm retains a good trade-off among quality, speed, and training difficulty. On a variety of datasets, both quantitative and qualitative evaluations of our simulation results demonstrate the effectiveness of our proposed method. In comparison to a typical baseline using total variation, our method achieves around $2$ dB improvement in PSNR and 0.2 in SSIM. We expect that our proposed PnP-KSVD algorithm can serve as a new baseline for video SCI reconstruction.
We consider a sparse deep ReLU network (SDRN) estimator obtained from empirical risk minimization with a Lipschitz loss function in the presence of a large number of features. Our framework can be applied to a variety of regression and classification problems. The unknown target function to estimate is assumed to be in a Sobolev space with mixed derivatives. Functions in this space only need to satisfy a smoothness condition rather than having a compositional structure. We develop non-asymptotic excess risk bounds for our SDRN estimator. We further derive that the SDRN estimator can achieve the same minimax rate of estimation (up to logarithmic factors) as one-dimensional nonparametric regression when the dimension of the features is fixed, and the estimator has a suboptimal rate when the dimension grows with the sample size. We show that the depth and the total number of nodes and weights of the ReLU network need to grow as the sample size increases to ensure a good performance, and also investigate how fast they should increase with the sample size. These results provide an important theoretical guidance and basis for empirical studies by deep neural networks.
We evaluate the effectiveness of semi-supervised learning (SSL) on a realistic benchmark where data exhibits considerable class imbalance and contains images from novel classes. Our benchmark consists of two fine-grained classification datasets obtained by sampling classes from the Aves and Fungi taxonomy. We find that recently proposed SSL methods provide significant benefits, and can effectively use out-of-class data to improve performance when deep networks are trained from scratch. Yet their performance pales in comparison to a transfer learning baseline, an alternative approach for learning from a few examples. Furthermore, in the transfer setting, while existing SSL methods provide improvements, the presence of out-of-class is often detrimental. In this setting, standard fine-tuning followed by distillation-based self-training is the most robust. Our work suggests that semi-supervised learning with experts on realistic datasets may require different strategies than those currently prevalent in the literature.
The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.
Self-supervised learning has been widely used to obtain transferrable representations from unlabeled images. Especially, recent contrastive learning methods have shown impressive performances on downstream image classification tasks. While these contrastive methods mainly focus on generating invariant global representations at the image-level under semantic-preserving transformations, they are prone to overlook spatial consistency of local representations and therefore have a limitation in pretraining for localization tasks such as object detection and instance segmentation. Moreover, aggressively cropped views used in existing contrastive methods can minimize representation distances between the semantically different regions of a single image. In this paper, we propose a spatially consistent representation learning algorithm (SCRL) for multi-object and location-specific tasks. In particular, we devise a novel self-supervised objective that tries to produce coherent spatial representations of a randomly cropped local region according to geometric translations and zooming operations. On various downstream localization tasks with benchmark datasets, the proposed SCRL shows significant performance improvements over the image-level supervised pretraining as well as the state-of-the-art self-supervised learning methods.
Graph neural networks (GNNs) are typically applied to static graphs that are assumed to be known upfront. This static input structure is often informed purely by insight of the machine learning practitioner, and might not be optimal for the actual task the GNN is solving. In absence of reliable domain expertise, one might resort to inferring the latent graph structure, which is often difficult due to the vast search space of possible graphs. Here we introduce Pointer Graph Networks (PGNs) which augment sets or graphs with additional inferred edges for improved model generalisation ability. PGNs allow each node to dynamically point to another node, followed by message passing over these pointers. The sparsity of this adaptable graph structure makes learning tractable while still being sufficiently expressive to simulate complex algorithms. Critically, the pointing mechanism is directly supervised to model long-term sequences of operations on classical data structures, incorporating useful structural inductive biases from theoretical computer science. Qualitatively, we demonstrate that PGNs can learn parallelisable variants of pointer-based data structures, namely disjoint set unions and link/cut trees. PGNs generalise out-of-distribution to 5x larger test inputs on dynamic graph connectivity tasks, outperforming unrestricted GNNs and Deep Sets.
Graph Neural Networks (GNNs) are information processing architectures for signals supported on graphs. They are presented here as generalizations of convolutional neural networks (CNNs) in which individual layers contain banks of graph convolutional filters instead of banks of classical convolutional filters. Otherwise, GNNs operate as CNNs. Filters are composed with pointwise nonlinearities and stacked in layers. It is shown that GNN architectures exhibit equivariance to permutation and stability to graph deformations. These properties provide a measure of explanation respecting the good performance of GNNs that can be observed empirically. It is also shown that if graphs converge to a limit object, a graphon, GNNs converge to a corresponding limit object, a graphon neural network. This convergence justifies the transferability of GNNs across networks with different number of nodes.
Graph neural networks (GNNs) are effective machine learning models for various graph learning problems. Despite their empirical successes, the theoretical limitations of GNNs have been revealed recently. Consequently, many GNN models have been proposed to overcome these limitations. In this survey, we provide a comprehensive overview of the expressive power of GNNs and provably powerful variants of GNNs.
We present a continuous formulation of machine learning, as a problem in the calculus of variations and differential-integral equations, very much in the spirit of classical numerical analysis and statistical physics. We demonstrate that conventional machine learning models and algorithms, such as the random feature model, the shallow neural network model and the residual neural network model, can all be recovered as particular discretizations of different continuous formulations. We also present examples of new models, such as the flow-based random feature model, and new algorithms, such as the smoothed particle method and spectral method, that arise naturally from this continuous formulation. We discuss how the issues of generalization error and implicit regularization can be studied under this framework.
This paper surveys the machine learning literature and presents machine learning as optimization models. Such models can benefit from the advancement of numerical optimization techniques which have already played a distinctive role in several machine learning settings. Particularly, mathematical optimization models are presented for commonly used machine learning approaches for regression, classification, clustering, and deep neural networks as well new emerging applications in machine teaching and empirical model learning. The strengths and the shortcomings of these models are discussed and potential research directions are highlighted.
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