Deep generative models have been recently proposed as modular datadriven priors to solve inverse problems. Linear inverse problems involve the reconstruction of an unknown signal (e.g. a tomographic image) from an underdetermined system of noisy linear measurements. Most results in the literature require that the reconstructed signal has some known structure, e.g. it is sparse in some known basis (usually Fourier or wavelet). Such prior assumptions can be replaced with pre-trained deep generative models (e.g. generative adversarial getworks (GANs) and variational autoencoders (VAEs)) with significant performance gains. This chapter surveys this rapidly evolving research area and includes empirical and theoretical results in compressed sensing for deep generative models.

We describe the new field of the mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of learning theory. These questions concern: the outstanding generalization power of overparametrized neural networks, the role of depth in deep architectures, the apparent absence of the curse of dimensionality, the surprisingly successful optimization performance despite the non-convexity of the problem, understanding what features are learned, why deep architectures perform exceptionally well in physical problems, and which fine aspects of an architecture affect the behavior of a learning task in which way. We present an overview of modern approaches that yield partial answers to these questions. For selected approaches, we describe the main ideas in more detail.

Convolutional neural networks (CNNs) are prevalent in computer vision, image, speech, and language processing applications, where they have been successfully applied to perform classification tasks at high accuracy rates. One of their main attractions is the ability to operate directly on raw input signals, such as images, and to extract salient features automatically from the raw data. The designer does not need to worry about which features to select to drive the classification process.

When the training data $\{\text{γ}\left(n\right),h_n\}$ is linearly separable, there will exist many separating hyperplanes that can discriminate the data into two classes. Some of the techniques we described in the previous chapters, such as logistic regression and perceptron, are able to find such separating hyperplanes.

We develop a sequential version of the importance sampling technique from Chapter 33 in order to respond to streaming data, thus leading to a sequential Monte Carlo solution. The algorithm will lead to the important class of particle filters. This chapter presents the basic data model and the main construction that enables recursive inference. Many of the inference and learning methods in subsequent chapters will possess a recursive structure, which is a fundamental property to enable them to continually learn in response to the arrival of sequential data measurements. Particle filters are particularly well suited for scenarios involving nonlinear models and non-Gaussian signals, and they have found applications in a wide range of areas where these two features (nonlinearity and non-Gaussianity) are prevalent, including in guidance and control, robot localization, visual tracking of objects, and finance.

The optimal Bayes classifier (52.8) requires knowledge of the conditional probability distribution $\mathbb{P}\left(\mathit{r}=r|\mathit{h}=h\right)$, which is generally unavailable. In this and the next few chapters, we describe data‐based generative methods that approximate the joint probability distribution $f_{\mathit{r},\mathit{h}}\left(r,h\right)$, or its components $\mathbb{P}\left(r=r\right)$ and $f_{h|r}\left(h|r\right)$, directly from the data.

We described several data-based methods for inference and learning in the previous chapters. These methods operate directly on the data to arrive at classification or inference decisions. One key challenge these methods face is that the available training data need not provide sufficient representation for the sample space.