In this chapter, we extend the methods from Chapter 2 to handle the case, where the number of features d is large. Least squares does not work well in this case. We introduce ridge regression and the lasso, which are two methods for high-dimensional regression. We also discuss the bias–variance tradeoff and the challenges in constructing confidence intervals in this setting.

So far, we have focused on predicting an outcome Y from a set of features X. In this chapter, we turn to causal inference where we ask: What would Y be if we set X to a particular value x? This question concerns the distribution of the outcome Y after some hypothetical intervention. Answering such causal questions requires new tools and stronger assumptions than prediction.

Regression and classification are used to produce a prediction $\widehat{Y}$ of an outcome Y. In many cases, we will also want a prediction set C that contains Y with probability $1-\alpha$. In this chapter, we discuss two methods for constructing predictions sets. The first is based on quantile regression. The second uses a method called conformal inference.

In this chapter, we discuss nonparametric regression, which does not assume that the regression function $\mu (x)$ is linear or even approximately linear. We only assume that $\mu (x)$ is a smooth function of x. This chapter describes some of these methods in the case where there is a single feature. Chapter 7 considers the case of multiple features.

This book aims to accelerate the inference of deep neural networks. Let us first consider why acceleration matters. Faster inference means quicker response times. For instance, in a speech recognition engine, transcription results can be delivered faster. In a translation engine, translations can be provided more quickly. This leads to an improved user experience.

In the previous chapters, we discussed methods to accelerate models by approximating pre-existing models. In this chapter, we describe how to obtain fast models by designing the architecture with acceleration in mind right from the beginning.

Low-rank approximation refers to approximating a matrix by the product of smaller matrices.

In this chapter, we introduce the concept of regression, which is a way to quantify the relationship between an outcome Y and a vector of features X. This relationship is expressed by the regression function, which is the mean of Y given X. This chapter discusses the main ideas and goals of regression analysis.

Quantization is the process of mapping continuous values to a smaller set of discrete values. A typical example is converting a floating-point number into an integer representation. For example, a matrix of 32-bit floating-point numbers

The previous chapters have given an overview of various theoretical and applied aspects of differential equations on graphs.We foresee both a broadening and deepening of the field in the near future.

In this chapter, we will discuss how to implement the graph processes we have defined elsewhere in this book. The fundamental implementation obstacle is that for practical applications, the graphs one desires to use can be very large.

An important topic which has seen a considerable amount of attention in recent years concerns the continuum limits of the graph-based models discussed in this book1.

In Sections 3.1, 3.2, and 3.3, we defined graph versions of the Allen–Cahn equation, MBO scheme, and MCF respectively. It was first speculated in Van Gennip et al. (2014) that these dynamics on graphs might be related.

Starting from this section, we only consider undirected graphs unless stated differently in specific situations. In the definition of ${\mathrm{GL}}_{\epsilon }$, we make the choices $\alpha (\epsilon )=1$ and $\mathcal{W}(u)=⟨W\circ u,1{⟩}_{\mathcal{V}}$.1 We will specify whether $W$ is a double-well or double-obstacle potential where this is relevant. Also, we recall from Remark 2.1.9 that we choose $q=1$.