Clustering is the process of examining a collection of “points,” and grouping the points into “clusters” according to some distance measure. The goal is that points in the same cluster have a small distance from one another, while points in different clusters are at a large distance from one another. A suggestion of what clusters might look like was seen in Fig. 1.1. However, there the intent was that there were three clusters around three different road intersections, but two of the clusters blended into one another because they were not sufficiently separated. Our goal in this chapter is to offer methods for discovering clusters in data. We are particularly interested in situations where the data is very large, and/or where the space either is high-dimensional, or the space is not Euclidean at all. We shall therefore discuss several algorithms that assume the data does not fit in main memory. However, we begin with the basics: the two general approaches to clustering and the methods for dealing with clusters in a non-Euclidean space.

We begin with the essence of data mining and a discussion of how data mining is treated by the various disciplines that contribute to this field. We cover “Bonferroni’s Principle,” which is really a warning about overusing the ability to mine data. We also summarize a few useful ideas that are not data mining per se, but are useful in understanding some important data-mining concepts. These include the TF.IDF measure of word importance, behavior of hash functions and indexes, and identities involving e, the base of natural logarithms. Finally, we give an outline of the topics covered in the balance of the book.

In this chapter we shall explore the idea of dimensionality reduction in more detail. We begin with a discussion of eigenvalues and their use in “principal component analysis” (PCA). We cover singular-value decomposition, a more powerful version of UV-decomposition. Finally, because we are always interested in the largest data sizes we can handle, we look at another form of decomposition, called CUR-decomposition, which is a variant of singular-value decomposition that keeps the matrices of the decomposition sparse if the original matrix is sparse.

Today, technology is driving disruptive change in the legal profession and the public is demanding lawyers offer more value and choice in how legal services are delivered. Given these pressures, tomorrow’s legal profession will be fundamentally different from the profession we know today. Against this backdrop, this chapter argues the next generation of lawyers need at least five categories of multidimensional knowledge and skills: collaboration; design; project management; problem-solving; and lifelong learning. The prevailing, traditional legal education model was not designed to teach these multidimensional skills. This chapter describes some of traditional legal education’s deficiencies, introduces the pedagogy of problem-based learning, and advocates a particular form of this pedagogy: project-based learning that involves real clients or community partners. Through project-based learning – a student-centred, active, and experiential learning model – students learn the fundamentals of law and legal practice while gaining the multidimensional knowledge and skills needed to navigate disruptive change. Project-based learning can prepare law students to actively shape the future of the profession – as opposed to merely reacting to change – by harnessing technology and interdisciplinary insights to improve legal systems and create better legal service models for the public.

In this chapter, we shall consider the design of neural nets, which are collections of perceptrons, or nodes, where the outputs of one rank (or layer of nodes becomes the inputs to nodes at the next layer. The last layer of nodes produces the outputs of the entire neural net. The training of neural nets with many layers requires enormous numbers of training examples, but has proven to be an extremely powerful technique, referred to as deep learning, when it can be used.We also consider several specialized forms of neural nets that have proved useful for special kinds of data. These forms are characterized by requiring that certain sets of nodes in the network share the same weights. Since learning all the weights on all the inputs to all the nodes of the network is in general a hard and time-consuming task, these special forms of network greatly simplify the process of training the network to recognize the desired class or classes of inputs. We shall study convolutional neural networks (CNNs), which are specially designed to recognize classes of images. We shall also study recurrent neural networks (RNNs) and long short-term memory networks (LSTMs), which are designed to recognize classes of sequences, such as sentences (sequences of words).