This chapter introduces the mathematics of data through the example of clustering, a fundamental technique in data analysis and machine learning. The chapter begins with a review of essential mathematical concepts, including matrix and vector algebra, differential calculus, optimization, and elementary probability, with practical Python examples. The chapter then delves into the k-means clustering algorithm, presenting it as an optimization problem and deriving Lloyd's algorithm for its solution. A rigorous analysis of the algorithm's convergence properties is provided, along with a matrix formulation of the k-means objective. The chapter concludes with an exploration of high-dimensional data, demonstrating through simulations and theoretical arguments how the "curse of dimensionality" can affect clustering outcomes.

This chapter focuses on the core concepts of optimization theory and its application in data science and AI. It begins with a review of differentiable functions of several variables, including the gradient and Hessian matrices, and key results like the Chain Rule and the Mean Value Theorem. The chapter then introduces optimality conditions for unconstrained optimization, explaining first-order and second-order conditions, and the role of convexity in ensuring global optimality. A detailed discussion of the gradient descent algorithm is provided, including its convergence analysis under different assumptions. The chapter concludes with an application to logistic regression, demonstrating how gradient descent is used to optimize the cross-entropy loss function in a supervised learning context. Practical Python examples are integrated throughout to illustrate the theoretical concepts.

After a discussion of best programming practices and a brief summary of basic features of the Python programming language, chapter 1 discusses several modern idioms. These include the use of list comprehensions, dictionaries, the for-else idiom, as well as other ways to iterate Pythonically. Throughout, the focus is on programming in a way which feels natural, i.e., working with the language (as opposed to working against the language). The chapter also includes basic information on how to make figures using Matplotlib, as well as advice on how to effectively use the NumPy library, with an emphasis on slicing, vectorization, and broadcasting. The chapter is rounded out by a physics project, which studies the visualization of electric fields, and a problem set.