Topological Data Science

This book begins from a proposition at once modest and disobedient: data have shape. The proposition does not claim that every dataset contains a hidden torus awaiting a sufficiently expensive package. It claims that observations acquire organisation when we declare which objects are near, incident, comparable, or compatible; and that this organisation may contain information not exhausted by coordinates, averages, or pairwise associations. The growth of topological data analysis has produced a fortunate abundance of theory, algorithms, and software. Abundance, however, creates its own poverty when definitions are separated from motivation, computation from inference, or striking diagrams from the scientific question that occasioned them. A persistence diagram is not an argument merely because its points decline to sit upon the diagonal. The argument must begin with representation and end with a claim whose jurisdiction is honestly stated. The governing principle of the volume is therefore fourfold. Every major construction first appears as an intuition, then as a formal definition, thereafter as an algorithm, and finally as an inference applied to data. This sequence is not ceremonial. Intuition without definition is suggestive but unstable; definition without computation may remain inert; computation without inference is a technical performance; inference without interpretation is an answer whose question has gone missing. No essential symbol is permitted to fall from the sky. Variables are introduced before they are used; displayed equations are derived and numbered; the meaning of each important expression is stated in prose. The reader will occasionally be asked to pause, predict, calculate, or object. Such invitations are not pedagogical decoration. Topology is learned by testing which distinctions survive transformation and which collapse under a counterexample. Figures serve as visual arguments. Their legends distinguish what is drawn, what can be inferred, and what remains merely plausible. Worked examples expose the chain between premises and result. Exercises complete the exposition rather than standing outside it, and full resolutions are supplied because an unexplained answer is only a rumour with arithmetic.

This book begins from a proposition at once modest and disobedient: data have shape. The proposition does not claim that every dataset contains a hidden torus awaiting a sufficiently expensive package. It claims that observations acquire organisation when we declare which objects are near, incident, comparable, or compatible; and that this organisation may contain information not exhausted by coordinates, averages, or pairwise associations. The growth of topological data analysis has produced a fortunate abundance of theory, algorithms, and software. Abundance, however, creates its own poverty when definitions are separated from motivation, computation from inference, or striking diagrams from the scientific question that occasioned them. A persistence diagram is not an argument merely because its points decline to sit upon the diagonal. The argument must begin with representation and end with a claim whose jurisdiction is honestly stated. The governing principle of the volume is therefore fourfold. Every major construction first appears as an intuition, then as a formal definition, thereafter as an algorithm, and finally as an inference applied to data. This sequence is not ceremonial. Intuition without definition is suggestive but unstable; definition without computation may remain inert; computation without inference is a technical performance; inference without interpretation is an answer whose question has gone missing. No essential symbol is permitted to fall from the sky. Variables are introduced before they are used; displayed equations are derived and numbered; the meaning of each important expression is stated in prose. The reader will occasionally be asked to pause, predict, calculate, or object. Such invitations are not pedagogical decoration. Topology is learned by testing which distinctions survive transformation and which collapse under a counterexample. Figures serve as visual arguments. Their legends distinguish what is drawn, what can be inferred, and what remains merely plausible. Worked examples expose the chain between premises and result. Exercises complete the exposition rather than standing outside it, and full resolutions are supplied because an unexplained answer is only a rumour with arithmetic.