Data Science Foundations: Geometry and Topology of Complex Hierarchic Systems and Big Data Analytics.

In the era of Big Data, the size and the dimension of data grow rapidly, and so does the complexity of the system. How to handle such a complex system is a key issue in today’s data science practice.

A typical data science book focuses mostly on introducing models and how to apply them to data, but Data Science Foundations is not typical. Besides introducing the methods, this book emphasizes the foundations for these methods, with focus on the mathematical and computational thinking process of readers. The mathematical facts behind the methods are made simple and accessible for readers. The book covers a large scope of interests, and fits the needs of readers from different areas of data science, ranging from statistics, to machine learning, to applied mathematics. Free R software is used to implement most examples and source codes are made available to readers.

The book mainly discusses problems arising from complex systems and the geometry and properties of these systems. The chapters are quite independent with each other, because the author tried to make the scenarios self-contained and cover as many scenarios as possible.

The first part of the book (Chapters 1 and 2) introduces several motivating examples on text analysis, including the study of anomaly detection in film scripts, stylistic continuation of composers, and methods for generating and synthesizing film, TV, and game scripts. Text analysis is certainly one of the most popular topics nowadays, with application areas across various scenarios, for example, knowledge mining, machine translation, and human-computer dialogue. One problem of text data is that the structure of texts (words, phrases, sentences, and paragraphs) is not well defined in ways that a computer can understand. Typical methods to handle such data often start with imposing some well-defined structure that best fits the data. A novel solution of using a hierarchical relation within the text is proposed and demonstrated to have some advantages, including being able to show the time dependency of scenes within scripts. The hierarchic relation is further lifted to a more general class of metrics: ultrametrics. The ultrametric has a slightly different definition from a regular metric: the triangle inequality of an ultrametric is defined as $d(x,z)\le \max (d(x,y),d(y,z))$, while in a regular metric space we only require $d(x,z)\le d(x,y)+d(y,z)$.

The second half of the book focuses on the scalability issue in modern data science. As the data size grows, two major challenges appear: (1) computational cost increases exponentially; (2) aggregated noise contamination may dominate the true signal. Chapter 5 mainly focuses on how to reduce the computational burden, with illustrations from search and discovery. A hierarchical clustering algorithm is demonstrated to achieve linear time cost increase when the data size scales if the ultrametric space is adopted, while the typical increase in speed of the computation cost of a hierarchical clustering algorithm is exponential. In terms of reducing noise and extracting signal, Chapter 6 explores the scaling of ultrametric through metric mapping. The mapping is a typical way in which ultra-high dimensional data are projected to a lower dimensional space and the metric is mapped at the same time from any types of ultrametric to Euclidean. Such a mapping builds a unified way of connecting regular metric space and ultrametric space; therefore, one may enjoy the good property coming with ultrametric space and still be able to apply general statistical methods designed for Euclidean space. More concrete examples of employing ultrametric spaces are discussed in Chapters 8 and 9, with focus on text analysis.

To sum up, big and complex data analysis is a growing field full of challenges and interesting problems. The author has provided the field a new view of the complex system, and solid mathematical foundations of corresponding methodologies. As a data scientist working with ultra-complex systems every day and a former mathematics researcher, I am excited to see some new approaches to these complex text analysis problems. It is worthwhile to mention that these approaches are derived from the literature of pure mathematics, which is often considered not directly applicable to real world problems. I will recommend the book to data scientists who want to extend their field of expertise, and to gain a different yet powerful view on complex systems.