Singularity of Data Analytic Operations

This book consists of the first three chapters in a longer work under preparation. In this book we apply mainly topological methods to study the (topological) singularities of functions defined (on dense subsets of) "data spaces" and taking values in spaces with nontrivial homology. The purpose is to gain insight into the numerical conditioning of statistical description, data summarization, and inference and learning methods. We prove general results that can often be used to bound below the dimension of singular sets. In future chapters we will apply our topological results to develop lower bounds on Hausdofff measure of the singular set and apply the theory to common data analytic operations.