Singularity of Data Analytic Operations
Steven P. Ellis
TL;DR
The work develops a general topological framework for understanding instability in statistical data analyses by modeling analysis procedures as data maps $\Phi: \mathcal{D} \dashrightarrow \mathsf{F}$. It defines singularities where limits fail to exist and uses test-patterns $\mathcal{T}$ and perfect-fit sets $\mathcal{P}$ to derive global obstructions to continuity via algebraic topology. The core contributions include a Sales Pitch that links local behavior near $\mathcal{T}$ to global singularity structure, lower bounds on the Hausdorff dimension and measure of the singular set, and a dilation-based, cone-fiber geometric approach to bound singularities in plane-fitting, linear location, and classification problems. The framework culminates in a main theorem that provides quantitative, geometry-driven limits on the size of singular sets in terms of their distance to perfect fits, with a detailed construction of tubular neighborhoods and conical fibers. Practically, these results illuminate why certain data-analytic methods must exhibit instability and guide the design of calibrated, robust alternatives. Overall, the paper offers a rigorous, topology-grounded lens for assessing and bounding the ill-conditioning of broad data-analytic methods.
Abstract
Statistical data by their very nature are indeterminate in the sense that if one repeats the process of collecting the data the new data set will be different from the original. But two data sets generated in the same way should ``tell the same story''. Therefore, a statistical method, a map $Φ$ taking a data set $x$ to a point in some space $\mathsf{F}$, should be stable at $x$: Small perturbations in $x$ should result in a small change in $Φ(x)$. Otherwise, $Φ$ is useless at $x$ or -- and this is important -- near $x$. So one doesn't want $Φ$ to have "singularities," data sets $x$ such that the the limit of $Φ(y)$ as $y$ approaches $x$ doesn't exist. (The same issue arises elsewhere in applied math.) We prove that broad classes of statistical methods have topological obstructions to continuity: They must have singularities. We derive broadly applicable lower bounds on the Hausdorff dimension, even Hausdorff measure, of the set of singularities of data maps. General results concerning severity of singularities are proved. For illustration, we show our results apply to plane fitting, measuring location of data on spheres, and to linear classification. This is not a "final" version, merely another attempt.