Statistical Depth Meets Machine Learning: Kernel Mean Embeddings and Depth in Functional Data Analysis

Abstract

Statistical depth is the act of gauging how representative a point is compared with a reference probability measure. The depth allows introducing rankings and orderings to data living in multivariate, or function spaces. Though widely applied and with much experimental success, little theoretical progress has been made in analysing functional depths. This article highlights how the common $h$-depth and related depths from functional data analysis can be viewed as a kernel mean embedding, widely used in statistical machine learning. This facilitates answers to several open questions regarding the statistical properties of functional depths. We show that (i) $h$-depth has the interpretation of a kernel-based method; (ii) several $h$-depths possess explicit expressions, without the need to estimate them using Monte Carlo procedures; (iii) under minimal assumptions, $h$-depths and their maximisers are uniformly strongly consistent and asymptotically Gaussian (also in infinite-dimensional spaces and for imperfectly observed functional data); and (iv) several $h$-depths uniquely characterise probability distributions in separable Hilbert spaces. In addition, we also provide a link between the depth and empirical characteristic function based procedures for functional data. Finally, the unveiled connections enable to design an extension of the $h$-depth towards regression problems.