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Statistical depth is the act of gauging how representative a point iscompared to a reference probability measure. The depth allows introducingrankings and orderings to data living in multivariate, or function spaces.Though widely applied and with much experimental success, little theoreticalprogress has been made in analysing functional depths. This article highlightshow the common $h$-depth and related statistical depths for functional data canbe viewed as a kernel mean embedding, a technique used widely in statisticalmachine learning. This connection facilitates answers to open questionsregarding statistical properties of functional depths, as well as it provides alink between the depth and empirical characteristic function based proceduresfor functional data.

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