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

arXiv: https://arxiv.org/abs/2105.12778 · HTML: https://mlsanity.com/paper/2105.12778/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 120 references that are in the map.
- First posted on arXiv 26 May 2021.

## Builds on (most cited first, 22 of 22)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [A Primer on the Signature Method in Machine Learning](https://mlsanity.com/paper/1603.03788/) (2016)
- [Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/1611.04488/) (2016)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [Signature moments to characterize laws of stochastic processes](https://mlsanity.com/paper/1810.10971/) (2018)
- [Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions](https://mlsanity.com/paper/1604.05251/) (2016)
- [Stein Point Markov Chain Monte Carlo](https://mlsanity.com/paper/1905.03673/) (2019)
- [A Kernel Two-Sample Test for Functional Data](https://mlsanity.com/paper/2008.11095/) (2020)
- [Probabilistic Integration: A Role in Statistical Computation?](https://mlsanity.com/paper/1512.00933/) (2015)
- [Choosing among notions of multivariate depth statistics](https://mlsanity.com/paper/2004.01927/) (2020)
- [Exact computation of the halfspace depth](https://mlsanity.com/paper/1411.6927/) (2014)
- [Lens depth function and k-relative neighborhood graph: versatile tools for ordinal data analysis](https://mlsanity.com/paper/1602.07194/) (2016)
- [On Mahalanobis distance in functional settings](https://mlsanity.com/paper/1803.06550/) (2018)
- [Post-Processing of MCMC](https://mlsanity.com/paper/2103.16048/) (2021)
- [On data depth in infinite dimensional spaces](https://mlsanity.com/paper/1402.2775/) (2014)
- [The DDG-classifier in the functional setting](https://mlsanity.com/paper/1501.00372/) (2015)
- [Classification with the pot-pot plot](https://mlsanity.com/paper/1608.02861/) (2016)
- [A unified framework on defining depth for point process using function smoothing](https://mlsanity.com/paper/2105.08893/) (2021)

## Built on by (most cited first, 6 of 6)

- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
- [Composite Goodness-of-fit Tests with Kernels](https://mlsanity.com/paper/2111.10275/) (2021)
- [Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD](https://mlsanity.com/paper/2607.24235/) (2026)
- [Bayes Hilbert Spaces for Posterior Approximation](https://mlsanity.com/paper/2304.09053/) (2023)
- [Model-Free Kernel Conformal Depth Measures Algorithm for Uncertainty Quantification in Regression Models in Separable Hilbert Spaces](https://mlsanity.com/paper/2506.08325/) (2025)
- [Dimension-invariant uniform consistency of the empirical spatial distribution function and its associated spatial depth estimator](https://mlsanity.com/paper/2607.16092/) (2026)

## Abstract (arXiv)

Statistical depth is the act of gauging how representative a point is compared to 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 statistical depths for functional data can be viewed as a kernel mean embedding, a technique used widely in statistical machine learning. This connection facilitates answers to open questions regarding statistical properties of functional depths, as well as it provides a link between the depth and empirical characteristic function based procedures for functional data.