# Eikonal depth: an optimal control approach to statistical depths

arXiv: https://arxiv.org/abs/2201.05274 · HTML: https://mlsanity.com/paper/2201.05274/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 65 references that are in the map.
- First posted on arXiv 14 Jan 2022.

## Builds on (most cited first, 19 of 19)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Robustness via curvature regularization, and vice versa](https://mlsanity.com/paper/1811.09716/) (2018)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Poisson Learning: Graph Based Semi-Supervised Learning At Very Low Label Rates](https://mlsanity.com/paper/2006.11184/) (2020)
- [Distributionally Robust Learning](https://mlsanity.com/paper/2108.08993/) (2021)
- [Analysis of p-Laplacian Regularization in Semi-Supervised Learning](https://mlsanity.com/paper/1707.06213/) (2017)
- [Adversarial Risk via Optimal Transport and Optimal Couplings](https://mlsanity.com/paper/1912.02794/) (2019)
- [Continuum limit of total variation on point clouds](https://mlsanity.com/paper/1403.6355/) (2014)
- [The Geometry of Adversarial Training in Binary Classification](https://mlsanity.com/paper/2111.13613/) (2021)
- [Adversarial Classification: Necessary conditions and geometric flows](https://mlsanity.com/paper/2011.10797/) (2020)
- [Choosing among notions of multivariate depth statistics](https://mlsanity.com/paper/2004.01927/) (2020)
- [A maximum principle argument for the uniform convergence of graph Laplacian regressors](https://mlsanity.com/paper/1901.10089/) (2019)
- [Large sample spectral analysis of graph-based multi-manifold clustering](https://mlsanity.com/paper/2107.13610/) (2021)
- [The limit shape of convex hull peeling](https://mlsanity.com/paper/1805.08278/) (2018)
- [A new analytical approach to consistency and overfitting in regularized empirical risk minimization](https://mlsanity.com/paper/1607.00274/) (2016)
- [Data depth and floating body](https://mlsanity.com/paper/1809.10925/) (2018)
- [Boundary Estimation from Point Clouds: Algorithms, Guarantees and Applications](https://mlsanity.com/paper/2111.03217/) (2021)
- [Tukey Depths and Hamilton-Jacobi Differential Equations](https://mlsanity.com/paper/2104.01648/) (2021)

## Built on by (most cited first, 1 of 1)

- [Hamilton-Jacobi equations on graphs with applications to semi-supervised learning and data depth](https://mlsanity.com/paper/2202.08789/) (2022)

## Abstract (arXiv)

Statistical depths provide a fundamental generalization of quantiles and medians to data in higher dimensions. This paper proposes a new type of globally defined statistical depth, based upon control theory and eikonal equations, which measures the smallest amount of probability density that has to be passed through in a path to points outside the support of the distribution: for example spatial infinity. This depth is easy to interpret and compute, expressively captures multi-modal behavior, and extends naturally to data that is non-Euclidean. We prove various properties of this depth, and provide discussion of computational considerations. In particular, we demonstrate that this notion of depth is robust under an aproximate isometrically constrained adversarial model, a property which is not enjoyed by the Tukey depth. Finally we give some illustrative examples in the context of two-dimensional mixture models and MNIST.