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Eikonal depth: An optimal control approach to statistical depths

  • *Corresponding author: Ryan Murray

    *Corresponding author: Ryan Murray
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  • 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 locations on the boundary of the support of the distribution or spatial infinity for distributions with unbounded support. 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 approximate 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.

    Mathematics Subject Classification: Primary: 62H05, 62G35; Secondary: 68Q87.

    Citation:

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  • Figure 1.  The figure shows, for $ \phi(s) = s $, the level sets of the eikonal depth of a distribution whose densities have compact support (uniform density on a square in Figure 1a) and support on all of $ \mathbb{R}^d $ (a mixture of Gaussians in Figure 1b). See Example 5 for more details on mixtures of Gaussian distributions

    Figure 2.  For $ \phi(s) = s^{\alpha} $, the figures show the effect of $ \alpha $ on the eikonal depth of a Gaussian mixture. Figure 2a shows the probability density for the Gaussian mixture. Figures 2b, 2c, and 2d show some level sets of the eikonal depth (for $ \alpha = 0.5 $, $ \alpha = 1 $, and $ \alpha = 2 $) of the probability distribution with density as depicted in 2a

    Figure 3.  The figure shows some level sets of the eikonal depth of a Gaussian mixture with $ \phi(s) = s^{1/2} $ and some level sets of the corresponding affine invariant version

    Figure 4.  Figure 4a shows some level sets of the eikonal depth for a Gaussian mixture model with means that are four standard deviations apart. The contribution of any of the components is negligible at two standard deviations from the other mean. Figure 4b shows some level sets for the eikonal depth for a Gaussian mixture model with means that are two standard derivations apart

    Figure 5.  The figure shows examples of the eikonal depth of some densities on manifolds. Figure 5c shows only the points in 5b with $ x \leq 0 $

    Figure 6.  The figure shows some representative points belonging to different level sets of the eikonal depth on the elements in MNIST labeled as 4's. The number of representatives in each depth range indicates the proportion of 4's in each range. We can observe that points lying deep in the distribution seem to be neat examples of a hand-written '4'

    Figure 7.  Figure 7a shows some representative points belonging to different level sets of the eikonal depth on the elements in MNIST labeled as 4's when the boundary of this set is defined using an approximation to the density. Figure 7b shows some of the elements on this boundary

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