Abstract

Math example $\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}$

Let ${\mathcal {F}}$ be a space of probability distributions equipped with densities. Consider univariate cases (Need to distinguish dimension of the domain; Wasserstein metric is based on quantile function, which doesn't exist in dimension >2. mostly focus on m=1) Let $\mathbf {d}$ (?) be a metric for ${\mathcal {F}}$ . L^p on cdf, L^p on densities, F-R, Wasserstein (connected to optimal transport, (almost) equal to L^2 metric on quantile function, W_2)

Densities as responses

Transformation based approaches

${\mathcal {F}}\to L^{2}$ , and do things there.. functional regression

one to one transformation

Fréchet regression

Works for this case only (predictors are Rp)

Wasserstein regression

Works when predictors are densities too. Pseudo-Riemanian, infinite-dimensional space, find tangent bundles

Fisher-Rao regression

Same idea as Wasserstein regression,

Other approaches

In the paper.

Densities as predictors

Refer to functional regression. Not much restrictions in this case