User:Lazydry/sandbox

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