Regression in Nonstandard Spaces with Fréchet and Geodesic Approaches
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One approach to tackle regression in nonstandard spaces is Fréchet regression, where the value of the regression function at each point is estimated via a Fréchet mean calculated from an estimated objective function. A second approach is geodesic regression, which builds upon fitting geodesics to observations by a least squares method. We compare these two approaches by using them to transform three of the most important regression estimators in statistics - linear regression, local linear regression, and trigonometric projection estimator - to settings where responses live in a metric space. The resulting procedures consist of known estimators as well as new methods. We investigate their rates of convergence in general settings and compare their performance in a simulation study on the sphere.
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