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The aim of this work is to establish that two recently published projectiontheorems, one dealing with a parametric generalization of relative entropy andanother dealing with R\'{e}nyi divergence, are equivalent under acorrespondence on the space of probability measures. Further, we demonstratethat the associated "Pythagorean" theorems are equivalent under thiscorrespondence. Finally, we apply Eguchi's method of obtaining Riemannianmetrics from general divergence functions to show that the geometry arisingfrom the above divergences are equivalent under the aforementionedcorrespondence.

On The Equivalence of Projections In Relative $\alpha$-Entropy and R\'{e}nyi Divergence