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Minimization Problems Based on Relative $α$-Entropy II: Reverse Projection

In part I of this two-part work, certain minimization problems based on a parametric family of relative entropies (denoted $\mathscr{I}_α$) were studied. Such minimizers were called forward $\mathscr{I}_α$-projections. Here, a complementary class of minimization problems leading to the so-called reverse $\mathscr{I}_α$-projections are studied. Reverse $\mathscr{I}_α$-projections, particularly on log-convex or power-law families, are of interest in robust estimation problems ($α>1$) and in constrained compression settings ($α<1$). Orthogonality of the power-law family with an associated linear family is first established and is then exploited to turn a reverse $\mathscr{I}_α$-projection into a forward $\mathscr{I}_α$-projection. The transformed problem is a simpler quasiconvex minimization subject to linear constraints.