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

In part I of this two-part work, certain minimization problems based on a parametric family of relative entropies (denoted Iα) were studied. Such minimizers were called forward Iα-projections. Here, a complementary class of minimization problems leading to the so-called reverse Iα-projections are studied. Reverse Iα-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 Iα-projection into a forward Iα-projection. The transformed problem is a simpler quasiconvex minimization subject to linear constraints.