Minimization Problems Based on Relative -Entropy II: Reverse Projection

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