Minimization Problems Based on Relative \(α\)-Entropy II: Reverse Projection - M Ashok Kumar; Sundaresan, Rajesh (arXiv.org, 2015)
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.