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

• Published in: arXiv.org
• Main Authors: M Ashok Kumar, Sundaresan, Rajesh
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 10.06.2015
• Subjects: Optimization; Orthogonality; Power law; Projection
• ISSN: 2331-8422

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