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Minimization Problems Based on Relative alpha-Entropy I: Forward Projection

Kumar, Ashok M and Sundaresan, Rajesh (2015) Minimization Problems Based on Relative alpha-Entropy I: Forward Projection. In: IEEE TRANSACTIONS ON INFORMATION THEORY, 61 (9). pp. 5063-5080.

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Official URL: http://dx.doi.org/10.1109/TIT.2015.2449311


Minimization problems with respect to a one-parameter family of generalized relative entropies are studied. These relative entropies, which we term relative alpha-entropies (denoted I-alpha), arise as redundancies under mismatched compression when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative alpha-entropies behave like squared Euclidean distance and satisfy the Pythagorean property. Minimizers of these relative alpha-entropies on closed and convex sets are shown to exist. Such minimizations generalize the maximum Renyi or Tsallis entropy principle. The minimizing probability distribution (termed forward I-alpha-projection) for a linear family is shown to obey a power-law. Other results in connection with statistical inference, namely subspace transitivity and iterated projections, are also established. In a companion paper, a related minimization problem of interest in robust statistics that leads to a reverse I-alpha-projection is studied.

Item Type: Journal Article
Additional Information: Copy right for this article belongs to the IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC, 445 HOES LANE, PISCATAWAY, NJ 08855-4141 USA
Keywords: Best approximant; exponential family; information geometry; Kullback-Leibler divergence; linear family; power-law family; projection; Pythagorean property; relative entropy; Renyi entropy; Tsallis entropy
Department/Centre: Division of Electrical Sciences > Electrical Communication Engineering
Date Deposited: 24 Sep 2015 04:49
Last Modified: 24 Sep 2015 04:49
URI: http://eprints.iisc.ac.in/id/eprint/52389

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