Deformed Statistics Formulation of the Information Bottleneck Method
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The theoretical basis for a candidate variational principle for the information bottleneck (IB) method is formulated within the ambit of the generalized nonadditive statistics of Tsallis. Given a nonadditivity parameter q , the role of the additive duality of nonadditive statistics ( q^*=2-q ) in relating Tsallis entropies for ranges of the nonadditivity parameter q < 1 and q > 1 is described. Defining X , X̃ , and Y to be the source alphabet, the compressed reproduction alphabet, and, the relevance variable respectively, it is demonstrated that minimization of a generalized IB (gIB) Lagrangian defined in terms of the nonadditivity parameter q^* self-consistently yields the nonadditive effective distortion measure to be the q -deformed generalized Kullback-Leibler divergence: D_K-L^q[p(Y|X)||p(Y|X̃)] . This result is achieved without enforcing any a-priori assumptions. Next, it is proven that the q^*-deformed nonadditive free energy of the system is non-negative and convex. Finally, the update equations for the gIB method are derived. These results generalize critical features of the IB method to the case of Tsallis statistics.
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