Metrics defined by Bregman divergences. II (Q1011862)

For any strictly convex function \(f\), the Bregman divergence \(B_f(x,y)\) is defined by \(B_f(x,y) : = f(x) - f(y) - (x-y)f'(y)\). (For the sake of simplicity the authors assume that all convex functions mentioned in the paper are \(C^{\infty}\)-smooth.) In general, \(B_f(x,y)\) is not symmetric. To overcome this, first the authors define a min-max procedure and illustrate its relationship to the typical min-max problem in convex programming. Second, they derive necessary and sufficient conditions that the min-max procedure lead to a metric in the case of scalars. Then the authors generalize this result to the case of vector spaces for two separate important cases. One is ``capacity'' associated with \(f(x):= x\log x\) and the other one is the min-max Itakura-Saito (IS) distance associated with \(f(x):= -\log x\). It is shown that the ``capacity'' to the power \(1/e\) is a metric and that the square root of the min-max IS distance is a metric.