The matrix geometric mean of parameterized, weighted arithmetic and harmonic means

作者：Kim Sejong; Lawson Jimmie^*; Lim Yongdo

来源：Linear Algebra and Its Applications, 2011, 435(9): 2114-2131.

DOI：10.1016/j.laa.2011.04.010

• Matrix mean
• Positive definite matrix
• Kullback-Leibler divergence
• Weighted mean

We define a new family of matrix means {L-mu(omega; A)}(mu is an element of R) where omega and A vary over all positive probability vectors in R-m and m-tuples of positive definite matrices resp. Each of these means interpolates between the weighted harmonic mean (mu = -infinity) and the arithmetic mean of the same weight (mu = infinity) with L mu <= L-nu for mu <= nu. Each has a variational characterization as the unique minimizer of the weighted sum for the symmetrized, parameterized Kullback-Leibler divergence. Furthermore, each can be realized as the common limit of the mean iteration by arithmetic and harmonic means (in the unparameterized case), or, more generally, the arithmetic and resolvent means. Other basic typical properties for a multivariable mean are derived.

• 出版日期2011-11-1