Geometric Matrix Midranges

The matrix geometric mean

An attractive candidate for the geometric mean of m positive definite matrices A1, . . . , Am is their Riemannian barycentre G. One of its important properties, monotonicity in the m arguments, has been established recently by J. Lawson and Y. Lim. We give a much simpler proof of this result, and prove some other inequalities. One of these says that, for every unitarily invariant norm, |||G||| ...

Constructing matrix geometric means

In this paper, we analyze the process of “assembling” new matrix geometric means from existing ones, and show what new means can be found, and what cannot be done because of group-theoretical obstructions. We show that for n = 4 a new matrix mean exists which is simpler to compute than the existing ones. Moreover, we show that for n > 4 the existing strategies of composing matrix means and taki...

Convolutional Geometric Matrix Completion

Geometric matrix completion (GMC) has been proposed for recommendation by integrating the relationship (link) graphs among users/items into matrix completion (MC) . Traditional GMC methods typically adopt graph regularization to impose smoothness priors for MC. Recently, geometric deep learning on graphs (GDLG) is proposed to solve the GMC problem, showing better performance than existing GMC m...

Ela Constructing Matrix Geometric Means

In this paper, we analyze the process of “assembling” new matrix geometric means from existing ones, through function composition or limit processes. We show that for n = 4 a new matrix mean exists which is simpler to compute than the existing ones. Moreover, we show that for n > 4 the existing proving strategies cannot provide a mean computationally simpler than the existing ones.

The matrix arithmetic-geometric mean inequality revisited