Recently a lot of work has been done on geometric means of positive definite matrices (see here and here for example). Has anyone extended this concept to larger sets of matrices (copositive, for instance)?
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asked Aug 14, 2013 at 9:45
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3 Answers
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The Karcher (or Cartan) mean is defined as a distance minimizer and works on other manifolds; for instance, people studied means of orthogonal matrices (for instance, http://epubs.siam.org/doi/abs/10.1137/S0895479801383877?journalCode=sjmael, http://dl.acm.org/citation.cfm?id=1721067), which have applications in optics and medical imaging. Maher Moakher, Pierre-Antoine Absil and Jonathan H. Manton worked on generic gradient descent methods that work in any Riemannian manifold. The standard examples are the usual arithmetic mean, the geometric mean on the positive cone, and the mean on the orthogonal (Stiefel) manifold, but maybe they have something more exotic in their "examples" sections.
The Ando-Li-Mathias paper (http://dx.doi.org/10.1016/j.laa.2003.11.019) defines by continuity means of semidefinite matrices as well, but that's not a very meaningful extension.
Bini and Iannazzo are working on means of Toeplitz and similarly structured matrices (http://poisson.phc.unipi.it/~maxreen/bruno/pdf/D.%20Bini,%20B.%20Iannazzo,%20B.%20Jeuris%20and%20R.%20Vandebril%20-%20Geometric%20means%20of%20structured%20matrices%20-%20Preprint.pdf). But that's more of a restriction than an extension.
That's all I know on extensions of the concept to other sets of matrices; as far as I know there is nothing specific on copositive.
answered Aug 14, 2013 at 10:09
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The cone of copositive matrices is convex, but it is not a symmetric cone. So the common notion of "geometric means" on this cone will look different from geometric means on symmetric spaces. See for example, Section 2 of this recent paper for some discussion of geometric means on symmetric spaces.
Even if you were able to define a Cartan mean on the set of copositive matrices, it'll probably be uncomputable, and therefore not so useful (primarily because testing membership in this cone is hard).
A related, interesting direction worth considering might be to instead consider the cone of double nonnegative matrices; this is (as you are I guess well aware) a subset of the CP matrices, while still being computationally tractable.
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Another viewpoint, some inequalities of definite positive matrix, for example arithmetic-geometric means inequality, harmonic-geometric mean inquality,... are some special cases of more general things. That is called connection. Let $M_n^+ = \{ A \in M_n | A\ \text{is a semi-definite matrix} \}$
An connection is a binary operation $\sigma: M^+_n \times M^+_n \rightarrow M^+_n$ sastisfies:
1, $A \le B, C \le D \Rightarrow A \sigma C \le B \sigma D$.
2, $C(A \sigma B) C \le (CAC) \sigma (CBC)$.
3, $\sigma$ is continuous.
Mean is a connection with $1 \sigma 1 = 1$.
Example: $A \nabla B = \dfrac{A+B}{2}$, geometric mean is a connection.
We can get many ralations between the connection and some matric inequalities, for eaxample trace inequalities: http://en.wikipedia.org/wiki/Trace_inequalities
You can search on internet with keywords connections matrix inequalities.
