# curvature of curves in the space of gaussians measures

I have a sequence of symmetric positive definite matrices in $GL(n)$ (in my case, covariance matrices of some gaussians) and vectors $R^n$ (the mean of these gaussians). I consider this sequence to be the discretization of a curve embedded in $GL(n)\times R^n$.

I would like to compute the embedding curvature of this curve at each of my sample points, using the Wasserstein geometry (see Wasserstein Geometry of Gaussian Measures), not information geometry (like The Riemannian Geometry of the Space of Positive-Definite Matrices ...).

I have some difficulties understanding the parameterization of the space of SPD matrices in the Wasserstein Geometry article above and thus, I am not able to compute the Levi-Civita connection and its projection on my curve with respect to the metric. Could anyone help me understand that (or anything else you feel I might not have understood - I feel I barely understand these things) ? Does anyone know of relevant work ?

Thanks!

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## 1 Answer

Your question is not specific enough about what you do not understand in the quoted paper; if you want help on this, you should at least explain what you understand and where the problem appears. Here is a little information about bibliography, that might help (or miss the point, I am not sure).

For an introduction to optimal transport and Wasserstein spaces, you can have a look at Villani's books ("Topics on ..." is more elementary, but the beginning of "... Old and New" is not as difficult to read as the size of the book might lead you to think, and I like it a lot). A more concise introduction can also be found in a nice little book by Nicola Gigli, a version of which seems to be at http://math.unice.fr/~gigli/Site_2/Publications_files/users_guide%20-%20final.pdf (but I am not sure this is exactly the text I read).

You should also now about Lott's paper "Some geometric calculations on Wasserstein space", Comm. Math. Phys. 277, p. 423-437, which computes the curvature of the Wasserstein space of a manifold.

Concerning the notion of curvature of a discretized curve, you might be interested in the concept of Menger's curvature, which applies in a very broad context.

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I read Villani's topics, and parts of its "Old and New", and Lott's paper. What I don't understand here is specific to matrices that is not addressed in the refs above: what parameterization is used to represent the matrices ? What is the basis to represent the tangent space (ie., symm. matrices) ? To compute the Christoffel symbols of the Levi-Civita connection, I need to be able to differentiate the metric along the vectors of a local basis... but which one ? Can I just take, for example for a 2x2 matrix, the basis consisting of the 3 matrices : e0=[1 0;0 0], e1=[0 1; 1 0], e2=[0 0; 0 1] ? – WhitAngl Aug 18 '12 at 19:33
I do not know the details of Takatsu's paper, I think you should ask a more precise question about this parametrization. Are you familiar with Lie groups? If not, here is probably the problem. Then a good read is the chapter 0 of Knapp's "Lie groups, beyond an introduction". – Benoît Kloeckner Aug 18 '12 at 20:57
I am not very comfortable with Lie groups, so I'll check this book. Thanks! :) – WhitAngl Aug 19 '12 at 6:20