Daniel-Packer
commented
4 days ago Hi @ttsesm ! Sorry for that late reply. I know I might be too late on this. However, in case I'm not, shoot me an email at daniel.the.packer@gmail.com, and we can schedule a zoom meeting to chat about it!
ttsesm
Dear @Daniel-Packer ,
I've recently discovered your work and thus I am interested about it. I am quite new to manifolds and/or optimal transport in general thus I am not sure whether I understood correctly your work but in any case I hope I can get some feedback.
Briefly, I am trying to compute the distance between gaussian distributions of different dimensions (in my case 2D and 3D) and find their coupling (matchings or the best correspondences between the two sets). Taking into account that and after doing some research in the literature, I considered using the Gromov-Wasserstein (GW) distance based on cost matrices computed on the Bures manifold (I've tried using different optimal transport libraries like POT or OTT for playing with these distances and manifolds).
In a more practical way, imagine that you have the following test case where I have a set of 3D gaussian distributions defined as ellipsoids by their mean value $μ$ (being a 1x3 vector) and their corresponding covariance matrices $Σ1$ (being a 3x3 matrix). Then I have a set of 2D gaussian distributions defined as ellipses again by their mean value $ν$ (being a 1x2 vector) and their corresponding covariance matrices $Σ2$ (being a 2x2 matrix). So in practice the ellipses are the projected ellipsoids in a plane through a linear map P, $\mathbb{R}^3 \rightarrow \mathbb{R}^2$.
So, in simple words I want to find which ellipsoid corresponds to which ellipse by using their mean and covariance information and minimizing the distance between them. In the ideal case scenario, the coupling will one-to-one, however in our real case scenario we will have outliers. Meaning that not all ellipsoids will match with all ellipses for that reason I considered using an approach based on unbalanced or partial Gromov-Wasserstein distance or something similar [1], [2]. Considering though that the two distributions are not in the same space we need somehow to bring them into the same space where we can apply the matching and find the coupling and the transport plan.
For this purpose we thought that a suitable approach would be to first compute the cost matrices of each set in the Bures Wasserstein manifold (this should create something similar to a weighted graph of all distributions with all distributions) and then compute their correspondence with the Gromov-Wasserstein distance. I've tried to play a bit with an oracle test ablation study of 50 ellipsoids with 50 ellipses where there is always a one-to-one correspondence, meaning the first 2D gaussian corresponds to the first 3D gaussian, the second with the second, the third with the third and so on. So at the end the coupling should be an identity matrix. However as you can see from the attached image I am not really getting the desired output.
Sometimes I am able to recover the correct solution either by using the pure GW or the entropic-GW (also playing a bit with the regularization parameters) as you can see in the graphs bellow:
but it is not robust, meaning that not always working especially if I introduce outliers. I guess the problem is that there are quite a few local minima where the optimization procedure is getting stuck.
Thus, my question would be whether this work you think that I would be suitable for such a task. I am not sure whether these low and upper bounds and the other modifications that you are introducing in this work would make any difference. Also if I understand it correctly you are working with discrete samples, i.e. points on the surface of the spheres, rather continuous distributions right?
Thank you for your time and I am looking forward to hearing back from you.