Coupling matrix manifolds assisted optimization for optimal transport problems

Optimal transport (OT) is a powerful tool for measuring the distance between two probability distributions. In this paper, we develop a new manifold named the coupling matrix manifold (CMM), where each point on CMM can be regarded as a transportation plan of the OT problem. We firstly explore the Riemannian geometry of CMM with the metric expressed by the Fisher information. These geometrical features of CMM have paved the way for developing numerical Riemannian optimization algorithms such as Riemannian gradient descent and Riemannian trust region algorithms, forming an essential optimization method for all types of OT problems. The proposed method is then applied to solve several OT problems studied by recent literature. For the classic OT problem and its entropy regularized variant, the OT solution generated from our method is comparable to that from the classic algorithms (i.e. Linear programming and Sinkhorn algorithms), while for other types of non-entropy regularized OT problems our method outperforms other state-of-the-art algorithms which don’t incorporate the geometric information of the OT feasible space

Surprising Implications of Differences in Locations Versus Differences in Means

Social science researchers depend on differences in means between experimental and control conditions to draw substantive conclusions. However, an alternative is to use differences in locations. For normal distributions, means and locations are the same, but for skew normal distributions, means and locations are different. If a difference in means and locations are similar, and in the same direction, the resulting substantive story may be similar. However, if a difference in means and locations are dissimilar, especially if they oppose directionally, the resulting substantive story may differ dramatically. We collected 51 data sets from online data repositories to check how often the differences in means versus locations are substantially different or are in different directions. Although the values depend on what one counts, the overall conclusion is that the two types of differences have a larger than trivial chance of disagreeing substantially. We suggest that when researchers report normal statistics (mean and standard deviation), they should report skew normal statistics (location, scale, and shape) too, against the nontrivial chance that the skew normal statistics imply a substantive story in opposition to that implied by the normal statistics

In Vivo Dynamics of Rac-Membrane Interactions

The small GTPase Rac cycles between the membrane and the cytosol as it is activated by nucleotide exchange factors (GEFs) and inactivated by GTPase-activating proteins (GAPs). Solubility in the cytosol is conferred by binding of Rac to guanine-nucleotide dissociation inhibitors (GDIs). To analyze the in vivo dynamics of Rac, we developed a photobleaching method to measure the dissociation rate constant (k(off)) of membrane-bound GFP-Rac. We find that k(off) is 0.048 s(−1) for wtRac and ∼10-fold less (0.004 s(−1)) for G12VRac. Thus, the major route for dissociation is conversion of membrane-bound GTP-Rac to GDP-Rac; however, dissociation of GTP-Rac occurs at a detectable rate. Overexpression of the GEF Tiam1 unexpectedly decreased k(off) for wtRac, most likely by converting membrane-bound GDP-Rac back to GTP-Rac. Both overexpression and small hairpin RNA-mediated suppression of RhoGDI strongly affected the amount of membrane-bound Rac but surprisingly had only slight effects on k(off). These results indicate that RhoGDI controls Rac function mainly through effects on activation and/or membrane association