5446f217e9504bc593ad9dcf2ec88dda-Supplemental.pdf

Python notebooks producing the figures of this paper are available at https://github.com/ Let F be the joint distribution on RTD obtained by first assigning a random vector =( 1| | T) via F and then applying the map to each of the components t, and let F 1,..., F T denote the corresponding marginal distributions on RD. Given F and t F t, define the second moment matrices = E[ >] 2 RTD TD and t = E[ t >t ] 2 RD D, and let r = rank() and rt = rank( t). Let M 2 Rr TD be a matrix whose rows form a basis of supp(F), and similarly let Nt 2 Rrt D be a matrix whose rows form a basis of supp(F t). Let N = diag(N1,..., NT), and define the Writing V =( V1| | VT), where Vt 2 Rrt d has rank dr, we can then construct the singular value decompositions Vt = Ut tW>t, with Ut 2 O(rt dt), t 2 Rdt dt and Wt 2 O(d dt), where dt = rank(Vt).