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In this talk, I will discuss my recent work on entropic optimal transport and Gromov-Wasserstein (GW) alignment. For the former, I will discuss stability of the entropic potentials with respect to the varying marginals and its statistical consequences. For the latter, I will discuss computational guarantees for entropic relaxation of the GW distance, which quantifies discrepancy between metric measure spaces, and limiting distributions for the empirical GW distance in several settings of interest. Again, key to our analysis is the stability estimates of the GW problem with respect to the auxiliary matrix that appears in the variational representation of the GW distance and varying marginals.
