Convergence rate of general entropic optimal transport costs

Convergence rate of entropy-regularized multi-marginal optimal transport costs

“…By using the singular values decomposition of the bilinear form obtained as an average of mixed second derivatives of the cost and a signature condition introduced in [Pas11], we are able to prove that E detaches quadratically from the set {E = 0} and this allows us to estimate the previous integral in the desired way as in [CPT23] and improve the results in [EN23] where only an upper bound depending on the quantization dimension of the solution to the unregularized problem is provided. Moreover, this slightly more flexible use of Minty's trick compared to [CPT23] allows us to obtain a lower bound also for degenerate cost functions in the two marginals setting. Given a κ depending on a signature condition (see (PS(κ))) on the second mixed derivatives of the cost, the lower bound can be summarized as follows:…”

“…In particular, we are going to extend the techniques introduced in [CPT23] for two marginals to the multi-marginal case which will also let us generalize the bounds in [CPT23] to the case of degenerate cost functions. For the two marginals and nondegenerate case, we also refer the reader to a very recent (and elegant) paper [MS23] where the authors push a little further the analysis of the convergence rate by disentangling the roles of ∫ cdγ and the relative entropy in the total cost and deriving convergence rate for both these terms.…”

“…Here we provide an improved, smaller, upper bound, which will depend only on the marginals, but not on the OT plans for the unregularized problem, and we also provide a lower bound depending on a signature condition on the mixed second derivatives of the cost function, that was introduced in [Pas12]. The main difficulty consists in adapting the estimates of [CPT23] to the local structure of the optimal plans described in [Pas12].…”

“…Our main findings can be summarized as follows: we establish two upper bounds, one valid for locally Lipschitz costs and a finer one valid for locally semiconcave costs. The proofs rely, as in [CPT23], on a multi-marginal variant of the block approximation introduced in [Car+17]. Notice that in this case the bound will depend only on the dimension of the support of the marginals.…”

“…Notice that in this case the bound will depend only on the dimension of the support of the marginals. Moreover, for locally semiconcave cost functions, by exploiting Alexandrov-type results as in [CPT23], we improve the upper bound by a 1/2 factor, obtaining the following inequality for some C * ∈ R + :…”