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Transfer Operators from Batches of Unpaired Points via Entropic Transport Kernels

13 February 2024
F. Beier
Hancheng Bi
Clément Sarrazin
Bernhard Schmitzer
Gabriele Steidl
    OT
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Abstract

In this paper, we are concerned with estimating the joint probability of random variables XXX and YYY, given NNN independent observation blocks (xi,yi)(\boldsymbol{x}^i,\boldsymbol{y}^i)(xi,yi), i=1,…,Ni=1,\ldots,Ni=1,…,N, each of MMM samples (xi,yi)=((xji,yσi(j)i))j=1M(\boldsymbol{x}^i,\boldsymbol{y}^i) = \bigl((x^i_j, y^i_{\sigma^i(j)}) \bigr)_{j=1}^M(xi,yi)=((xji​,yσi(j)i​))j=1M​, where σi\sigma^iσi denotes an unknown permutation of i.i.d. sampled pairs (xji,yji)(x^i_j,y_j^i)(xji​,yji​), j=1,…,Mj=1,\ldots,Mj=1,…,M. This means that the internal ordering of the MMM samples within an observation block is not known. We derive a maximum-likelihood inference functional, propose a computationally tractable approximation and analyze their properties. In particular, we prove a Γ\GammaΓ-convergence result showing that we can recover the true density from empirical approximations as the number NNN of blocks goes to infinity. Using entropic optimal transport kernels, we model a class of hypothesis spaces of density functions over which the inference functional can be minimized. This hypothesis class is particularly suited for approximate inference of transfer operators from data. We solve the resulting discrete minimization problem by a modification of the EMML algorithm to take addional transition probability constraints into account and prove the convergence of this algorithm. Proof-of-concept examples demonstrate the potential of our method.

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