Transfer Operators from Batches of Unpaired Points via Entropic Transport Kernels
This addresses a specific data alignment issue in statistical inference, particularly for transfer operators, but appears incremental as it builds on existing entropic transport and maximum-likelihood frameworks.
The paper tackles the problem of estimating joint probability distributions from batches of unpaired data points with unknown internal permutations, proposing a maximum-likelihood inference functional using entropic optimal transport kernels and proving recovery of the true density as the number of blocks increases. It demonstrates the method's potential through proof-of-concept examples.
In this paper, we are concerned with estimating the joint probability of random variables $X$ and $Y$, given $N$ independent observation blocks $(\boldsymbol{x}^i,\boldsymbol{y}^i)$, $i=1,\ldots,N$, each of $M$ samples $(\boldsymbol{x}^i,\boldsymbol{y}^i) = \bigl((x^i_j, y^i_{σ^i(j)}) \bigr)_{j=1}^M$, where $σ^i$ denotes an unknown permutation of i.i.d. sampled pairs $(x^i_j,y_j^i)$, $j=1,\ldots,M$. This means that the internal ordering of the $M$ 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 $Γ$-convergence result showing that we can recover the true density from empirical approximations as the number $N$ 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.