# Transfer Operators from Batches of Unpaired Points via Entropic Transport Kernels

arXiv: https://arxiv.org/abs/2402.08425 · HTML: https://mlsanity.com/paper/2402.08425/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 33 references that are in the map.
- First posted on arXiv 13 Feb 2024.

## Builds on (most cited first, 6 of 6)

- [Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions](https://mlsanity.com/paper/1604.05251/) (2016)
- [Towards a mathematical theory of trajectory inference](https://mlsanity.com/paper/2102.09204/) (2021)
- [Strictly proper kernel scores and characteristic kernels on compact spaces](https://mlsanity.com/paper/1712.05279/) (2017)
- [The Linearized Hellinger--Kantorovich Distance](https://mlsanity.com/paper/2102.08807/) (2021)
- [From Optimal Transport to Discrepancy](https://mlsanity.com/paper/2002.01189/) (2020)
- [Domain decomposition for entropy regularized optimal transport](https://mlsanity.com/paper/2001.10986/) (2020)

## Built on by (most cited first, 2 of 2)

- [Entropic Optimal Transport Eigenmaps for Nonlinear Alignment and Joint Embedding of High-Dimensional Datasets](https://mlsanity.com/paper/2407.01718/) (2024)
- [Measure-Theoretic Time-Delay Embedding](https://mlsanity.com/paper/2409.08768/) (2024)

## Abstract (arXiv)

In this paper, we are concerned with estimating the joint probability of random variables X and Y, given N independent observation blocks (boldsymbolxi,boldsymbolyi), i=1,ldots,N, each of M samples (boldsymbolxi,boldsymbolyi) = bigl((xij, yiσi(j)) bigr)j=1M, where σi denotes an unknown permutation of i.i.d. sampled pairs (xij,yji), 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 Gamma-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.