标题： 通过非配对点批次的熵传输核算子 显示英文标题

标题： Transfer Operators from Batches of Unpaired Points via Entropic Transport Kernels

摘要： 本文关注于估计随机变量$X$和$Y$的联合概率，已知$N$个独立的观测块 $(\boldsymbol{x}^i,\boldsymbol{y}^i)$,$i=1,\ldots,N$，每个块包含$M$个样本 $(\boldsymbol{x}^i,\boldsymbol{y}^i) = \bigl((x^i_j, y^i_{\sigma^i(j)}) \bigr)_{j=1}^M$，其中$\sigma^i$表示 i.i.d. 的未知排列。 采样的对组$(x^i_j,y_j^i)$, $j=1,\ldots,M$。 这意味着在一个观测块内$M$个样本的内部顺序未知。 我们推导出一个最大似然推理泛函，提出了一种计算上易于处理的近似方法，并分析了它们的性质。 特别是，我们证明了一个$\Gamma$-收敛结果，表明当块的数量$N$趋向于无穷大时，我们可以从经验近似中恢复真实的密度。 利用熵最优传输核，我们构建了一类关于数据密度函数假设空间的模型，该推理泛函可以在这些假设空间上被最小化。 这个假设类特别适合于从数据中近似推导转移算子。 通过修改EMML算法以考虑额外的转移概率约束来解决由此产生的离散最小化问题，并证明了该算法的收敛性。 概念验证示例展示了我们方法的潜力。 显示英文摘要

摘要： 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 $(\boldsymbol{x}^i,\boldsymbol{y}^i) = \bigl((x^i_j, y^i_{\sigma^i(j)}) \bigr)_{j=1}^M$, where $\sigma^i$ 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 $\Gamma$-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.