The separate tasks of denoising, conditional expectation and manifold learning can often be posed in a common setting of finding the conditional expectations arising from a product of two random variables. This paper focuses on this more general problem and describes an operator theoretic approach to estimating the conditional expectation. Kernel integral operators are used as a compactification tool, to set up the estimation problem as a linear inverse problem in a reproducing kernel Hilbert space. This equation is shown to have solutions that are stable to numerical approximation, thus guaranteeing the convergence of data-driven implementations. The overall technique is easy to implement, and their successful application to some real-world problems are also shown.

翻译：去噪、条件期望与流形学习这三项不同任务，通常可归结为寻找由两个随机变量乘积所诱导的条件期望这一共同框架。本文聚焦于这一更泛化的问题，描述了一种基于算子理论的条件期望估计方法。通过使用核积分算子作为压缩工具，将估计问题转化为再生核希尔伯特空间中的线性逆问题。研究表明，该方程的解对数值逼近具有稳定性，从而保证了基于数据驱动的实现方法的收敛性。所提技术易于实现，并展示了其在若干实际问题中的成功应用。