The separate tasks of denoising, least squares 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 allow 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、最小二乘期望以及流形学习这三个独立任务通常可以归结为寻找由两个随机变量乘积产生的条件期望这一共同框架下的问题。本文聚焦于这个更一般的议题，并提出了一种基于算子理论的估计条件期望的方法。我们采用核积分算子作为紧化工具，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，该方程存在允许数值近似的解，从而保证了数据驱动实现的收敛性。所提出的整体技术易于实现，并展示了其在若干实际问题中的成功应用。