Conditional expectation using compactification operators
This provides a unified framework for various statistical and machine learning tasks, though it appears incremental as it builds on existing operator methods.
The paper tackles the general problem of estimating conditional expectations from products of random variables, which unifies tasks like denoising and manifold learning, by developing an operator-theoretic approach using kernel integral operators as a compactification tool, with data-driven implementations shown to converge and be applicable to real-world problems.
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.