# Conditional expectation using compactification operators

arXiv: https://arxiv.org/abs/2306.10592 · HTML: https://mlsanity.com/paper/2306.10592/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 61 references that are in the map.
- First posted on arXiv 18 Jun 2023.

## Builds on (most cited first, 5 of 5)

- [Optimal Rates for Regularized Conditional Mean Embedding Learning](https://mlsanity.com/paper/2208.01711/) (2022)
- [A variational approach to the consistency of spectral clustering](https://mlsanity.com/paper/1508.01928/) (2015)
- [Spectral convergence of diffusion maps: improved error bounds and an alternative normalisation](https://mlsanity.com/paper/2006.02037/) (2020)
- [Nonparametric approximation of conditional expectation operators](https://mlsanity.com/paper/2012.12917/) (2020)
- [Manifold learning with bi-stochastic kernels](https://mlsanity.com/paper/1711.06711/) (2017)

## Built on by (most cited first, 1 of 1)

- [Image denoising as a conditional expectation](https://mlsanity.com/paper/2505.21546/) (2025)

## Abstract (arXiv)

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.