# Recursive Estimation of Conditional Kernel Mean Embeddings

arXiv: https://arxiv.org/abs/2302.05955 · HTML: https://mlsanity.com/paper/2302.05955/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 58 references that are in the map.
- First posted on arXiv 12 Feb 2023.

## Builds on (most cited first, 8 of 8)

- [A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings](https://mlsanity.com/paper/2002.03689/) (2020)
- [Universal Bayes consistency in metric spaces](https://mlsanity.com/paper/1906.09855/) (2019)
- [A Rigorous Theory of Conditional Mean Embeddings](https://mlsanity.com/paper/1912.00671/) (2019)
- [Universal consistency and rates of convergence of multiclass prototype algorithms in metric spaces](https://mlsanity.com/paper/2010.00636/) (2020)
- [Semi-Parametric Uncertainty Bounds for Binary Classification](https://mlsanity.com/paper/1903.09790/) (2019)
- [Vector-Valued Least-Squares Regression under Output Regularity Assumptions](https://mlsanity.com/paper/2211.08958/) (2022)
- [Computing Functions of Random Variables via Reproducing Kernel Hilbert Space Representations](https://mlsanity.com/paper/1501.06794/) (2015)
- [Exact Distribution-Free Hypothesis Tests for the Regression Function of Binary Classification via Conditional Kernel Mean Embeddings](https://mlsanity.com/paper/2103.05126/) (2021)

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

- [Kernel Mean Embedding Topology: Weak and Strong Forms for Stochastic Kernels and Implications for Model Learning](https://mlsanity.com/paper/2502.13486/) (2025)

## Abstract (arXiv)

Kernel mean embeddings, a widely used technique in machine learning, map probability distributions to elements of a reproducing kernel Hilbert space (RKHS). For supervised learning problems, where input-output pairs are observed, the conditional distribution of outputs given the inputs is a key object. The input dependent conditional distribution of an output can be encoded with an RKHS valued function, the conditional kernel mean map. In this paper we present a new recursive algorithm to estimate the conditional kernel mean map in a Hilbert space valued L2 space, that is in a Bochner space. We prove the weak and strong L2 consistency of our recursive estimator under mild conditions. The idea is to generalize Stone's theorem for Hilbert space valued regression in a locally compact Polish space. We present new insights about conditional kernel mean embeddings and give strong asymptotic bounds regarding the convergence of the proposed recursive method. Finally, the results are demonstrated on three application domains: for inputs coming from Euclidean spaces, Riemannian manifolds and locally compact subsets of function spaces.