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 $L_2$ space, that is in a Bochner space. We prove the weak and strong $L_2$ 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.

翻译：核均值嵌入是机器学习中广泛应用的技术，它将概率分布映射到再生核希尔伯特空间（RKHS）的元素。对于观测到输入-输出对的监督学习问题，给定输入条件下输出的条件分布是一个关键研究对象。依赖于输入的输出条件分布可以用一个RKHS值函数（即条件核均值映射）来编码。本文提出了一种新的递归算法，用于在希尔伯特空间值的$L_2$空间（即博赫纳空间）中估计条件核均值映射。我们在温和条件下证明了所提递归估计量的弱一致性与强$L_2$一致性。其核心思想是将斯通定理推广到局部紧波兰空间中的希尔伯特空间值回归问题。我们提出了关于条件核均值嵌入的新见解，并给出了所提递归方法收敛性的强渐近界。最后，通过在三个应用领域（输入来自欧几里得空间、黎曼流形以及函数空间的局部紧子集）的实例验证了该方法的有效性。