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Reproducing kernel Hilbert spaces in the mean field limit

  • *Corresponding author: Christian Fiedler

    *Corresponding author: Christian Fiedler 
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  • Kernel methods, being supported by a well-developed theory and coming with efficient algorithms, are among the most popular and successful machine learning techniques. From a mathematical point of view, these methods rest on the concept of kernels and function spaces generated by kernels, so–called reproducing kernel Hilbert spaces. Motivated by recent developments of learning approaches in the context of interacting particle systems, we investigate kernel methods acting on data with many measurement variables. We show the rigorous mean field limit of kernels and provide a detailed analysis of the limiting reproducing kernel Hilbert space. Furthermore, several examples of kernels, that allow a rigorous mean field limit, are presented.

    Mathematics Subject Classification: 46E22, 82B40, 74A25, 82C40.


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  • Figure 1.  Commutative diagram summarizing the relation between mean field limit (MFL) of a sequence of kernels $ (k_M)_M $ and their corresponding reproducing kernel Hilbert spaces. Here, $ f_M $ denotes an element of the space $ H_M $, and $ k,H_k $ indicate the MFL of $ (k_M)_M $ and $ (H_M)_M $, respectively. The mean field limits are given in Theorem 3.2 and Theorem 4.4., respectively

    Figure 2.  Commutative diagram on the relation of canonical feature map of $ k^{[M]} $ and KMEs

    Figure 3.  Diagram illustration of the relations of double sum kernel, KME and MFL

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