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info.json
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{
"abstract": "In this paper, we study the minimax estimation of the Bochner integral \\[ \\mu_k(P) := \\int_\\mathcal{X} k(\\cdot,x)\\, dP(x), \\] also called the <em>kernel mean embedding</em>, based on random samples drawn i.i.d. from $P$, where $k:\\mathcal{X}\\times\\mathcal{X}\\rightarrow \\mathbb{R}$ is a positive definite kernel. Various estimators (including the empirical estimator), $\\hat{\\theta}_n$ of $\\mu_k(P)$ are studied in the literature wherein all of them satisfy $\\|\\hat{\\theta}_n-\\mu_k(P)\\|_{\\mathcal{H}_k}=O_P(n^{-1/2})$ with $\\mathcal{H}_k$ being the reproducing kernel Hilbert space induced by $k$. The main contribution of the paper is in showing that the above mentioned rate of $n^{-1/2}$ is minimax in $\\|\\cdot\\|_{\\mathcal{H}_k}$ and $\\|\\cdot\\|_{L^2(\\mathbb{R}^d)}$-norms over the class of discrete measures and the class of measures that has an infinitely differentiable density, with $k$ being a continuous translation- invariant kernel on $\\mathbb{R}^d$. The interesting aspect of this result is that the minimax rate is independent of the smoothness of the kernel and the density of $P$ (if it exists).",
"authors": [
"Ilya Tolstikhin",
"Bharath K. Sriperumbudur",
"Krikamol Mu",
"et"
],
"id": "17-032",
"issue": 86,
"pages": [
1,
47
],
"title": "Minimax Estimation of Kernel Mean Embeddings",
"volume": 18,
"year": 2017
}