{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/density-estimation-in-infinite-dimensional","title":"Density Estimation in Infinite Dimensional Exponential Families","arxiv_id":"1312.3516","date":"2013-12-12","proceeding":null,"authors":["Bharath Sriperumbudur","Kenji Fukumizu","Arthur Gretton","Aapo Hyvärinen","Revant Kumar"],"abstract":"In this paper, we consider an infinite dimensional exponential family,\n$\\mathcal{P}$ of probability densities, which are parametrized by functions in\na reproducing kernel Hilbert space, $H$ and show it to be quite rich in the\nsense that a broad class of densities on $\\mathbb{R}^d$ can be approximated\narbitrarily well in Kullback-Leibler (KL) divergence by elements in\n$\\mathcal{P}$. The main goal of the paper is to estimate an unknown density,\n$p_0$ through an element in $\\mathcal{P}$. Standard techniques like maximum\nlikelihood estimation (MLE) or pseudo MLE (based on the method of sieves),\nwhich are based on minimizing the KL divergence between $p_0$ and\n$\\mathcal{P}$, do not yield practically useful estimators because of their\ninability to efficiently handle the log-partition function. Instead, we propose\nan estimator, $\\hat{p}_n$ based on minimizing the \\emph{Fisher divergence},\n$J(p_0\\Vert p)$ between $p_0$ and $p\\in \\mathcal{P}$, which involves solving a\nsimple finite-dimensional linear system. When $p_0\\in\\mathcal{P}$, we show that\nthe proposed estimator is consistent, and provide a convergence rate of\n$n^{-\\min\\left\\{\\frac{2}{3},\\frac{2\\beta+1}{2\\beta+2}\\right\\}}$ in Fisher\ndivergence under the smoothness assumption that $\\log\np_0\\in\\mathcal{R}(C^\\beta)$ for some $\\beta\\ge 0$, where $C$ is a certain\nHilbert-Schmidt operator on $H$ and $\\mathcal{R}(C^\\beta)$ denotes the image of\n$C^\\beta$. We also investigate the misspecified case of $p_0\\notin\\mathcal{P}$\nand show that $J(p_0\\Vert\\hat{p}_n)\\rightarrow \\inf_{p\\in\\mathcal{P}}J(p_0\\Vert\np)$ as $n\\rightarrow\\infty$, and provide a rate for this convergence under a\nsimilar smoothness condition as above. Through numerical simulations we\ndemonstrate that the proposed estimator outperforms the non-parametric kernel\ndensity estimator, and that the advantage with the proposed estimator grows as\n$d$ increases.","url_abs":"http://arxiv.org/abs/1312.3516v4","url_pdf":"http://arxiv.org/pdf/1312.3516v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"density-estimation-in-infinite-dimensional","repo_url":"https://github.com/karlnapf/kernel_exp_family","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"BSD-3-Clause"}}],"tasks":[{"task_slug":"density-estimation","task_name":"Density Estimation"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1312.3516","atlas_url":"https://app.syntology.ai/?focus=1312.3516","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}