{"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/kernel-thinning","title":"Kernel Thinning","arxiv_id":"2105.05842","date":"2021-05-12","proceeding":null,"authors":["Raaz Dwivedi","Lester Mackey"],"abstract":"We introduce kernel thinning, a new procedure for compressing a distribution $\\mathbb{P}$ more effectively than i.i.d. sampling or standard thinning. Given a suitable reproducing kernel $\\mathbf{k}_{\\star}$ and $\\mathcal{O}(n^2)$ time, kernel thinning compresses an $n$-point approximation to $\\mathbb{P}$ into a $\\sqrt{n}$-point approximation with comparable worst-case integration error across the associated reproducing kernel Hilbert space. The maximum discrepancy in integration error is $\\mathcal{O}_d(n^{-1/2}\\sqrt{\\log n})$ in probability for compactly supported $\\mathbb{P}$ and $\\mathcal{O}_d(n^{-\\frac{1}{2}} (\\log n)^{(d+1)/2}\\sqrt{\\log\\log n})$ for sub-exponential $\\mathbb{P}$ on $\\mathbb{R}^d$. In contrast, an equal-sized i.i.d. sample from $\\mathbb{P}$ suffers $\\Omega(n^{-1/4})$ integration error. Our sub-exponential guarantees resemble the classical quasi-Monte Carlo error rates for uniform $\\mathbb{P}$ on $[0,1]^d$ but apply to general distributions on $\\mathbb{R}^d$ and a wide range of common kernels. Moreover, the same construction delivers near-optimal $L^\\infty$ coresets in $\\mathcal O(n^2)$ time. We use our results to derive explicit non-asymptotic maximum mean discrepancy bounds for Gaussian, Mat\\'ern, and B-spline kernels and present two vignettes illustrating the practical benefits of kernel thinning over i.i.d. sampling and standard Markov chain Monte Carlo thinning, in dimensions $d=2$ through $100$.","url_abs":"https://arxiv.org/abs/2105.05842v10","url_pdf":"https://arxiv.org/pdf/2105.05842v10.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":"kernel-thinning","repo_url":"https://github.com/microsoft/goodpoints","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"jax","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2105.05842","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2105.05842"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. Samples come from repositories linked to the paper, official or community; repo_kind says which.","repos":[{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/microsoft/goodpoints","reach":null}],"summary":{"ran_fixture":1},"by_repo_kind":{},"repo_kind_vocabulary":{"official":"The archive marks this repository official for the paper","named_in_paper":"The archive records that the paper mentions this repository; it is not marked official","listed":"In the archive's code links for this paper, not marked official and not recorded as mentioned in the paper","found_in_text":"Syntology found this repository in the paper's own text; whether it is the authors' implementation is not asserted","community":"Not in the archive's code links for this paper; a community repository Syntology harvested"},"n_pointer_only_for_licence":1,"samples":[{"code_sha256_prefix":"9a913678c56785e8","entry":"compress","repo":null,"repo_kind":null,"path":null,"file_url":null,"link_basis":"identical_code_first_harvested_elsewhere","language":"python","status":"ran_fixture","verification_level":1,"contract_check":"RAISES","metamorphic_tier":"invariant","behaviour_fingerprint":false,"licence":null,"inline_ok":false,"mcp_get_code":{"code_sha256":"9a913678c56785e8"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}