{"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/strong-self-concordance-and-sampling","title":"Strong Self-Concordance and Sampling","arxiv_id":"1911.05656","date":"2019-11-13","proceeding":null,"authors":["Aditi Laddha","Yin Tat Lee","Santosh Vempala"],"abstract":"Motivated by the Dikin walk, we develop aspects of an interior-point theory for sampling in high dimension. Specifically, we introduce a symmetric parameter and the notion of strong self-concordance. These properties imply that the corresponding Dikin walk mixes in $\\tilde{O}(n\\bar{\\nu})$ steps from a warm start in a convex body in $\\mathbb{R}^{n}$ using a strongly self-concordant barrier with symmetric self-concordance parameter $\\bar{\\nu}$. For many natural barriers, $\\bar{\\nu}$ is roughly bounded by $\\nu$, the standard self-concordance parameter. We show that this property and strong self-concordance hold for the Lee-Sidford barrier. As a consequence, we obtain the first walk to mix in $\\tilde{O}(n^{2})$ steps for an arbitrary polytope in $\\mathbb{R}^{n}$. Strong self-concordance for other barriers leads to an interesting (and unexpected) connection -- for the universal and entropic barriers, it is implied by the KLS conjecture.","url_abs":"https://arxiv.org/abs/1911.05656v2","url_pdf":"https://arxiv.org/pdf/1911.05656v2.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"strong-self-concordance-and-sampling","repo_url":"https://github.com/ethz-randomwalk/polytopewalk","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1911.05656","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}