{"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/intrinsic-persistent-homology-via-density","title":"Intrinsic persistent homology via density-based metric learning","arxiv_id":"2012.07621","date":"2020-12-11","proceeding":null,"authors":["Ximena Fernández","Eugenio Borghini","Gabriel Mindlin","Pablo Groisman"],"abstract":"We address the problem of estimating topological features from data in high dimensional Euclidean spaces under the manifold assumption. Our approach is based on the computation of persistent homology of the space of data points endowed with a sample metric known as Fermat distance. We prove that such metric space converges almost surely to the manifold itself endowed with an intrinsic metric that accounts for both the geometry of the manifold and the density that produces the sample. This fact implies the convergence of the associated persistence diagrams. The use of this intrinsic distance when computing persistent homology presents advantageous properties such as robustness to the presence of outliers in the input data and less sensitiveness to the particular embedding of the underlying manifold in the ambient space. We use these ideas to propose and implement a method for pattern recognition and anomaly detection in time series, which is evaluated in applications to real data.","url_abs":"https://arxiv.org/abs/2012.07621v3","url_pdf":"https://arxiv.org/pdf/2012.07621v3.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":"intrinsic-persistent-homology-via-density","repo_url":"https://github.com/ximenafernandez/intrinsicPH","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"anomaly-detection","task_name":"Anomaly Detection"},{"task_slug":"metric-learning","task_name":"Metric Learning"},{"task_slug":"time-series-1","task_name":"Time Series"},{"task_slug":"time-series","task_name":"Time Series Analysis"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2012.07621","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2012.07621"}},"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/ximenafernandez/intrinsicPH","reach":null}],"summary":{"ran_draft_wrong":2,"ran_honours":1},"by_repo_kind":{"official":{"samples":3,"ran":3,"repositories":1}},"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":3,"samples":[{"code_sha256_prefix":"497f7b9e912b24a3","entry":"DTM","repo":"ximenafernandez/intrinsicPH","repo_kind":"official","path":"src/DTM_filtrations.py","file_url":"https://github.com/ximenafernandez/intrinsicPH/blob/HEAD/src/DTM_filtrations.py","link_basis":"first_harvest_node","language":"python","status":"ran_draft_wrong","verification_level":1,"contract_check":"MISDECLARED","metamorphic_tier":"invariant","behaviour_fingerprint":true,"licence":"GPL-3.0","inline_ok":false,"mcp_get_code":{"code_sha256":"497f7b9e912b24a3"}},{"code_sha256_prefix":"86e845654b113a19","entry":"WeightedRipsFiltrationValue","repo":"ximenafernandez/intrinsicPH","repo_kind":"official","path":"src/DTM_filtrations.py","file_url":"https://github.com/ximenafernandez/intrinsicPH/blob/HEAD/src/DTM_filtrations.py","link_basis":"first_harvest_node","language":"python","status":"ran_honours","verification_level":1,"contract_check":"HONOURS","metamorphic_tier":"well_formed","behaviour_fingerprint":true,"licence":"GPL-3.0","inline_ok":false,"mcp_get_code":{"code_sha256":"86e845654b113a19"}},{"code_sha256_prefix":"b0f9db3edc72671b","entry":"gudhi_to_ripser","repo":"ximenafernandez/intrinsicPH","repo_kind":"official","path":"src/DTM_filtrations.py","file_url":"https://github.com/ximenafernandez/intrinsicPH/blob/HEAD/src/DTM_filtrations.py","link_basis":"first_harvest_node","language":"python","status":"ran_draft_wrong","verification_level":1,"contract_check":"OUTPUT_MISDECLARED","metamorphic_tier":"deterministic","behaviour_fingerprint":false,"licence":"GPL-3.0","inline_ok":false,"mcp_get_code":{"code_sha256":"b0f9db3edc72671b"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}