{"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/subspace-clustering-in-high-dimensions-phase","title":"Subspace clustering in high-dimensions: Phase transitions & Statistical-to-Computational gap","arxiv_id":"2205.13527","date":"2022-05-26","proceeding":null,"authors":["Luca Pesce","Bruno Loureiro","Florent Krzakala","Lenka Zdeborová"],"abstract":"A simple model to study subspace clustering is the high-dimensional $k$-Gaussian mixture model where the cluster means are sparse vectors. Here we provide an exact asymptotic characterization of the statistically optimal reconstruction error in this model in the high-dimensional regime with extensive sparsity, i.e. when the fraction of non-zero components of the cluster means $\\rho$, as well as the ratio $\\alpha$ between the number of samples and the dimension are fixed, while the dimension diverges. We identify the information-theoretic threshold below which obtaining a positive correlation with the true cluster means is statistically impossible. Additionally, we investigate the performance of the approximate message passing (AMP) algorithm analyzed via its state evolution, which is conjectured to be optimal among polynomial algorithm for this task. We identify in particular the existence of a statistical-to-computational gap between the algorithm that require a signal-to-noise ratio $\\lambda_{\\text{alg}} \\ge k / \\sqrt{\\alpha} $ to perform better than random, and the information theoretic threshold at $\\lambda_{\\text{it}} \\approx \\sqrt{-k \\rho \\log{\\rho}} / \\sqrt{\\alpha}$. Finally, we discuss the case of sub-extensive sparsity $\\rho$ by comparing the performance of the AMP with other sparsity-enhancing algorithms, such as sparse-PCA and diagonal thresholding.","url_abs":"https://arxiv.org/abs/2205.13527v2","url_pdf":"https://arxiv.org/pdf/2205.13527v2.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":"subspace-clustering-in-high-dimensions-phase","repo_url":"https://github.com/lucpoisson/subspaceclustering","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"clustering","task_name":"Clustering"},{"task_slug":"high","task_name":"Vocal Bursts Intensity Prediction"}],"methods":[{"method_slug":"amp","method_name":"AMP"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2205.13527","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2205.13527"}},"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":"deterministic:regex_extraction","url":"https://github.com/lucpoisson/SubspaceClustering","reach":{"status":"ok"}},{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/lucpoisson/subspaceclustering","reach":{"status":"ok"}}],"summary":{"ran_fixture":2,"ran_honours":2,"ran_draft_wrong":2},"by_repo_kind":{"official":{"samples":6,"ran":6,"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":6,"samples":[{"code_sha256_prefix":"d543322a8472f885","entry":"SPCA","repo":"lucpoisson/SubspaceClustering","repo_kind":"official","path":"Miscellaneous/Mix_functions.py","file_url":"https://github.com/lucpoisson/SubspaceClustering/blob/HEAD/Miscellaneous/Mix_functions.py","link_basis":"first_harvest_node","language":"python","status":"ran_fixture","verification_level":1,"contract_check":"RAISES","metamorphic_tier":"invariant","behaviour_fingerprint":true,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"d543322a8472f885"}},{"code_sha256_prefix":"2726c48f2f089f87","entry":"damp","repo":"lucpoisson/SubspaceClustering","repo_kind":"official","path":"AMP_SE/SE_AMP_functions.py","file_url":"https://github.com/lucpoisson/SubspaceClustering/blob/HEAD/AMP_SE/SE_AMP_functions.py","link_basis":"first_harvest_node","language":"python","status":"ran_honours","verification_level":1,"contract_check":"HONOURS","metamorphic_tier":"invariant","behaviour_fingerprint":true,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"2726c48f2f089f87"}},{"code_sha256_prefix":"7e8c095002d297ff","entry":"fu","repo":"lucpoisson/SubspaceClustering","repo_kind":"official","path":"AMP_SE/SE_AMP_functions.py","file_url":"https://github.com/lucpoisson/SubspaceClustering/blob/HEAD/AMP_SE/SE_AMP_functions.py","link_basis":"first_harvest_node","language":"python","status":"ran_honours","verification_level":1,"contract_check":"HONOURS","metamorphic_tier":"invariant","behaviour_fingerprint":true,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"7e8c095002d297ff"}},{"code_sha256_prefix":"70bf2053de2ee569","entry":"fv","repo":"lucpoisson/SubspaceClustering","repo_kind":"official","path":"AMP_SE/SE_AMP_functions.py","file_url":"https://github.com/lucpoisson/SubspaceClustering/blob/HEAD/AMP_SE/SE_AMP_functions.py","link_basis":"first_harvest_node","language":"python","status":"ran_fixture","verification_level":1,"contract_check":"RAISES","metamorphic_tier":"invariant","behaviour_fingerprint":true,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"70bf2053de2ee569"}},{"code_sha256_prefix":"d7d19f41999d4a69","entry":"get_instance","repo":"lucpoisson/SubspaceClustering","repo_kind":"official","path":"Miscellaneous/Mix_functions.py","file_url":"https://github.com/lucpoisson/SubspaceClustering/blob/HEAD/Miscellaneous/Mix_functions.py","link_basis":"first_harvest_node","language":"python","status":"ran_draft_wrong","verification_level":1,"contract_check":"OUTPUT_MISDECLARED","metamorphic_tier":"deterministic","behaviour_fingerprint":true,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"d7d19f41999d4a69"}},{"code_sha256_prefix":"298ccfd11ab76ae9","entry":"oracle","repo":"lucpoisson/SubspaceClustering","repo_kind":"official","path":"Miscellaneous/Mix_functions.py","file_url":"https://github.com/lucpoisson/SubspaceClustering/blob/HEAD/Miscellaneous/Mix_functions.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":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"298ccfd11ab76ae9"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}