{"doi":"10.1214/22-ejp758","title":"On the non-asymptotic concentration of heteroskedastic Wishart-type matrix","abstract":"This paper focuses on the non-asymptotic concentration of the heteroskedastic Wishart-type matrices. Suppose Z is a p1-by-p2random matrix and Zij∼N(0,σij2) independently, we prove the expected spectral norm of Wishart matrix deviations (i.e., EZZ⊤−EZZ⊤) is upper bounded by (1+ϵ)2σCσR+σC2+Cσ Rσ∗ log(p1∧p2)+Cσ∗2log(p 1∧p2), where σC2:=maxj ∑i=1p1σ ij2, σR2:=maxi ∑j=1p2σ ij2 and σ∗2:=maxi,jσij2. A minimax lower bound is developed that matches this upper bound. Then, we derive the concentration inequalities, moments, and tail bounds for the heteroskedastic Wishart-type matrix under more general distributions, such as sub-Gaussian and heavy-tailed distributions. Next, we consider the cases where Z has homoskedastic columns or rows (i.e., σij≈σi or σij≈σj) and derive the rate-optimal Wishart-type concentration bounds. Finally, we apply the developed tools to identify the sharp signal-to-noise ratio threshold for consistent clustering in the heteroskedastic clustering problem.","journal":"Electronic Journal of Probability","year":2022,"id":280155,"datarank":0.0,"base_score":0.0,"endowment":0.0,"self_citation_contribution":0.0,"citation_network_contribution":0.0,"self_endowment_contribution":0.0,"citer_contribution":0.0,"corpus_percentile":null,"corpus_rank":null,"citation_count":11,"citer_count":0,"citers_with_citation_signal":0,"citers_with_endowment":0,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":0.9551,"is_data_producer":false,"deposit_databanks":null,"is_oa":true,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":"2022-01-01","fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":476861,"name":"Rungang Han","orcid":"0000-0002-3208-1031","position":1,"is_corresponding":false},{"id":392362,"name":"Anru R. Zhang","orcid":"0000-0002-8721-5252","position":2,"is_corresponding":false},{"id":322350,"name":"Tianxi Cai","orcid":"0000-0002-5379-2502","position":0,"is_corresponding":true}],"reference_count":44,"raw_metadata":null,"created_at":"2026-07-19T00:28:59.341002Z","pmid":null,"pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"fair_f":null,"fair_a":null,"fair_i":null,"fair_r":null,"fair_zscore":null,"fair_rationale":null,"fair_model":null,"fair_agent_version":null,"fair_fulltext_source":null,"fair_has_llm":null,"fair_computed_at":null,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}