{"doi":"10.1137/20m1342124","title":"Doubly Stochastic Normalization of the Gaussian Kernel Is Robust to Heteroskedastic Noise","abstract":"A fundamental step in many data-analysis techniques is the construction of an affinity matrix describing similarities between data points. When the data points reside in Euclidean space, a widespread approach is to form an affinity matrix by the Gaussian kernel with pairwise distances, and to follow with a certain normalization (e.g., the row-stochastic normalization or its symmetric variant). We demonstrate that the doubly stochastic normalization of the Gaussian kernel with zero main diagonal (i.e., no self-loops) is robust to heteroskedastic noise. That is, the doubly stochastic normalization is advantageous in that it automatically accounts for observations with different noise variances. Specifically, we prove that in a suitable high-dimensional setting where heteroskedastic noise does not concentrate too much in any particular direction in space, the resulting (doubly stochastic) noisy affinity matrix converges to its clean counterpart with rate $m^{-1/2}$, where $m$ is the ambient dimension. We demonstrate this result numerically and show that, in contrast, the popular row-stochastic and symmetric normalizations behave unfavorably under heteroskedastic noise. Furthermore, we provide examples of simulated and experimental single-cell RNA sequence data with intrinsic heteroskedasticity, where the advantage of the doubly stochastic normalization for exploratory analysis is evident.","journal":"SIAM Journal on Mathematics of Data Science","year":2021,"id":210860,"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":2,"citer_count":0,"citers_with_citation_signal":0,"citers_with_endowment":0,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":0.9491,"is_data_producer":false,"deposit_databanks":null,"is_oa":true,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":"2021-01-01","fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":556919,"name":"Ronald R. Coifman","orcid":"0000-0001-7336-7784","position":1,"is_corresponding":false},{"id":69468,"name":"Yuval Kluger","orcid":"0000-0002-3035-071X","position":2,"is_corresponding":false},{"id":554800,"name":"Boris Landa","orcid":null,"position":0,"is_corresponding":true}],"reference_count":63,"raw_metadata":null,"created_at":"2026-07-18T23:52:16.049481Z","pmid":"34124607","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":[]}