{"doi":"10.1093/molbev/msaa130","title":"Gradients Do Grow on Trees: A Linear-Time<i>O</i>(<i>N</i>)-Dimensional Gradient for Statistical Phylogenetics","abstract":"Calculation of the log-likelihood stands as the computational bottleneck for many statistical phylogenetic algorithms. Even worse is its gradient evaluation, often used to target regions of high probability. Order O(N)-dimensional gradient calculations based on the standard pruning algorithm require O(N2) operations, where N is the number of sampled molecular sequences. With the advent of high-throughput sequencing, recent phylogenetic studies have analyzed hundreds to thousands of sequences, with an apparent trend toward even larger data sets as a result of advancing technology. Such large-scale analyses challenge phylogenetic reconstruction by requiring inference on larger sets of process parameters to model the increasing data heterogeneity. To make these analyses tractable, we present a linear-time algorithm for O(N)-dimensional gradient evaluation and apply it to general continuous-time Markov processes of sequence substitution on a phylogenetic tree without a need to assume either stationarity or reversibility. We apply this approach to learn the branch-specific evolutionary rates of three pathogenic viruses: West Nile virus, Dengue virus, and Lassa virus. Our proposed algorithm significantly improves inference efficiency with a 126- to 234-fold increase in maximum-likelihood optimization and a 16- to 33-fold computational performance increase in a Bayesian framework.","journal":"Molecular Biology and Evolution","year":2020,"id":63581,"datarank":0.7355940628826314,"base_score":3.784189633918261,"endowment":3.784189633918261,"self_citation_contribution":0.5676284450877392,"citation_network_contribution":0.16796561779489222,"self_endowment_contribution":0.5676284450877392,"citer_contribution":0.16796561779489222,"corpus_percentile":null,"corpus_rank":null,"citation_count":43,"citer_count":13,"citers_with_citation_signal":7,"citers_with_endowment":7,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":0.9453,"is_data_producer":false,"deposit_databanks":null,"is_oa":true,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":"2020-01-01","fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":326458,"name":"Zhenyu Zhang","orcid":"0000-0001-5570-090X","position":1,"is_corresponding":false},{"id":336580,"name":"Andrew J. Holbrook","orcid":"0000-0002-3558-200X","position":2,"is_corresponding":false},{"id":336581,"name":"Akihiko Nishimura","orcid":"0000-0002-6932-2513","position":3,"is_corresponding":false},{"id":88111,"name":"Guy Baele","orcid":"0000-0002-1915-7732","position":4,"is_corresponding":false},{"id":34991,"name":"Andrew Rambaut","orcid":"0000-0003-4337-3707","position":5,"is_corresponding":false},{"id":36828,"name":"Philippe Lemey","orcid":"0000-0003-2826-5353","position":6,"is_corresponding":false},{"id":32104,"name":"Marc A. Suchard","orcid":"0000-0001-9818-479X","position":7,"is_corresponding":false},{"id":108749,"name":"Xiang Ji","orcid":"0000-0002-7243-0865","position":0,"is_corresponding":true}],"reference_count":72,"raw_metadata":{"citation_network_status":"fetched"},"created_at":"2026-07-18T21:11:24.000992Z","pmid":"32458974","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":[]}