{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T07:57:40Z","timestamp":1774425460877,"version":"3.50.1"},"reference-count":33,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2022,1,13]],"date-time":"2022-01-13T00:00:00Z","timestamp":1642032000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Appl. Math. Stat."],"abstract":"<jats:p>Testing for independence plays a fundamental role in many statistical techniques. Among the nonparametric approaches, the distance-based methods (such as the distance correlation-based hypotheses testing for independence) have many advantages, compared with many other alternatives. A known limitation of the distance-based method is that its computational complexity can be high. In general, when the sample size is <jats:italic>n<\/jats:italic>, the order of computational complexity of a distance-based method, which typically requires computing of all pairwise distances, can be <jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>2<\/jats:sup>). Recent advances have discovered that in the <jats:italic>univariate<\/jats:italic> cases, a fast method with <jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic>\u2009log\u2009 <jats:italic>n<\/jats:italic>) computational complexity and <jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic>) memory requirement exists. In this paper, we introduce a test of independence method based on random projection and distance correlation, which achieves nearly the same power as the state-of-the-art distance-based approach, works in the <jats:italic>multivariate<\/jats:italic> cases, and enjoys the <jats:italic>O<\/jats:italic>(<jats:italic>nK<\/jats:italic>\u2009log\u2009 <jats:italic>n<\/jats:italic>) computational complexity and <jats:italic>O<\/jats:italic>(\u2009max{<jats:italic>n<\/jats:italic>, <jats:italic>K<\/jats:italic>}) memory requirement, where <jats:italic>K<\/jats:italic> is the number of random projections. Note that saving is achieved when <jats:italic>K<\/jats:italic> &amp;lt; <jats:italic>n<\/jats:italic>\/\u2009log\u2009 <jats:italic>n<\/jats:italic>. We name our method a Randomly Projected Distance Covariance (RPDC). The statistical theoretical analysis takes advantage of some techniques on the random projection which are rooted in contemporary machine learning. Numerical experiments demonstrate the efficiency of the proposed method, relative to numerous competitors.<\/jats:p>","DOI":"10.3389\/fams.2021.779841","type":"journal-article","created":{"date-parts":[[2022,1,13]],"date-time":"2022-01-13T06:41:32Z","timestamp":1642056092000},"update-policy":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":12,"title":["A Statistically and Numerically Efficient Independence Test Based on Random Projections and Distance Covariance"],"prefix":"10.3389","volume":"7","author":[{"given":"Cheng","family":"Huang","sequence":"first","affiliation":[]},{"given":"Xiaoming","family":"Huo","sequence":"additional","affiliation":[]}],"member":"1965","published-online":{"date-parts":[[2022,1,13]]},"reference":[{"key":"B1","doi-asserted-by":"crossref","first-page":"1518","DOI":"10.1126\/science.1205438","article-title":"Detecting Novel Associations in Large Data Sets","volume":"334","author":"David","year":"2011","journal-title":"Science"},{"key":"B2","doi-asserted-by":"publisher","first-page":"879","DOI":"10.1214\/aos\/1176345528","article-title":"On Nonparametric Measures of Dependence for Random Variables","author":"Schweizer","year":"1981","journal-title":"Ann Stat"},{"key":"B3","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1007\/s00184-008-0229-9","article-title":"A Measure of Mutual Complete Dependence","volume":"71","author":"Siburg","year":"2010","journal-title":"Metrika"},{"key":"B4","first-page":"63","article-title":"Measuring Statistical Dependence with hilbert-schmidt Norms","author":"Gretton","year":"2005"},{"key":"B5","doi-asserted-by":"publisher","first-page":"1236","DOI":"10.1214\/09-aoas312","article-title":"Brownian Distance Covariance","volume":"3","author":"Sz\u00e9kely","year":"2009","journal-title":"Ann Appl Stat"},{"key":"B6","doi-asserted-by":"publisher","first-page":"2769","DOI":"10.1214\/009053607000000505","article-title":"Measuring and Testing Dependence by Correlation of Distances","volume":"35","author":"G\u00e1borSz\u00e9kely","year":"2007","journal-title":"Ann Stat"},{"key":"B7","first-page":"116","article-title":"On Quantifying Dependence: a Framework for Developing Interpretable Measures","volume":"28","author":"Matthew","year":"2013","journal-title":"Stat Sci"},{"key":"B8","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1080\/00401706.2015.1054435","article-title":"Fast Computing for Distance Covariance","volume":"58","author":"Huo","year":"2016","journal-title":"Technometrics"},{"key":"B9","doi-asserted-by":"publisher","first-page":"916","DOI":"10.1198\/016214505000000097","article-title":"Multivariate Nonparametric Tests of independence","volume":"100","author":"Taskinen","year":"2005","journal-title":"J Am Stat Assoc"},{"key":"B10","doi-asserted-by":"publisher","first-page":"3284","DOI":"10.1214\/12-aop803","article-title":"Distance Covariance in Metric Spaces","volume":"41","author":"Lyons","year":"2013","journal-title":"Ann Probab"},{"key":"B11","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1080\/01621459.1963.10500830","article-title":"Probability Inequalities for Sums of Bounded Random Variables","volume":"58","author":"Hoeffding","year":"1963","journal-title":"J Am Stat Assoc"},{"key":"B12","doi-asserted-by":"publisher","first-page":"2263","DOI":"10.1214\/13-aos1140","article-title":"Equivalence of Distance-Based and RKHS-Based Statistics in Hypothesis Testing","volume":"41","author":"Sejdinovic","year":"2013","journal-title":"Ann Stat"},{"key":"B13","volume-title":"Approximation Theorems of Mathematical Statistics (Wiley Series in Probability and Statistics)","author":"Serfling","year":"1980"},{"key":"B14","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1214\/aoms\/1177728786","article-title":"Some Theorems on Quadratic Forms Applied in the Study of Analysis of Variance Problems, I. 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