BBP phase transition

E898469

mathematical phenomenon phase transition in random matrix theory

The BBP phase transition is a phenomenon in random matrix theory describing the critical point at which an eigenvalue associated with a low-rank perturbation separates from the bulk spectrum, leading to a detectable “spike.”

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Statements (46)

Predicate	Object
instanceOf	mathematical phenomenon ⓘ phase transition in random matrix theory ⓘ
alsoKnownAs	Baik–Ben Arous–Péché phase transition NERFINISHED ⓘ
appliesTo	Wigner-type random matrices ⓘ low-rank perturbations of large random matrices ⓘ sample covariance matrices ⓘ spiked random matrix models ⓘ
characterizedBy	change in eigenvector alignment with the signal direction ⓘ eigenvalue leaving the support of the limiting spectral distribution ⓘ transition between undetectable and detectable spikes ⓘ
concerns	fluctuations of the top eigenvalue near the critical point ⓘ largest eigenvalue behavior under rank-one perturbations ⓘ threshold for eigenvalue separation from the bulk ⓘ
describes	critical point where an eigenvalue separates from the bulk spectrum ⓘ detectability threshold for a spike in spiked random matrix models ⓘ emergence of an outlier eigenvalue due to low-rank perturbation ⓘ
field	mathematical statistics ⓘ probability theory ⓘ random matrix theory ⓘ
hasAspect	critical regime with nonstandard fluctuation behavior ⓘ subcritical regime where spike is not detectable ⓘ supercritical regime where an outlier eigenvalue appears ⓘ
hasImplication	design of optimal tests for spiked alternatives ⓘ fundamental detection limits in noisy matrix models ⓘ limits of PCA for weak signals in high dimensions ⓘ
introducedIn	paper by Baik, Ben Arous, and Péché on eigenvalues of deformed random matrices ⓘ
involves	low-rank deformation of invariant random matrix ensembles ⓘ spectral properties of large random matrices ⓘ
mathematicalContext	asymptotic regime of large matrix dimension ⓘ limit of eigenvalue distributions ⓘ
namedAfter	Gérard Ben Arous NERFINISHED ⓘ Jinho Baik NERFINISHED ⓘ Sandrine Péché NERFINISHED ⓘ
relatedTo	Marchenko–Pastur law NERFINISHED ⓘ Tracy–Widom distribution NERFINISHED ⓘ eigenvalue outliers ⓘ phase transitions in high-dimensional statistics ⓘ principal component analysis ⓘ signal detection in high dimensions ⓘ spiked covariance model ⓘ
usedIn	finance and portfolio theory ⓘ genomics and population genetics ⓘ high-dimensional hypothesis testing ⓘ statistical signal processing ⓘ wireless communications ⓘ
yearIntroduced	2005 ⓘ

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random matrix theory → hasKeyConcept → BBP phase transition ⓘ