Edge universality of separable covariance matrices

Yang, Fan

doi:10.1214/19-ejp381

Cited by 32 publications

(43 citation statements)

References 66 publications

(163 reference statements)

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“…The BBP transition has been observed in many random matrix ensembles with finite rank perturbations. Without attempting to be comprehensive, we mention the references [14,15,24,36,37,42] on deformed Wigner matrices, [3,5,6,12,25,33,41] on spiked sample covariance matrices, [17,49,51] on spiked separable covariance matrices, and [8,9,10,47] on several other deformed random matrix ensembles. In our setting, the SCC matrix C r X r Y can be regarded as a finite rank perturbation of the SCC matrix in the null case with r " 0.…”

Section: Introductionmentioning

confidence: 99%

Sample canonical correlation coefficients of high-dimensional random vectors with finite rank correlations

Ma¹,

Yang²

2021

Preprint

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Consider two random vectors r x P R p and r y P R q of the forms rx " Az `C1{2 1 x and r y " Bz `C1{2 2 y, where x P R p , y P R q and z P R r are independent random vectors with i.i.d. entries of zero mean and unit variance, C1 and C2 are p ˆp and q ˆq deterministic population covariance matrices, and A and B are p ˆr and q ˆr deterministic factor loading matrices. With n independent observations of pr x, r yq, we study the sample canonical correlations between rx and r y. We consider the high-dimensional setting with finite rank correlations, that is, p{n Ñ c1 and q{n Ñ c2 as n Ñ 8 for some constants c1 P p0, 1q and c2 P p0, 1 ´c1q, and r is a fixed integer. Let t1 ě t2 ě ¨¨¨ě tr ě 0 be the squares of the nontrivial population canonical correlation coefficients between rx and r y, and let r λ1 ě r λ2 ě ¨¨¨ě r λp^q ě 0 be the squares of the sample canonical correlation coefficients. If the entries of x, y and z are i.i.d. Gaussian, then the following dichotomy has been shown in [7] for a fixed threshold tc P p0, 1q: for any 1 ď i ď r, if ti ă tc, then r λi converges to the right-edge λ`of the limiting eigenvalue spectrum of the sample canonical correlation matrix, and moreover, n 2{3 p r λi ´λ`q converges weakly to the Tracy-Widom law; if ti ą tc, then r λi converges to a deterministic limit θi P pλ`, 1q that is determined by c1, c2 and ti. In this paper, we prove that these results hold universally under the sharp fourth moment conditions on the entries of x, y and z. Moreover, we prove the results in full generality, in the sense that they also hold for near-degenerate ti's and for ti's that are close to the threshold tc. Finally, we also provide almost sharp convergence rates for the sample canonical correlation coefficients under a general a-th moment assumption.

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Section: Introductionmentioning

confidence: 99%

Sample canonical correlation coefficients of high-dimensional random vectors with finite rank correlations

Ma¹,

Yang²

2021

Preprint

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“…The BBP transition has been observed and studied in many random matrix ensembles deformed by low-rank perturbations. Without attempting to be comprehensive, we refer the reader to [13,14,22,40,41,49] on deformed Wigner matrices, [1,2,3,10,23,35,48] on spiked sample covariance matrices, [15,56,58] on spiked separable covariance matrices, and [6,7,8,54] on several other types of deformed random matrix ensembles. In this paper, we study the SCC matrix C r X r Y , which can be regarded as a low-rank perturbation of the SCC matrix in the null case with r " 0.…”

Section: Introductionmentioning

confidence: 99%

Limiting distribution of the sample canonical correlation coefficients of high-dimensional random vectors

Yang¹

2021

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We consider two high-dimensional random vectors rx P R p and r y P R q with finite rank correlations. More precisely, suppose that rx " x `Az and r y " y `Bz, for independent random vectors x P R p , y P R q and z P R r with i.i.d. entries of mean zero and variance one, and two deterministic factor loading matrices A P R pˆr and B P R qˆr . With n i.i.d. observations of pr x, r yq, we study the sample canonical correlations between rx and r y. In this paper, we focus on the high-dimensional setting with a rank-r correlation. Let t1 ě t2 ě ¨¨¨ě tr be the squares of the population canonical correlation coefficients between rx and r y, and r λ1 ě r λ2 ě ¨¨¨ě r λr be the squares of the largest r sample canonical correlation coefficients. Under certain moment assumptions on the entries of x, y and z, we show that there exists a threshold tc P p0, 1q such that the following dichotomy happens. If ti ă tc, n 2{3 p r λi ´λ`q converges in law to the Tracy-Widom distribution, where λ`is the right-edge of the bulk spectrum of the sample canonical correlation matrix; if ti ą tc, ? np r λi ´θiq converges in law to a centered normal distribution, where θi ą λ`is a fixed outlier location determined by ti. Our results extend the ones in [5] for Gaussian vectors rx and r y. Moreover, we find that the variance of the limiting distribution of ? np r λi ´θiq also depends on the fourth cumulants of the entries of x, y and z, a phenomenon that cannot be observed in the Gaussian case.

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“…We copy Lemma 6.5 in Lee and Yin (2014) which is powerful to handle the expectation of S j 1 k 1 S j 2 k 2 • • • S jsks . One may also refer to Lemma 7.5 of Yang (2019) where the generalized Green function entries are involved.…”

Section: Proof Of Theorem 321 and Theorem 333mentioning

confidence: 99%

“…This method has been applied to prove local laws for di↵erent random matrix models, see Ding and Yang (2020b), Han et al (2018), Han et al (2016), Yang (2019. One of the challenges in the procedure is to remove the condition that the third moment vanishes, which relies heavily on the model structure.…”

Section: Introductionmentioning

confidence: 99%

“…One of the challenges in the procedure is to remove the condition that the third moment vanishes, which relies heavily on the model structure. This has been completely solved for the models in Han et al (2018); Knowles and Yin (2017) where a further expansion on the Green function was performed together with a careful parity analysis, but for the separable model studied in Yang (2019), the third moment vanishing condition is not so easy to remove, thus there are some additional conditions put on the third moment together with the spectral domain.…”

Section: Introductionmentioning

confidence: 99%

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The extreme eigenvalues of two types of random matrices

Zhang¹

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The fluctuations of extreme eigenvalues of a large random matrix model is a central topic in random matrix theory, motivated by applications in principle component analysis, factor analysis, or signal detection problems. This thesis establishes asymptotic distributions for the largest eigenvalues of two types of matrices under the high dimensional setting where the dimension goes to infinity proportionally with the sample size.I would like to express my deep and sincere attitude to my supervisor, Professor Pan Guangming, for his invaluable guidance and unfailing encouragement throughout my Ph.D. study. He led me to the scientific research, raised many interesting questions to me, and helped me a lot with his wide knowledge. I feel very fortunate for having the opportunity to work with him. I would like to thank Division of Mathematical Sciences of Nanyang Technological university for providing me the scholarships for pursuing the research study. I am thankful to my thesis committee members, Prof. Xiang Liming and Prof. Feng Qu for their help and guidance. I also appreciate many teachers, Prof.

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Edge universality of separable covariance matrices

Cited by 32 publications

References 66 publications

Sample canonical correlation coefficients of high-dimensional random vectors with finite rank correlations

Sample canonical correlation coefficients of high-dimensional random vectors with finite rank correlations

Limiting distribution of the sample canonical correlation coefficients of high-dimensional random vectors

The extreme eigenvalues of two types of random matrices

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