Asymptotic null and non-null distributions of test statistics for redundancy in high-dimensional canonical correlation analysis

Oda, Ryoya; Yanagihara, Hirokazu; Fujikoshi, Yasunori

doi:10.1142/s2010326319500011

Cited by 5 publications

(3 citation statements)

References 10 publications

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“…So far, there are only a handful of works devoted to this topic. Fujikoshi in [20] derived the asymptotic distributions of the canonical correlation coefficients when q is fixed while p is proportional to n. Oda et al in [39] considered the problem of testing for redundancy in high dimensional canonical correlation analysis. Recently, with certain sparsity assumption, the theoretical results and potential applications of high-dimensional sparse CCA have been discussed in [23,22].…”

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confidence: 99%

Canonical correlation coefficients of high-dimensional Gaussian vectors: Finite rank case

Bao¹,

Hu²,

Pan³

et al. 2019

Ann. Statist.

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Consider a Gaussian vector z = (x , y ) , consisting of two subvectors x and y with dimensions p and q respectively. With n independent observations of z, we study the correlation between x and y, from the perspective of the Canonical Correlation Analysis. We investigate the high-dimensional case: both p and q are proportional to the sample size n. Denote by Σuv the population crosscovariance matrix of random vectors u and v, and denote by Suv the sample counterpart. The canonical correlation coefficients between x and y are known as the square roots of the nonzero eigenvalues of the canonical correlation matrix Σ −1 xx ΣxyΣ −1 yy Σyx. In this paper, we focus on the case that Σxy is of finite rank k, i.e. there are k nonzero canonical correlation coefficients, whose squares are denoted by r1 ≥ · · · ≥ r k > 0. We study the sample counterparts of ri, i = 1, . . . , k, i.e. the largest k eigenvalues of the sample canonical correlation matrix S −1 xx SxyS −1 yy Syx, denoted by λ1 ≥ · · · ≥ λ k . We show that there exists a threshold rc ∈ (0, 1), such that for each i ∈ {1, . . . , k}, when ri ≤ rc, λi converges almost surely to the right edge of the limiting spectral distribution of the sample canonical correlation matrix, denoted by d+. When ri > rc, λi possesses an almost sure limit in (d+, 1], from which we can recover ri's in turn, thus provide an estimate of the latter in the high-dimensional scenario. We also obtain the limiting distribution of λi's under appropriate normalization. Specifically, λi possesses Gaussian type fluctuation if ri > rc, and follows Tracy-Widom distribution if ri < rc. Some applications of our results are also discussed.

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confidence: 99%

Canonical correlation coefficients of high-dimensional Gaussian vectors: Finite rank case

Bao¹,

Hu²,

Pan³

et al. 2019

Ann. Statist.

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show abstract

“…Assuming some sparsity conditions, the theory of high-dimensional sparse CCA and its applications were discussed in [26,27]. In [46], the asymptotic null and non-null distributions of some test statistics for redundancy are derived for high-dimensional CCA. In [38], the authors studied the asymptotic behaviours of the likelihood ratio processes of CCA under the null hypothesis of no spikes and the alternative hypothesis of a single spike.…”

Section: Related Workmentioning

confidence: 99%

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

Yang¹

2021

Preprint

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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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“…Under certain sparsity assumptions, the theory of high-dimensional sparse CCA and it applications have been discussed in [28,29]. In [40], the authors derived asymptotic null and non-null distributions of several test statistics for tests of redundancy in high-dimensional CCA. In [35], the authors studied the asymptotic behaviors of the likelihood ratio processes of CCA under the null hypothesis of no spikes and the alternative hypothesis of a single spike.…”

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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Asymptotic null and non-null distributions of test statistics for redundancy in high-dimensional canonical correlation analysis

Cited by 5 publications

References 10 publications

Canonical correlation coefficients of high-dimensional Gaussian vectors: Finite rank case

Canonical correlation coefficients of high-dimensional Gaussian vectors: Finite rank case

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

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

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