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

2021/02/05 by Zongming Ma, Fan Yang, Ma, Zongming +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2102.03297

openalex publication_date 2021/02/05 · openalex created_date 2021/02/15 · openalex updated_date 2026/07/28

Abstract

Consider two random vectors \widetilde\mathbf x ∈ \mathbb R^p and \widetilde\mathbf y ∈ \mathbb R^q of the forms \widetilde\mathbf x=A\mathbf z+\mathbf C₁^1/2\mathbf x and \widetilde\mathbf y=B\mathbf z+\mathbf C₂^1/2\mathbf y, where \mathbf x∈ \mathbb R^p, \mathbf y∈ \mathbb R^q and \mathbf z∈ \mathbb R^r are independent vectors with i.i.d. entries of mean 0 and variance 1, \mathbf C₁ and \mathbf C₂ are p × p and q× q deterministic covariance matrices, and A and B are p× r and q× r deterministic matrices. With n independent observations of (\widetilde\mathbf x,\widetilde\mathbf y), we study the sample canonical correlations between \widetilde\mathbf x and \widetilde\mathbf y. We consider the high-dimensional setting with finite rank correlations. Let t₁≥ t₂ ≥ ⋯≥ t_r be the squares of the nontrivial population canonical correlation coefficients, and let \widetildeλ₁ ≥\widetildeλ₂≥⋯≥\widetildeλ_{p\wedge q} be the squares of the sample canonical correlation coefficients. If the entries of \mathbf x, \mathbf y and \mathbf z are i.i.d. Gaussian, then the following dichotomy has been shown in [7] for a fixed threshold t_c ∈(0, 1): for 1≤ i ≤ r, if t_i < t_c, then \widetildeλ_i converges to the right-edge λ₊ of the limiting eigenvalue spectrum of the sample canonical correlation matrix; if t_i>t_c, then \widetildeλ_i converges to a deterministic limit θ_i ∈ (λ₊,1) determined by t_i. In this paper, we prove that these results hold universally under the sharp fourth moment conditions on the entries of \mathbf x and \mathbf y. Moreover, we prove the results in full generality, in the sense that they also hold for near-degenerate t_i's and for t_i's that are close to the threshold t_c.

Citations

• On the distribution of the largest eigenvalue in principal components analysis

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