We study the asymptotic spectral properties of high-dimensional Spearman correlation matrices for scale-mixture data. We consider observations of the form $x_t=\sigma_t \xi_t \in \mathbb{R}^N,$ where the coordinates of $\xi_t$ are i.i.d.\ and the scalar mixture variable $\sigma_t$ is shared by all coordinates. Under natural symmetry assumptions, the coordinates of $x_t$ are pairwise uncorrelated in both the Pearson and Spearman sense. Nevertheless, they are not independent when the mixture variable is non-degenerate. We show that this higher-order dependence survives the rank transformation and leaves a nontrivial spectral signature. In the proportional regime $N/T\to q\in(0,\infty),$ the empirical spectral distribution of the Spearman correlation matrix converges almost surely to a generalized Mar\v{c}enko--Pastur law governed by the limiting distribution of an effective rank variance. We also formulate a broader latent-variable extension, which covers, in particular, some scale-mixture models with correlated directional components. We discuss solvable examples and numerical approximations, motivated in part by heavy-tailed data in robust multivariate statistics, econometrics, and finance.
We provide limit theory for the trace of the squared sample correlation matrix $\mathbf R$, constructed from $n$ observations of a $p$-dimensional random vector with iid components. If the entries have finite fourth moment and $p$ and $n$ grow proportionally, it is known that $\operatorname{tr}({\mathbf R}^2)$ satisfie...
We study the smallest nonzero eigenvalue of the sample correlation matrix $\mathbf{R}_n$ formed from a $p_n \times n$ data matrix with i.i.d. real entries $\xi$ of mean zero and unit variance, in the high-dimensional regime $p_n / n \to \phi \in (0, \infty) \setminus \{1\}$. In the tall regime $\phi>1$, we prove almost...
Ze-Qin Lin, Guamgming Pan, Hao-Zhu Zhao et al.· 1 citation
The set of $n\times n$ correlation matrices, known as the elliptope, has volume decaying at the super-exponential rate $\exp\{-\tfrac14 n^2\log n\}$. We characterize where this vanishing volume concentrates. A uniform draw is entrywise close to the identity yet globally far from it and nearly singular: its maximum abso...
It is shown that for online / streaming algorithms the story is very different, and constant aspect ratio is insufficient for nonzero overlap, in stark contrast to ordinary high dimensional PCA, where nonzero overlap is possible at constant $n/d and improves as $n/d$ increases.
We investigate the geometric fluctuations of principal subspaces for high-dimensional covariance matrices through the squared Frobenius $\sin\Theta$ distance between the sample and population eigenspaces associated with the $r_p$ largest eigenvalues. An explicit first-order expansion and a central limit theorem are est...
We prove a sharp concentration inequality for the spectral norm of sparse random tensors with independent Bernoulli entries. Let $T$ be an order-$k$ tensor of dimension $n\times\cdots\times n$ with independent Bernoulli$(p)$ entries, where $k$ is fixed. For any $c,r>0$, we show that $\|T-\mathbb E T\|\le C_{k,r,c}\sqrt...
Zhixin Zhou, Yizhe Zhu· 0 citations
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