Consider two random vectors $\wt{\bx} \in \R^p$ and $\wt{\by} \in \mathbb R^q$ of the forms $\wt{\bx}=A \mathbf z + \bC_1^{1/2}\mathbf x $ and $\wt{\by}=B \mathbf z + \bC_2^{1/2}\mathbf y $, where $\mathbf x \in \R^p$, $\mathbf y \in \R^q$ and $\mathbf z\in \R^r$ are independent random vectors with i.i.d. entries of zero mean and unit variance, $\bC_1$ and $\bC_2$ are $p \times p$ and $q\times q$ deterministic population covariance matrices, and $A$ and $B$ are $p \times r$ and $q\times r$ deterministic factor loading matrices. With $n$ independent observations of $(\wt{\mathbf x},\wt{\mathbf y})$, we study the sample canonical correlations between $\wt{\bx} $ and $\wt{\by}$. We consider the high-dimensional setting with finite rank correlations, that is, ${p}/{n}\to c_1$ and ${q}/{n}\to c_2$ as $n\to \infty$ for some constants $c_1\in (0,1)$ and $c_2\in (0,1-c_1)$, and $r$ is a fixed integer. Let $t_1\ge t_2 \ge \cdots\ge t_r\ge 0$ be the squares of the nontrivial population canonical correlation coefficients between $\wt {\bx}$ and $\wt{\by}$, and let $\wt\lambda_1 \ge \wt\lambda_2\ge \cdots \ge \wt\lambda_{p\wedge q}\ge 0$ 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 \cite{CCA} for a fixed threshold $t_c \in(0, 1)$: for any $1\le i \le r$, if $t_i < t_c$, then $\wt\lambda_i$ converges to the right-edge $\lambda_+$ of the limiting eigenvalue spectrum of the sample canonical correlation matrix, and moreover, $n^{2/3}(\wt\lambda_i-\lambda_+)$ converges weakly to the Tracy-Widom law; if $t_i>t_c$, then $\wt\lambda_i$ converges to a deterministic limit $\theta_i \in (\lambda_+, 1)$ that is determined by $c_1$, $c_2$ and $t_i$. In this paper, we prove that these results hold universally under the sharp fourth moment conditions on the entries of $\mathbf x$, $\mathbf y$ and $\mathbf z$. 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 $t_c$. Finally, we also provide almost sharp convergence rates for the sample canonical correlation coefficients under a general $a$-th moment assumption.

We give a new type of Schur-Weyl duality for the representations of a family of quantum subgroups and their centralizer algebra. We define and classify singly-generated, Yang-Baxter relation planar algebras. We present the skein theoretic construction of a new parameterized planar algebra. We construct infinitely many new subfactors and unitary fusion categories, and compute their trace formula as a closed-form expression, in terms of Young diagrams.

We give the formulation of a Riemannian Newton algorithm for solving a class of nonlinear eigenvalue problems by minimizing a total energy function subject to the orthogonality constraint. Under some mild assumptions, we establish the global and quadratic convergence of the proposed method. Moreover, the positive definiteness condition of the Riemannian Hessian of the total energy function at a solution is derived. Some numerical tests are reported to illustrate the efficiency of the proposed method for solving large-scale problems.

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