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edited Jun 29, 2021 at 16:19

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Let $d,m \to \infty$ (integers) with $m/d \to \rho \in (0, \infty)$. Let $C=(1/d)C_0$$C$ be a $d \times d$ psd matrix with $\|C_0\|_{op} = \mathcal O(1)$$trace(C)=\mathcal O(1)$, and let $w_1,\ldots,w_m$ be iid uniformly distributed on the unit-sphere in $\mathbb R^d$. Consider the quartic form $$ F := \frac{1}{m}\sum_{j,\ell=1}^m (w_j^\top w_\ell)(w_j^\top C w_\ell). $$

Question. What are good probabilistic lower and upper-bounds for $F$ only in terms of $\rho$ and the eigenvalues of $C_0$$C$ ?

For example, the solution for the case where $C_0$$C$ is diagonal will already be very helpful.

Isotropic example

Thanks to this post https://mathoverflow.net/a/334219/78539, we know that if $C_0 = (1/d) I_d$$C = (1/d) I_d$, then $F = m^{-1}\|WW^\top\|_F^2 = m^{-1}\sum_{j}\lambda_j(W W^\top)^2\overset{a.s}{\to} \langle \lambda^2\rangle_{\text{MP}(1/\rho)}$ (if I haven't made some scaling errors), where $\text{MP}(\gamma)$ is the Marchenko-Pastur law with parameter $\gamma$.

Let $d,m \to \infty$ (integers) with $m/d \to \rho \in (0, \infty)$. Let $C=(1/d)C_0$ be a $d \times d$ psd matrix with $\|C_0\|_{op} = \mathcal O(1)$, and let $w_1,\ldots,w_m$ be iid uniformly distributed on the unit-sphere in $\mathbb R^d$. Consider the quartic form $$ F := \frac{1}{m}\sum_{j,\ell=1}^m (w_j^\top w_\ell)(w_j^\top C w_\ell). $$

Question. What are good probabilistic lower and upper-bounds for $F$ only in terms of $\rho$ and the eigenvalues of $C_0$ ?

For example, the solution for the case where $C_0$ is diagonal will already be very helpful.

Isotropic example

Thanks to this post https://mathoverflow.net/a/334219/78539, we know that if $C_0 = (1/d) I_d$, then $F = m^{-1}\|WW^\top\|_F^2 = m^{-1}\sum_{j}\lambda_j(W W^\top)^2\overset{a.s}{\to} \langle \lambda^2\rangle_{\text{MP}(1/\rho)}$, where $\text{MP}(\gamma)$ is the Marchenko-Pastur law with parameter $\gamma$.

Let $d,m \to \infty$ (integers) with $m/d \to \rho \in (0, \infty)$. Let $C$ be a $d \times d$ psd matrix with $trace(C)=\mathcal O(1)$, and let $w_1,\ldots,w_m$ be iid uniformly distributed on the unit-sphere in $\mathbb R^d$. Consider the quartic form $$ F := \frac{1}{m}\sum_{j,\ell=1}^m (w_j^\top w_\ell)(w_j^\top C w_\ell). $$

Question. What are good probabilistic lower and upper-bounds for $F$ only in terms of $\rho$ and the eigenvalues of $C$ ?

For example, the solution for the case where $C$ is diagonal will already be very helpful.

Isotropic example

Thanks to this post https://mathoverflow.net/a/334219/78539, we know that if $C = (1/d) I_d$, then $F = m^{-1}\|WW^\top\|_F^2 = m^{-1}\sum_{j}\lambda_j(W W^\top)^2\overset{a.s}{\to} \langle \lambda^2\rangle_{\text{MP}(1/\rho)}$ (if I haven't made some scaling errors), where $\text{MP}(\gamma)$ is the Marchenko-Pastur law with parameter $\gamma$.