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dohmatob
dohmatob
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Asymptotics of $(1/n)w^\top$w^\top G^2 w$, where $w$ is a unit-vector, $G:=X^T(XX^T+t I_n)^{-1}X$, $t > 0$, and $X$ is an $n\times d$ gaussian random matrix

Let $X$ be an random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $w$ be a unit-vector in $\mathbb R^d$. With $\lambda>0$, and define $G:=X^\top(XX^\top + \lambda I_n)^{-1}X$. Finally, defined $\alpha := (1/d)w^\top G^2 w$$\alpha := w^\top G^2 w$.

Question. In the limit $n,d \to \infty$ with $n/d \to \rho \in (0,\infty)$, what is the limitting value of $\alpha$ as a function of $\lambda$ and $\rho$ ?

A useful subcase is when $\lambda \to 0^+$.

Question. What is the value of $\lim_{\lambda \to 0^+}\lim_{n,d \to \infty \\ n/d \to \rho}\alpha$ as a function of $\rho$ ?

Asymptotics of $(1/n)w^\top G^2 w$, where $w$ is a unit-vector, $G:=X^T(XX^T+t I_n)^{-1}X$, $t > 0$, and $X$ is an $n\times d$ gaussian random matrix

Let $X$ be an random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $w$ be a unit-vector in $\mathbb R^d$. With $\lambda>0$, and define $G:=X^\top(XX^\top + \lambda I_n)^{-1}X$. Finally, defined $\alpha := (1/d)w^\top G^2 w$.

Question. In the limit $n,d \to \infty$ with $n/d \to \rho \in (0,\infty)$, what is the limitting value of $\alpha$ as a function of $\lambda$ and $\rho$ ?

A useful subcase is when $\lambda \to 0^+$.

Question. What is the value of $\lim_{\lambda \to 0^+}\lim_{n,d \to \infty \\ n/d \to \rho}\alpha$ as a function of $\rho$ ?

Asymptotics of $w^\top G^2 w$, where $w$ is a unit-vector, $G:=X^T(XX^T+t I_n)^{-1}X$, $t > 0$, and $X$ is an $n\times d$ gaussian random matrix

Let $X$ be an random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $w$ be a unit-vector in $\mathbb R^d$. With $\lambda>0$, and define $G:=X^\top(XX^\top + \lambda I_n)^{-1}X$. Finally, defined $\alpha := w^\top G^2 w$.

Question. In the limit $n,d \to \infty$ with $n/d \to \rho \in (0,\infty)$, what is the limitting value of $\alpha$ as a function of $\lambda$ and $\rho$ ?

A useful subcase is when $\lambda \to 0^+$.

Question. What is the value of $\lim_{\lambda \to 0^+}\lim_{n,d \to \infty \\ n/d \to \rho}\alpha$ as a function of $\rho$ ?

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dohmatob
dohmatob
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  • 76

Asymptotic limit Asymptotics of $(1/n)w^\top G^2 w$, where $w$ is a fixed unit-vector, $G:=X^\top=X^T(XX^\top+tXX^T+t I_n)^{-1}X$, $t > 0$, and $X$ is an $n\times d$ gaussian random matrix

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dohmatob
dohmatob
  • 6.9k
  • 1
  • 18
  • 76

Asymptotic limit of $(1/n)w^\top G^2 w$ where $w$ is a fixed unit-vector, $G:=X^\top(XX^\top+t I_n)^{-1}X$, $t > 0$, and $X$ is gaussian random matrix

Let $X$ be an random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $w$ be a unit-vector in $\mathbb R^d$. With $\lambda>0$, and define $G:=X^\top(XX^\top + \lambda I_n)^{-1}X$. Finally, defined $\alpha := (1/d)w^\top G^2 w$.

Question. In the limit $n,d \to \infty$ with $n/d \to \rho \in (0,\infty)$, what is the limitting value of $\alpha$ as a function of $\lambda$ and $\rho$ ?

A useful subcase is when $\lambda \to 0^+$.

Question. What is the value of $\lim_{\lambda \to 0^+}\lim_{n,d \to \infty \\ n/d \to \rho}\alpha$ as a function of $\rho$ ?