Let $X_N$ be a $N \times N$ symmetric matrix such that the probability distributions of all its independent entries (not necessarily identically distributed) satisfy the log-sobolev-inequality with constant $\frac{c}{N}$. Let its eigenvalues be $\lambda_1 \leq \lambda_2 \leq..\leq \lambda_N$. Then apparently the following are true for any Lipschitz function $f$ with Lipschitz constant $\vert f \vert$,
$P( \vert Tr[ f(X_N)] - \mathbb{E}[Tr[f(X_N) ] ] \vert \geq \delta N ) \leq 2 e^{-\frac{ N^2 \delta^2 }{4c \vert f\vert^2 } }$
$P( \vert f( \lambda_k (X_N) ) - \mathbb{E}[f(\lambda_k (X_N)) ] \vert \geq \delta ) \leq 2 e^{-\frac{ N \delta^2 }{4c \vert f\vert^2 } }$
- Apparently the above follow trivially from the Herbst's theorem but I can't see how. A naive and direct application of it on the functions $ Tr[ f(X_N)]$ and $f( \lambda_k (X_N) )$ gives me different constants. It would be great if I can see a calculation of this.
The definitions I am using,
For a function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ the Lipschitz constant, $\vert f \vert := sup_{x \neq y, x,y \in \mathbb{R}^n } \frac{ \vert f(x) - f(y) \vert }{ \vert \vert x - y \vert \vert _2 } $
A probability measure $P$ on $\mathbb{R}$ is said to satisfy the log-sobolev-inequality with constant $c$ if for any $f \in L^2(P)$, we have, $\int f^2 log \left [ \frac{ f^2 }{\int f^2 dP } \right ] dP \leq 2c \int \vert f'\vert ^2 dP $
Herbst's Theorem : If the probability measure $P$ satisfies the log-sobolev-inequality on $\mathbb{R}^M$ with constant $c$ and $f$ is a Lipschitz function on $\mathbb{R}^M$ with Lipscitz constant $\vert f\vert$. Then for all $\lambda \in \mathbb{R}$ and $\delta > 0$ we have,
$\mathbb{E}_P [ e^{\lambda (f - \mathbb{E}_P[f] ) }] \leq e^{\frac {c \lambda ^2 \vert f \vert ^2 }{2}}$
$P [ \vert f - \mathbb{E}_P[f] \vert \geq \delta ] \leq 2 e^{-\frac {\delta^2 }{2c \vert f\vert ^2}}$