subgaussian

[[concept]]

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Definition

Vectors

If $X$ is mean-zero $L$ subgaussian, then $\mathbb{P}[|X|\ge t]\le 2\exp \left(-\frac{t^2}{2L^2}\right)$

Matrices

Sub-Gaussian Matrix

A zero-mean symmetric random matrix $Q$ is sub-Gaussian with parameter $V\in {\mathcal{S}}_{+}^{d\times d}$ (positive) if ${\Psi }_Q(\lambda )⪯e^{{\lambda }^{2}V/2}\text{for all }\lambda \in \mathbb{R}$ Where ${\Psi }_Q$ is the moment generating function.

Example

Suppose $Q=\epsilon B$ where $\epsilon \in \{\pm 1\}$ and $B$ is a fixed symmetric $d\times d$ matrix. Then ${\mathbb{E}}_{}\left[e^{\lambda Q}\right]={\sum }_{k=0}^{\infty }\frac{{\lambda }^{2k}}{(2k)!}{B}^{2k}⪯{\sum }_{k=1}^{\infty }\frac{1}{k!}{\left(\frac{{\lambda }^2{B}^2}{2}\right)}^k=e^{{\lambda }^2{B}^2/2}$ Thus $Q$ is subgaussian with $V=B^2=\text{Var}(Q)$

More generally, if $Q=gB$ where $g$ is ${\sigma }^2$ subgaussian with mean 0, then $v={\sigma }^2{B}^2$

Example

Now consider, $Q=\epsilon C$, where $C\in {\mathcal{S}}^{d\times d}$ is a random matrix and $\epsilon \in \{\pm 1\}$. Suppose the spectral norm $||C|{|}_2\le b$. Then, fixing $C$, we see that

\mathbb{E}_{\varepsilon}\left[e^{\lambda\varepsilon C}\right] &\preceq \exp\left( \frac{\lambda^2}{2}C^2 \right) \\ \lvert \lvert C \rvert \rvert _{2} \leq b \implies \exp\left( \frac{\lambda^2}{2}C^2 \right) &\preceq \exp\left( \frac{\lambda^2}{2}b^2 I \right) \\ \implies \Psi_{Q}(\lambda) &\preceq \exp\left( \frac{\lambda^2}{2}b^2 I \right)\quad \forall\,\lambda \in \mathbb{R} \end{align}$$ ie, $Q$ is a [[subgaussian matrix]] with $V = b^2I$

References

References

High-Dimensional Statistics - A Non-Asymptotic Viewpoint

• pg 25, Theorem 2.6 (equivalent defintions)

See Also

• subgaussian random vector
• subexponential matrix
• subgaussian matrices are subexponential matrices

Mentions

Mentions

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