subgaussian

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Definition

Vectors

If $X$ is mean-zero $L$ subgaussian, then

P [| X | \geq t] \leq 2 \exp (- \frac{t^{2}}{2 L^{2}})

Matrices

Sub-Gaussian Matrix

A zero-mean symmetric random matrix $Q$ is sub-Gaussian with parameter $V \in S_{+}^{d \times d}$ (positive) if

Ψ_{Q} (λ) ⪯ e^{λ^{2} V / 2} for all λ \in R

Where $Ψ_{Q}$ is the moment generating function.

Example

Suppose $Q = ε B$ where $ε \in {\pm 1}$ and $B$ is a fixed symmetric $d \times d$ matrix. Then

E_{} [e^{λ Q}] = \sum_{k = 0}^{\infty} \frac{λ^{2 k}}{(2 k)!} B^{2 k} ⪯ \sum_{k = 1}^{\infty} \frac{1}{k!} {(\frac{λ^{2} B^{2}}{2})}^{k} = e^{λ^{2} B^{2} / 2}

Thus $Q$ is subgaussian with $V = B^{2} = Var (Q)$

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

Example

Now consider, $Q = ε C$ , where $C \in S^{d \times d}$ is a random matrix and $ε \in {\pm 1}$ . Suppose the spectral norm $| | C | |_{2} \leq b$ . Then, fixing $C$ , we see that

\begin{aligned} E_{ε} [e^{λ ε C}] & ⪯ \exp (\frac{λ^{2}}{2} C^{2}) \\ | | C | |_{2} \leq b ⟹ \exp (\frac{λ^{2}}{2} C^{2}) & ⪯ \exp (\frac{λ^{2}}{2} b^{2} I) \\ ⟹ Ψ_{Q} (λ) & ⪯ \exp (\frac{λ^{2}}{2} b^{2} I) \forall λ \in R \end{aligned}

ie, $Q$ is a subgaussian matrix with $V = b^{2} I$

References

High-Dimensional Statistics - A Non-Asymptotic Viewpoint

pg 25, Theorem 2.6 (equivalent defintions)

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