Concentration dimension explained

In mathematics - specifically, in probability theory - the concentration dimension of a Banach space-valued random variable is a numerical measure of how "spread out" the random variable is compared to the norm on the space.

Definition

Let (B, || ||) be a Banach space and let X be a Gaussian random variable taking values in B. That is, for every linear functional ℓ in the dual space B^∗, the real-valued random variable ⟨ℓ, X⟩ has a normal distribution. Define

\sigma(X) = \sup \left\{ \left. \sqrt{\operatorname{E} [\langle \ell, X \rangle ^{2} ]} \right| \ell \in B ^{\ast} , \| \ell \| \leq 1 \right\}.

Then the concentration dimension d(X) of X is defined by

d(X) = (\operatorname{E)/([\|) X \| ^{2} ]}{\sigma(X) ^{2} }.

Examples

• If B is n-dimensional Euclidean space Rⁿ with its usual Euclidean norm, and X is a standard Gaussian random variable, then σ(X) = 1 and E[||''X''||²] = n, so d(X) = n.
• If B is Rⁿ with the supremum norm, then σ(X) = 1 but E[||''X''||²] (and hence d(X)) is of the order of log(n).

References

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