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Random ε-nets and embeddings in $ℓ^{N}_{∞}$

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Gordon Y., Litvak A., Pajor A., Tomczak-Jaegermann N.

Studia Mathematica

2007

tom 178

nr 1

91-98

We show that, given an n-dimensional normed space X, a sequence of $N = (8/ε)^{2n}$ independent random vectors $(X_{i})_{i=1}^{N}$, uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map $Γ: ℝ → ℝ^{N}$ defined by $Γx = (⟨x,X_{i}⟩)_{i=1}^{N}$ embeds X in $ℓ^{N}_{∞}$ with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into $ℓ_{∞}^{N}$ with asymptotically best possible relation between N, n, and ε.

Ograniczanie wyników

1 Studia Mathematica

1 Gordon Y.

1 Litvak A.

1 Pajor A.

1 Tomczak-Jaegermann N.

1 2007

Wyniki wyszukiwania

Random ε-nets and embeddings in $ℓ^{N}_{∞}$