A Spectral Gap Precludes Low-Dimensional Embeddings
نویسنده
چکیده
We prove that there is a universal constant C > 0 with the following property. Suppose that n ∈ N and that A = (aij) ∈ Mn(R) is a symmetric stochastic matrix. Denote the second-largest eigenvalue of A by λ2(A). Then for any finite-dimensional normed space (X, ∥ · ∥) we have ∀x1, . . . , xn ∈ X, dim(X) > 1 2 exp ( C 1− λ2(A) √ n ( ∑n i=1 ∑n j=1 ∥xi − xj∥ 2 ∑n i=1 ∑n j=1 aij∥xi − xj∥ ) 1 2 )
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