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Comparison of Metric Spectral Gaps

Assaf Naor (2014)

Analysis and Geometry in Metric Spaces

Let A = (aij) ∊ Mn(ℝ) be an n by n symmetric stochastic matrix. For p ∊ [1, ∞) and a metric space (X, dX), let γ(A, dpx) be the infimum over those γ ∊ (0,∞] for which every x1, . . . , xn ∊ X satisfy [...] Thus γ (A, dpx) measures the magnitude of the nonlinear spectral gap of the matrix A with respect to the kernel dpX : X × X →[0,∞). We study pairs of metric spaces (X, dX) and (Y, dY ) for which there exists Ψ: (0,∞)→(0,∞) such that γ (A, dpX) ≤Ψ (A, dpY ) for every symmetric stochastic A ∊ Mn(ℝ)...

Constant Distortion Embeddings of Symmetric Diversities

David Bryant, Paul F. Tupper (2016)

Analysis and Geometry in Metric Spaces

Diversities are like metric spaces, except that every finite subset, instead of just every pair of points, is assigned a value. Just as there is a theory of minimal distortion embeddings of fiite metric spaces into L1, there is a similar, yet undeveloped, theory for embedding finite diversities into the diversity analogue of L1 spaces. In the metric case, it iswell known that an n-point metric space can be embedded into L1 withO(log n) distortion. For diversities, the optimal distortion is unknown....

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