Expanders with respect to Hadamard spaces and random graphs

Abstract

Abstract. It is shown that there exists a sequence of 3-regular graphs {G n } ∞ n=1 and a Hadamard space X such that {G n } ∞ n=1forms an expander sequence with respect to X, yet random regular graphs are not expanders with respect to X. This answers a question of [NS11]. {G n } ∞ n=1 are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear time constant factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of martingale methods.

On Lipschitz extension from finite subsets

“…The metric d given in (69) satisfies d((x, 0), d(y, 0)) = ω 0 ( x − y 1 ) and d((x, 1), (y, 1)) = ω 1 ( x − y 1 ). As explained in Remark 5.5 of [MN15], it follows that both of the metric spaces (F n 2 × {0}, d) and (F n 2 × {1}, d) embed into ℓ 1 with O(1) distortion. We do not know whether the metric space (F n 2 × F 2 , d) admits an embedding into ℓ 1 with O(1) distortion.…”

Super-expanders and warped cones

“…Generalisations. We remark that the notion of expanders (non-linear spectral gaps) has been extensively studied also in the case when X is a metric space (see, for instance, [13], and references therein). Theorem 1 may also find applications in this context, because it remains true under very mild assumptions about space X, which are satisfied by large classes of metric spaces, notably CAT(0) spaces considered in [13].…”

Extension, Separation and Isomorphic Reverse Isoperimetry

“…A useful property[211, Lemma 5.4] of this truncated L 1 metric is c L 1 .L 6M embeds back into L 1 with bi-Lipschitz distortion O.1/. Theorem 122 gives a different proof of this since if X D `n 1 , then by (1.38) the right-hand side of (4.1) is equal to min¹2; kx yk 1 º=.2/.…”