Probabilistic clustering of high dimensional norms

Abstract

Separating decompositions of metric spaces are an important randomized clustering paradigm that was formulated by Bartal in [Bar96] and is defined as follows. Given a metric space (X, dX ), its modulus of separated decomposability, denoted SEP(X, dX ), is the infimum over those σ ∈ (0, ∞] such that for every finite subset S ⊆ X and every ∆ > 0 there exists a distribution over random partitions P of S into sets of diameter at most ∆ such that for every x, y ∈ S the probability that both x and y do not fall into the same cluster of the random partition P is at most σdX (x, y)/∆.Here we obtain new bounds on SEP(X, · X ) when (X, · X ) is a finite dimensional normed space, yielding, as a special Our new bounds on the modulus of separated decomposability rely on extremal results for orthogonal hyperplane projections of convex bodies, specifically using the work [BN02] of Barthe and the author. This yields additional refined estimates, an example of which is that for every n ∈ N and k ∈ {1, . . . , n} we have SEP ( n 2 ) k k log(en/k), where ( n 2 ) k denotes the subset of R n consisting of all those vectors that have at most k nonzero entries, equipped with the Euclidean metric.The above statements have implications to the Lipschitz extension problem through its connection to random partitions that was developed by Lee and the author in [LN04,LN05]. Given a metric space (X, dX ), let e(X) denote the infimum over those K ∈ (0, ∞] such that for every * Supported by BSF grant 2010021, the Packard Foundation and the Simons Foundation. The research that is presented here was conducted under the auspices of the Simons Algorithms and Geometry (A&G) Think Tank.† A full version of this extended abstract, titled "Separating decompositions in high dimensions, extremal hyperplane projections, and Lipschitz extension" is forthcoming. It contains several additional results and extensions of the work that is presented here, but also this extended abstract covers some material that does not appear in the full version.

Extension, Separation and Isomorphic Reverse Isoperimetry

“…As [32] computes (1.41) exactly, the implicit constant factors in (1.42) can be evaluated, but in the present context such precision is of secondary importance. While (1.42) follows from [32] (see the deduction in [227]), we will give a self-contained proof of (1.42) in Section 6 as a special case of a more general result that we will use for other purposes as well. In the range q 2 .2; 1/, a different approach to computing (1.41) was found in [157].…”

Extension, separation and isomorphic reverse isoperimetry

“…As [BN02] computes (41) exactly, the implicit constant factors in (42) can be evaluated, but in the present context such precision is of secondary importance. While (42) follows from [BN02] (see the deduction in [Nao17a]), we will give a self-contained proof of (42) in Section 6 as a special case of a more general result that we will use for other purposes as well. In the range q ∈ (2, ∞), a different approach to computing (41) was found in [KRZ04].…”

Impossibility of almost extension

“…The lower bound on e(ℓ n ∞ ) in ( 14) is a classical combination of [Sob41] and [Lin64], and the upper bound on e(ℓ n ∞ ) in ( 14) is due to [Nao17a]. Remarkably, even the Euclidean case X = ℓ n 2 remains a tantalizing mystery, with the currently best known bounds being…”