Comparison of Metric Spectral Gaps

Abstract

Let A = (a ij ) ∈ Mn(R) be an n by n symmetric stochastic matrix. For p ∈ [ , ∞) and a metric space (X, d X

Metric Dimension Reduction: A Snapshot of the Ribe Program

“…All of the available examples of n-point subsets of 1 for which the 1 analogue of the JL lemma fails (namely if k = O(log n), then they do not embed with O(1) distortion into k 1 ) actually embed into the real line R with O(1) average distortion; this follows from [221]. Specifically, the examples in [58,151] are the shortest-path metric on planar graphs, and the example in Theorem 23 is O(1)-doubling, and both of these classes of metric spaces are covered by [221]; see also [188,Section 7] for generalizations. Thus, the various known proofs which demonstrate that the available examples cannot be embedded into a low dimensional subspace of 1 argue that any such lowdimensional embedding must highly distort some distance, but this is not so for a typical distance.…”

“…This definition is implicit in [105], and appeared explicitly in [196]; see [177,188,178] for a detailed treatment. It suffices to note here that if (H, · H ) is a Hilbert space and p = 2, then by expanding the squares one directly sees that γ(A, · 2 H ) = 1/(1 − λ 2 (A)) is the reciprocal of the spectral gap of A.…”

“…We will clarify what we mean by this later; a detailed discussion appears in [17]. The need to make the interpolation-based proof in [190] constructive/algorithmic led us to merge the argument in [190] with the proof of a theorem from [188], rather than quoting and using the latter as a "black box" as we did in [190]. In doing so, we realized that for the purpose of obtaining only the weaker bound (70) one could more efficiently combine [188] and [190] so as to skip the use of complex interpolation and to obtain the estimate (70) as well as its nonlinear Rayleigh quotient counterpart.…”

“…The need to make the interpolation-based proof in [190] constructive/algorithmic led us to merge the argument in [190] with the proof of a theorem from [188], rather than quoting and using the latter as a "black box" as we did in [190]. In doing so, we realized that for the purpose of obtaining only the weaker bound (70) one could more efficiently combine [188] and [190] so as to skip the use of complex interpolation and to obtain the estimate (70) as well as its nonlinear Rayleigh quotient counterpart. Thus, despite superficial differences, the argument below amounts to unravelling the proofs in [188,190] and removing steps that are needed elsewhere but not for (70).…”

“…In doing so, we realized that for the purpose of obtaining only the weaker bound (70) one could more efficiently combine [188] and [190] so as to skip the use of complex interpolation and to obtain the estimate (70) as well as its nonlinear Rayleigh quotient counterpart. Thus, despite superficial differences, the argument below amounts to unravelling the proofs in [188,190] and removing steps that are needed elsewhere but not for (70). At present, we do not have a proof of the stronger inequality (69) that differs from its proof in [190], and the interpolation-based approach of [190] is used for more refined algorithmic results in the forthcoming work [18].…”