Bounding the support of a measure from its marginal moments

Abstract

Abstract. Given all moments of the marginals of a measure μ on R n , one provides (a) explicit bounds on its support and (b) a numerical scheme to compute the smallest box that contains the support of μ.

Structural Analysis of Laplacian Spectral Properties of Large-Scale Networks

“…In [4], Lasserre developed an approach to find bounds on the support of an unknown density function when only a sequence of its moments is available. In order to adapt Problem 1 to May 21, 2018 DRAFT this framework, we need to introduce some definitions.…”

“…In what follows, we propose a solution to Problem 1 using a technique proposed by Lasserre in [4]. In that paper, the following problem was addressed: In the context of Problem 1, we have access to a truncated sequence of five spectral moments, (m k (L G )) 1≤k≤5 , corresponding to the unknown spectral density function ρ G and given by the expressions (2), (4), and (6).…”

“…Therefore, a solution to Problem 2 would directly provide an upper bound on the spectral gap, α ≥ λ 2 , and a lower bound on the spectral radius, β ≤ λ n . We now describe a numerical scheme proposed in [4] to solve Problem 2. This solution is based on a series of semidefinite programs in one variable.…”

“…In this section, we have presented an optimization-based approach to compute optimal bounds on the Laplacian spectral gap and spectral radius from a truncated sequence of Laplacian spectral moments. The truncated sequence of spectral moments (m k (L G )) 5 k=1 can be written in terms of local structural measurements using (2) and (4). Hence, the above methodology allows to compute bounds on the spectral radius and spectral gap of the Laplacian matrix given a collection of local structural features of the network.…”

Sharp pointwise gradient estimates for Riesz potentials with a bounded density

“…Remark that in contrast to the algebraic character of the classical Markov inequalities [5], (14) has a different, transcentental nature. The analogous two-dimensional L-problem is much less explored, recent works point out some direct applications of this problem to tomography, geophysics, the problem in particular has to do with the distribution of pairs of random variables or the logarithmic potential of a planar domain, see [11], [10], [16], [12].…”