The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

Abstract

In this paper we improve the bounds for the Carathéodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, $$\mathbb {R}^n$$ R n , and $$[0,1]^n$$ [ 0 , 1 ] n . We also treat moment problems with small gaps. We find that for every $$\varepsilon >0$$ ε > 0 and $$d\in \mathbb {N}$$ d ∈ N there is a $$n\in \mathbb {N}$$ n ∈ N such that we can construct a moment functional $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ d → R which needs at least $$(1-\varepsilon )\cdot \left( {\begin{matrix} n+d\\ n\end{matrix}}\right) $$ ( 1 - ε ) · n + d n atoms $$l_{x_i}$$ l x i . Consequences and results for the Hankel matrix and flat extension are gained. We find that there are moment functionals $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le 2d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ 2 d → R which need to be extended to the worst case degree 4d, $$\tilde{L}:\mathbb {R}[x_1,\cdots ,x_n]_{\le 4d}\rightarrow \mathbb {R}$$ L ~ : R [ x 1 , ⋯ , x n ] ≤ 4 d → R , in order to have a flat extension.

The multidimensional truncated Moment Problem: Shape and Gaussian Mixture Reconstruction from Derivatives of Moments

“…And this was even done with the restriction a 1 = • • • = a k = a > 0. In [dDK19] we proved new lower bounds for the Carathéodory numbers for Dirac measures which grow asymptotically close to the Richter upper bound. Now we show that for Gaussian mixtures the same lower bounds hold even when arbitrary variances are allowed.…”

“…So C M A2,4 ≥ 6, i.e., there is a moment sequence/functional on R[x 1 , x 2 ] ≤4 which can be represented by a sum of 6 Gaussians but not less. The upper bound for the Dirac measures in the projective case is also 6 [Rez92] In [dDK19] the point evaluation on the grid was extended to higher dimensions and improved lower bounds where found. In fact, the following result was shown.…”

“…, A k ∈ R n×n we examine the number k of mixtures required to represent a moment sequence s, i.e., its minimal number, the (mixture) Carathéodory number C M A (s). Based on very recent results on the Carathéodory number C A (number of Dirac delta measures, i.e., point evaluations) in [RS18,dDS18a,dDS18b] and especially [dDK19] we derive new lower bounds and asymptotic limits for the case of mixtures as well. We show that a non-zero (polynomial) function p with finitely many zeros Z(p) gives a moment sequence s, resp.…”

“…[dD19],[dDK19], Theorem 6.22, and Example 6.21 explicitly provide the following.Corollary 6.23. Let d ∈ N and X = R. For the one-dimensional Gaussian (and log-normal) measures with A = R[x] ≤d we have Example 6.21 and Theorem 6.22 gives the lower bound and [dD19, Cor.…”