Recovering Lower-Dimensional Semialgebraic Support of a Measure from its Moments

Ruben Karapetyan; Shenyuan Ma; Ales Wodecki; Jakub Marecek

math.STSep 27, 2026

Recovering Lower-Dimensional Semialgebraic Support of a Measure from its Moments

Authors: Ruben Karapetyan, Shenyuan Ma, Ales Wodecki, Jakub Marecek

Organizations: Czech Technical University in Prague

Abstract

Recovering probability measures from their moments has numerous applications, esp. in connection with the method of moments in statistics and optimization. In the setting where measure need not be finitely atomic, but its support is known to be compact and semialgebraic with codimension at least one, the problem is still open. We combine moment-matrix kernel information with the Christoffel--Darboux kernel to provide a discrete approximation of the support. To validate the proposed approach, we test our algorithm on analytically computed moments and pseudo-moments arising from polynomial optimization problems without unique global minimizers. This complements well-known recent work on recovery of measures with algebraic support, where the kernel of a moment matrix can reveal polynomials vanishing on the support, and on recovery of sufficiently regular full-dimensional supports, where estimators constructed by thresholding the Christoffel--Darboux kernel are known to converge asymptotically to the support.

Figures & tables

	Algorithm 1			CD, ambient grid
Experiment	$d_{v}$	Samples	$d_{\mathrm{H}}$	$d_{g}$	Samples	$d_{\mathrm{H}}$
Moments in closed form ( Section 5.2 )
Parabola $y=x^{2},\ 0\leq x\leq 1$	2	$1.1\times 10^{4}$	0.014	4	$10^{6}$	0.095
Hyperbola $xy=1,\ 0.5\leq x\leq 1$	2	4945	0.002	4	$10^{6}$	0.114
Hyperbola 2 $x^{2}-y^{2}=1,\ 1\leq x\leq 2$	2	$1.4\times 10^{4}$	0.003	4	$10^{6}$	0.096
3D cone (Section 5.2 ) $z^{2}-x^{2}-y^{2}=0$ , $z\in[-1,-0.5]\cup[0.5,1]$	6	$2\times 10^{4}$	0.090	8	$10^{6}$	0.240

Table 1: Support recovery by Algorithm 1 and by CD thresholding on an ambient grid. Here

d_{v}

and

d_{g}

are the kernel degrees, Samples is the number of points at which the kernel is evaluated, and

d_{\mathrm{H}}

is the Hausdorff distance to the true support, computed against a fine discretization of

\supp\mu

. The grid baseline uses the higher degree and more samples in every row, and has the larger error in every row.

Figure 1 : The cone, with moments in closed form. Left: values of $\kappa_{6}$ at $20000$ points sampled uniformly on the recovered carrier $\{x_{3}^{2}-x_{2}^{2}-x_{1}^{2}=0\}$ , on a logarithmic color scale; the dichotomy appears as dark inner annuli and bright rims. Right: the $10600$ points accepted at $\tau_{6}=82.7$ ; both connected components are recovered.

Figure 2 : Number of evaluation points needed to reach Hausdorff errors $d_{\mathrm{H}}<1.0$ and $d_{\mathrm{H}}<0.6$ . The carrier estimator uses $\kappa_{1}$ on $\mathcal{Z}$ ; the grid baseline is allowed $\kappa_{3}$ in $\mathbb{R}^{2n}$ . Failed runs are plotted at $10^{8}$ for display only; they indicate failure within the tested budget and are not estimates of the number of samples needed.

Figure 3 : The parabola $y=x^{2}$ , $0\leq x\leq 1$ , with moments in closed form. 1) The CD kernel of degree $4$ on the grid. 2) Points of $\mathcal{Z}$ accepted by Algorithm 1 , $d_{\mathrm{H}}=0.014$ . 3) Grid points accepted at threshold $80$ ; the estimate has large gaps, $d_{\mathrm{H}}=0.095$ . 4) Grid points accepted at the higher threshold $500$ ; the gaps are filled, but points far from the support are admitted, $d_{\mathrm{H}}=0.17$ .

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4 : Thresholding the CD kernel $\kappa_{1}$ on the variety and on the grid. Left: values of $\kappa_{1}$ on the $1000\times 1000$ grid. Middle: points of the variety accepted at $\tau_{1}=3.36$ . Right: grid points accepted at $\tau_{1}=3.36$ .

Figure 5 : The CD kernels $\kappa_{2}$ and $\kappa_{3}$ , evaluated and thresholded on the $1000\times 1000$ grid. Left: values of $\kappa_{2}$ . Middle left: grid points accepted at $\tau_{2}=3.36$ . Middle right: values of $\kappa_{3}$ . Right: grid points accepted at $\tau_{3}=5.0$ .

Figure 6 : The CD kernel $\kappa_{2}$ for the semicircle. Left: values of $\kappa_{2}$ on the $1000\times 1000$ grid, on a logarithmic color scale; the kernel separates the two halves of the circle, although both have the same vanishing ideal. Middle: the $1.1\times 10^{4}$ accepted points of the variety, $d_{\mathrm{H}}(S^{\mathrm{variety}}_{2,\tau_{2}},S)=0.05$ . Right: the $2\times 10^{5}$ accepted grid points, $d_{\mathrm{H}}(S^{\mathrm{grid}}_{2,\tau_{2}},S)=0.69$ .

Figure 7 : The CD kernel $\kappa_{4}$ for the semicircle. Left: values of $\kappa_{4}$ on the $1000\times 1000$ grid, on a logarithmic color scale. Right: the accepted grid points, $d_{\mathrm{H}}(S^{\mathrm{grid}}_{4,\tau_{4}},S)=0.19$ .

Figure 8 : CD kernel $\kappa_{2}$ and the accepted variety points. Left: $\kappa_{2}$ evaluated on the variety, projected onto dimensions 1 and 2. Middle left: The accepted points projected onto dimensions 1 and 2. Middle right: $\kappa_{2}$ evaluated on the variety, projected onto dimensions 3 and 4. Right: The accepted points projected onto dimensions 3 and 4.

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Mixture models, such as Gaussian mixture models, are widely used in machine learning to represent complex data distributions. A key challenge, especially in high-dimensional settings, is to determine the mixture order and estimate the mixture parameters. We study the problem of approximating a target measure, available only through finitely many of its moments, by a mixture of distributions from a parametric family (e.g., Gaussian, exponential, Poisson), with approximation quality measured by the 2-Wasserstein or the total variation distance. Unlike many existing approaches, the parameter set is not assumed to be finite; it is modeled as a compact basic semi-algebraic set. We introduce a hierarchy of semidefinite relaxations with asymptotic convergence to the desired optimal value. In addition, when a certain rank condition is satisfied, the convergence is even finite and recovery of an optimal mixing measure is obtained. We also present an application to clustering, where our framework serves either as a stand-alone method or as a preprocessing step that yields both the number of clusters and strong initial parameter estimates, thereby accelerating convergence of standard (local) clustering algorithms.

Abstract

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