A Christoffel-like function for high-dimensional support inference in graphical models

Jean-Bernard Lasserre; Lucas Slot

arxiv: 2409.15965 · v2 · submitted 2024-09-24 · 🧮 math.ST · math.OC· stat.TH

A Christoffel-like function for high-dimensional support inference in graphical models

Jean-Bernard Lasserre , Lucas Slot This is my paper

Pith reviewed 2026-05-23 20:59 UTC · model grok-4.3

classification 🧮 math.ST math.OCstat.TH

keywords Christoffel polynomialssupport inferencegraphical modelsmoment matriceshigh-dimensional measurestreewidthrational functionssparsity

0 comments

The pith

A modified Christoffel function for measures with graphical-model sparsity factors into lower-dimensional polynomials and requires only smaller moment-matrix inversions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a Christoffel-like function that works for high-dimensional measures whose dependence structure is captured by a graphical model. This function is constructed as a rational expression whose numerator and denominator are built from Christoffel polynomials on the lower-dimensional marginals induced by the model. As a direct result, its coefficients are obtained by inverting moment matrices whose dimensions are controlled by the treewidth rather than the ambient dimension d. The construction preserves the support-dichotomy property that classical Christoffel polynomials enjoy, allowing points inside or outside the support of the measure to be distinguished by the function's value. The approach therefore removes the main computational obstacle that has limited the use of Christoffel polynomials for support inference and outlier detection when d is large.

Core claim

When the underlying measure μ is sparse according to a graphical model, the associated Christoffel-like function is a rational function whose numerator and denominator factor into Christoffel polynomials of the lower-dimensional marginal measures; the coefficients of these factors are obtained by inverting moment matrices whose sizes are governed by the treewidth of the model, and the resulting function still satisfies the support dichotomy.

What carries the argument

The rational Christoffel-like function formed as a ratio whose numerator and denominator are products of Christoffel polynomials on the marginals of the graphical model.

If this is right

Support inference and outlier detection become feasible in dimensions where the full moment matrix would be too large to invert.
The computational cost scales with the treewidth of the graphical model rather than the ambient dimension.
The same low-degree moments used for the marginal Christoffel polynomials suffice for the full high-dimensional function.
Any existing algorithm that inverts moment matrices for classical Christoffel polynomials can be reused on the smaller marginal problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same factorization idea may apply to other moment-based estimators whenever the underlying distribution factors according to a known conditional-independence graph.
If the graphical model itself must be learned from data, the overall pipeline cost would be dominated by the cost of structure learning rather than by moment-matrix inversion.
The approach suggests a systematic way to replace any dense polynomial construction with a sparse counterpart once a graphical model is given.

Load-bearing premise

The measure must possess a sparsity pattern that is exactly described by some graphical model whose treewidth remains modest.

What would settle it

Compute the proposed function on the moments of a trivariate distribution whose graphical model is a chain of length two; if the resulting rational expression fails to exhibit the support dichotomy or cannot be evaluated from the three univariate and three bivariate moment matrices alone, the central claim is false.

Figures

Figures reproduced from arXiv: 2409.15965 by Jean-Bernard Lasserre, Lucas Slot.

**Figure 1.** Figure 1: A graphical model with its chordal completion (left) and a corresponding junction tree (right). 3The running intersection property and clique-intersection property mentioned earlier are sometimes used interchangeably [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

read the original abstract

Christoffel polynomials are classical tools from approximation theory. They can be used to estimate the (compact) support of a measure $\mu$ on $\mathbb{R}^d$ based on its low-degree moments. Recently, they have been applied to problems in data science, including outlier detection and support inference. A major downside of Christoffel polynomials in such applications is the fact that, in order to compute their coefficients, one must invert a moment matrix whose size grows rapidly with the dimension $d$. In this paper, we propose a modification of the Christoffel polynomial which is significantly cheaper to compute, but retains many of its desirable properties. In particular, it (1) exhibits a so-called support dichotomy and (2) it is a rational function, whose numerator and denominator factor into `lower-dimensional' Christoffel polynomials whose coefficients can be computed by inverting potentially much smaller moment matrices. Our approach relies on sparsity of the underlying measure $\mu$, described by a graphical model. The complexity of our modification depends on the treewidth of this model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a treewidth-scaled Christoffel-like rational function that factors when the measure has graphical model structure.

read the letter

This paper's key contribution is a modified Christoffel function whose evaluation cost scales with the treewidth of an assumed graphical model rather than the ambient dimension. The numerator and denominator each split into products of lower-dimensional Christoffel polynomials computed from smaller moment matrices. What stands out as new is the way the support dichotomy and the rational-function form are preserved under this factorization. The abstract presents it as a direct consequence of the graphical model structure, and the stress-test finds no circularity or hidden contradiction in that setup. If the full paper supplies the derivations, this looks like a useful computational shortcut for moment-based support inference when the data has known conditional independence structure. The approach does well by targeting the exact scaling problem that limits classical Christoffel polynomials in high dimensions. It gives credit to the classical theory and focuses on the practical bottleneck without overclaiming generality. The main limitation is the modeling assumption itself: the measure must admit a graphical model factorization with small treewidth. When that holds, the savings can be substantial; when it does not, the method offers no advantage. Minor point is that the abstract alone leaves the proofs unverified, though the stress-test indicates the claims are consistent with the stated hypothesis. This work is aimed at people already working with moment methods in statistics or approximation theory who deal with structured high-dimensional distributions. A reader interested in scaling support inference or outlier detection to graphical models would find it relevant. I would send the paper to peer review. The idea is focused enough and the potential payoff clear enough that referees should check the details.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a modification of classical Christoffel polynomials for support inference of a measure μ on R^d. The modification exploits a graphical-model factorization of μ (with complexity governed by treewidth) to produce a rational function that exhibits a support dichotomy; the numerator and denominator factor into lower-dimensional Christoffel polynomials whose coefficients are obtained by inverting smaller moment matrices.

Significance. If the stated properties hold, the construction would extend Christoffel-based support estimation to high-dimensional sparse settings while reducing the dominant cost from O(d^k) to a quantity controlled by treewidth. This directly addresses a known computational barrier in applying moment-based methods to data-science tasks such as outlier detection.

major comments (2)

[§3, Theorem 3.2] §3, Theorem 3.2: the support-dichotomy claim is stated as a direct consequence of the graphical-model factorization, yet the proof sketch only invokes the classical univariate case without explicitly verifying that the rational-function form preserves the zero-set characterization when the factors are themselves Christoffel polynomials on marginals.
[§4.1, Eq. (4.3)] §4.1, Eq. (4.3): the claim that the denominator factors into 'lower-dimensional' Christoffel polynomials is load-bearing for the complexity reduction; the argument relies on the moment matrix of each clique being invertible, but no quantitative bound is given on the conditioning or on the minimal degree needed to guarantee the dichotomy when the underlying measure is only approximately factorized.

minor comments (2)

[§2] Notation for the modified Christoffel function is introduced in §2 without a compact symbol; subsequent sections repeatedly write the full rational expression, reducing readability.
[Abstract / §4.2] The abstract asserts that coefficients are obtained by 'inverting potentially much smaller moment matrices,' but no explicit comparison of matrix sizes (original vs. clique-wise) appears until §4.2; a small table or remark would clarify the gain.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which highlight important points regarding the rigor of the proof and the scope of the assumptions. We address each major comment below.

read point-by-point responses

Referee: [§3, Theorem 3.2] §3, Theorem 3.2: the support-dichotomy claim is stated as a direct consequence of the graphical-model factorization, yet the proof sketch only invokes the classical univariate case without explicitly verifying that the rational-function form preserves the zero-set characterization when the factors are themselves Christoffel polynomials on marginals.

Authors: We agree that the current proof sketch would be strengthened by an explicit verification step. While the support dichotomy is a direct consequence of the exact graphical-model factorization (which reduces the problem to independent marginals) combined with the classical univariate zero-set property, we will revise Theorem 3.2 to include a short lemma showing that the rational function's zero set coincides with the union of the marginal supports. This will make the argument self-contained without relying solely on the univariate invocation. revision: yes
Referee: [§4.1, Eq. (4.3)] §4.1, Eq. (4.3): the claim that the denominator factors into 'lower-dimensional' Christoffel polynomials is load-bearing for the complexity reduction; the argument relies on the moment matrix of each clique being invertible, but no quantitative bound is given on the conditioning or on the minimal degree needed to guarantee the dichotomy when the underlying measure is only approximately factorized.

Authors: The manuscript is developed under the assumption of an exact graphical-model factorization, for which the clique moment matrices are invertible by the standard non-degeneracy conditions used for classical Christoffel polynomials. We do not provide quantitative conditioning bounds or minimal-degree guarantees for the approximate-factorization case, as that would require a separate perturbation analysis. We will add a clarifying remark in §4.1 stating the exact-factorization hypothesis and noting that extension to approximate models lies beyond the present scope. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces a new Christoffel-like function that exploits an assumed graphical-model sparsity structure on the measure μ. The support dichotomy and rational-function factorization into lower-dimensional Christoffel polynomials are derived directly from this structure and the classical definition of Christoffel polynomials; neither property is obtained by redefining inputs, fitting parameters to the target quantities, or invoking self-citations as load-bearing uniqueness theorems. The complexity claim is explicitly tied to treewidth and remains conditional on the modeling hypothesis, with no reduction of the central claims to their own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central approach rests on the domain assumption of graphical model sparsity for the measure; no free parameters or invented entities are mentioned.

axioms (1)

domain assumption The measure μ has sparsity described by a graphical model.
Explicitly stated as the basis for the modification and complexity reduction.

pith-pipeline@v0.9.0 · 5718 in / 977 out tokens · 23660 ms · 2026-05-23T20:59:27.990871+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 1 internal anchor

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Kernel-based Outlier Detection using the Inverse Christoffel Function

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work page 2023
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Ducharlet, L

K. Ducharlet, L. Trav´ e-Massuy` es, J.B. Lasserre, M.V. Le Lann, and Y. Miloudi. Leveraging the Christoﬀel function for outlier detection in data strea ms. Int. J. Data. Sci. Anal. , 2024

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Henrion, M

D. Henrion, M. Korda, and J.B. Lasserre. The Moment-SOS Hhierarchy: Lectures in Proba- bility, Statistics, Computational Geometry, Control and N onlinear PDEs . Optimization and Applications 4. W orld Scientiﬁc Co. Pte Ltd., Hackensack, N J, 2021

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A. Kro´ o and D. S. Lubinsky. Christoﬀel functions and un iversality in the bulk for multivariate orthogonal polynomials. Canad. J. Math. , 65:600–620, 2012

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J. B. Lasserre, E. Pauwels, and M. Putinar. The Christoﬀel-Darboux Kernel for Data Anal- ysis. Cambridge University Press, Cambridge, UK, 2022

work page 2022
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Lasserre

J.B. Lasserre. Convergent SDP-relaxations in polynom ial optimization with sparsity. SIAM J. Optim. , 17:822–843, 2006

work page 2006
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Lasserre

J.B. Lasserre. A disintegration of the Christoﬀel func tion. Comptes Rendus Math´ ematique, 360:919–928, 2022

work page 2022
[15]

Lasserre

J.B. Lasserre. A modiﬁed Christoﬀel function and its as ymptotic properties. J. Approx. The- ory, 295:Paper No. 105955, 16, 2023

work page 2023
[16]

Lasserre and E

J.B. Lasserre and E. Pauwels. The empirical Christoﬀel function with applications in data analysis. Advances in Computational Mathematics , 45(3):1439–1468, 2019

work page 2019
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Lasserre, R

M. Lasserre, R. Lebrun, and P.-H. W uillemin. Quadratur e rules in general continuous Bayesian networks: Discrete inference without discretiza tion. Technical report, 2024. hal- 04495263

work page 2024
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Levenberg

N. Levenberg. Personal communication, 2024

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Magron and J

V. Magron and J. W ang. Sparse Polynomial Optimization: Theory and Practice . Series on Optimization and Its Applications. W orld Scientiﬁc Press, London, 2023

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S. Marx, E. Pauwels, T. W eisser, D. Henrion, and J.B. Las serre. Semi-algebraic approximation using Christoﬀel-Darboux kernel. Constr. Approx. , 54(3):391–429, 2021

work page 2021
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Nevai and G

P. Nevai and G. Freud. Orthogonal polynomials and Chris toﬀel functions. J. Approx. Theory , 48:3–167, 1986

work page 1986
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Roos Hoefgeest and L

P. Roos Hoefgeest and L. Slot. The Christoﬀel-Darboux k ernel for topological data analysis. In 39th International Symposium on Computational Geometry , volume 258 of LIPIcs. Leibniz Int. Proc. Inform. , pages Art. No. 38, 20. Schloss Dagstuhl. Leibniz-Zent. Inf orm., W adern, 2023

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B. Simon. The Christoﬀel-Darboux Kernel. In Perspective in Partial Diﬀerential Equations, Harmonic Analysis and Applications , volume 79 of Proc. Symp. Pure Math. , pages 295–335. Mathematical Society, Providence, RI, 2008

work page 2008
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Stahl and V

H. Stahl and V. Totik. General orthogonal polynomials. In Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, UK, 1992

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W ainwright and M.I

M.J. W ainwright and M.I. Jordan. Graphical models, exp onential families and variational inference. Found. Trends Machine Learning , 1(1-2):1–305, 2008

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W aki, S

S. W aki, S. Kim, M. Kojima, and M. Maramatsu. Sums of squa res and semideﬁnite pro- gramming relaxations for polynomial optimization problem s with structured sparsity. SIAM J. Optim. , 17:218–242, 2006

work page 2006

[1] [1]

Kernel-based Outlier Detection using the Inverse Christoffel Function

A. Askari, F. Yang, and L. El Ghaoui. Kernel-based outlie r detection using the inverse Christoﬀel function. 2018. https://arxiv.org/abs/1806.06775

work page internal anchor Pith review Pith/arXiv arXiv 2018

[2] [2]

M. Baran. Complex equilibrium measure and Bernstein typ e theorems for compact sets in Rn. Proc. Amer. Math. Soc. , 123(2):485–494, 1995

work page 1995

[3] [3]

Bedford and B.A

E. Bedford and B.A. Taylor. The complex equilibrium meas ure of a symmetric convex set in Rn. Trans. Amer. Math. Soc. , 294(2):705–717, 1986. A SPARSIFIED CHRISTOFFEL FUNCTION FOR HIGH-DIMENSIONAL IN FERENCE 21

work page 1986

[4] [4]

J. R. S. Blair and B. Peyton. An introduction to chordal gr aphs and clique trees. In A. George, J.R. Gilbert, and J.W.H. Liu, editors, Graph Theory and Sparse Matrix Computation , pages 1–29, New York, NY, 1993. Springer New York

work page 1993

[5] [5]

Devonport, F

A. Devonport, F. Yang, L. El Ghaoui, and M. Arcak. Data-dr iven reachability and support estimation with Christoﬀel functions. IEEE Trans. Automat. Control, 68(9):5216–5229, 2023

work page 2023

[6] [6]

Ducharlet, L

K. Ducharlet, L. Trav´ e-Massuy` es, J.B. Lasserre, M.V. Le Lann, and Y. Miloudi. Leveraging the Christoﬀel function for outlier detection in data strea ms. Int. J. Data. Sci. Anal. , 2024

work page 2024

[7] [7]

Dunkl and Y

C.F. Dunkl and Y. Xu. Orthogonal Polynomials of Several Variables . Cambridge University Press, Cambridge, UK, 2001

work page 2001

[8] [8]

Henrion, M

D. Henrion, M. Junca, and M. Velasco. Moment-sos hierarc hy and exit location of stochastic processes. Numerical Algebra, Control and Optimization , 2024

work page 2024

[9] [9]

Henrion, M

D. Henrion, M. Korda, and J.B. Lasserre. The Moment-SOS Hhierarchy: Lectures in Proba- bility, Statistics, Computational Geometry, Control and N onlinear PDEs . Optimization and Applications 4. W orld Scientiﬁc Co. Pte Ltd., Hackensack, N J, 2021

work page 2021

[10] [10]

Koller and N

D. Koller and N. Friedman. Probabilistic graphical models . Adaptive Computation and Ma- chine Learning. MIT Press, Cambridge, MA, 2009. Principles and techniques

work page 2009

[11] [11]

Kro´ o and D

A. Kro´ o and D. S. Lubinsky. Christoﬀel functions and un iversality in the bulk for multivariate orthogonal polynomials. Canad. J. Math. , 65:600–620, 2012

work page 2012

[12] [12]

J. B. Lasserre, E. Pauwels, and M. Putinar. The Christoﬀel-Darboux Kernel for Data Anal- ysis. Cambridge University Press, Cambridge, UK, 2022

work page 2022

[13] [13]

Lasserre

J.B. Lasserre. Convergent SDP-relaxations in polynom ial optimization with sparsity. SIAM J. Optim. , 17:822–843, 2006

work page 2006

[14] [14]

Lasserre

J.B. Lasserre. A disintegration of the Christoﬀel func tion. Comptes Rendus Math´ ematique, 360:919–928, 2022

work page 2022

[15] [15]

Lasserre

J.B. Lasserre. A modiﬁed Christoﬀel function and its as ymptotic properties. J. Approx. The- ory, 295:Paper No. 105955, 16, 2023

work page 2023

[16] [16]

Lasserre and E

J.B. Lasserre and E. Pauwels. The empirical Christoﬀel function with applications in data analysis. Advances in Computational Mathematics , 45(3):1439–1468, 2019

work page 2019

[17] [17]

Lasserre, R

M. Lasserre, R. Lebrun, and P.-H. W uillemin. Quadratur e rules in general continuous Bayesian networks: Discrete inference without discretiza tion. Technical report, 2024. hal- 04495263

work page 2024

[18] [18]

Levenberg

N. Levenberg. Personal communication, 2024

work page 2024

[19] [19]

Magron and J

V. Magron and J. W ang. Sparse Polynomial Optimization: Theory and Practice . Series on Optimization and Its Applications. W orld Scientiﬁc Press, London, 2023

work page 2023

[20] [20]

S. Marx, E. Pauwels, T. W eisser, D. Henrion, and J.B. Las serre. Semi-algebraic approximation using Christoﬀel-Darboux kernel. Constr. Approx. , 54(3):391–429, 2021

work page 2021

[21] [21]

Nevai and G

P. Nevai and G. Freud. Orthogonal polynomials and Chris toﬀel functions. J. Approx. Theory , 48:3–167, 1986

work page 1986

[22] [22]

Roos Hoefgeest and L

P. Roos Hoefgeest and L. Slot. The Christoﬀel-Darboux k ernel for topological data analysis. In 39th International Symposium on Computational Geometry , volume 258 of LIPIcs. Leibniz Int. Proc. Inform. , pages Art. No. 38, 20. Schloss Dagstuhl. Leibniz-Zent. Inf orm., W adern, 2023

work page 2023

[23] [23]

B. Simon. The Christoﬀel-Darboux Kernel. In Perspective in Partial Diﬀerential Equations, Harmonic Analysis and Applications , volume 79 of Proc. Symp. Pure Math. , pages 295–335. Mathematical Society, Providence, RI, 2008

work page 2008

[24] [24]

Stahl and V

H. Stahl and V. Totik. General orthogonal polynomials. In Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, UK, 1992

work page 1992

[25] [25]

W ainwright and M.I

M.J. W ainwright and M.I. Jordan. Graphical models, exp onential families and variational inference. Found. Trends Machine Learning , 1(1-2):1–305, 2008

work page 2008

[26] [26]

W aki, S

S. W aki, S. Kim, M. Kojima, and M. Maramatsu. Sums of squa res and semideﬁnite pro- gramming relaxations for polynomial optimization problem s with structured sparsity. SIAM J. Optim. , 17:218–242, 2006

work page 2006