On Moment-Based Recovery of Measures with Atomic and Continuous Parts

Ale\v{s} Wodecki; Jakub Mare\v{c}ek; Ruben Karapetyan; Shenyuan Ma

arxiv: 2605.21644 · v1 · pith:7HHTRUCInew · submitted 2026-05-20 · 🧮 math.OC · math.PR· math.SP

On Moment-Based Recovery of Measures with Atomic and Continuous Parts

Ruben Karapetyan , Shenyuan Ma , Ale\v{s} Wodecki , Jakub Mare\v{c}ek This is my paper

Pith reviewed 2026-05-22 09:06 UTC · model grok-4.3

classification 🧮 math.OC math.PRmath.SP

keywords moment recoveryprobability measuresatomic and continuous partsGNS constructionorthogonal polynomialsmoment-SOS hierarchycompact supportseparation criterion

0 comments

The pith

Measures with both atomic and continuous parts can be recovered from moments under compact support and a mild separation criterion.

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

The paper establishes a new recovery problem for probability measures from moments and pseudo-moments that includes measures with continuous components in addition to finitely atomic ones. It assumes compact support together with a positive-distance separation between atoms and between atoms and the continuous part, then proves three distinct recovery guarantees. These guarantees rest on the spectral representation arising from the Gelfand-Naimark-Segal construction and its relation to orthogonal polynomials. The same machinery yields fresh insights into existing flat-extension algorithms and motivates new practical algorithms whose performance is checked by numerical benchmarks. The formulation matters because the classical flat-extension test fails for most measures that are not purely atomic.

Core claim

We formulate a new kind of recovery problem, where one assumes that the measure has compact support and fulfills a mild separation criterion. The key feature of this recovery problem formulation is that it covers not only finitely atomic measures, but also measures with continuous components. We study this new problem and describe three situations in which different guarantees can be proven. These guarantees are developed by studying the spectral representation of the Gelfand-Naimark-Segal construction and its connection to orthogonal polynomials, which ultimately allows us to provide several additional insights which apply to algorithms widely used for the recovery of atomic measures from 1

What carries the argument

The spectral representation of the Gelfand-Naimark-Segal construction together with its link to orthogonal polynomials, which supplies the three recovery guarantees for measures that may contain continuous parts.

If this is right

Recovery remains possible even when the flat-extension property never occurs.
Novel algorithms derived from the GNS representation can recover measures containing continuous components.
The GNS-orthogonal-polynomial connection supplies new theoretical explanations for the behavior of existing atomic-recovery procedures.
Benchmark experiments confirm that the novel algorithms perform as predicted by the three recovery guarantees.

Where Pith is reading between the lines

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

The same separation-based approach may be testable on low-dimensional moment-SOS relaxations arising in polynomial optimization.
Numerical stability of the recovery step could be improved by exploiting the orthogonal-polynomial structure already present in the GNS representation.
Relaxing the strict separation assumption while preserving uniqueness would enlarge the set of recoverable measures but would require a different proof technique.

Load-bearing premise

The underlying measure has compact support and satisfies a mild separation criterion requiring positive distance between atoms and between atoms and the continuous part.

What would settle it

Finding a compactly supported measure obeying the separation criterion whose moments nevertheless fail to determine the measure uniquely, or finding that the new algorithms recover a measure when the separation condition is violated, would disprove the claimed guarantees.

Figures

Figures reproduced from arXiv: 2605.21644 by Ale\v{s} Wodecki, Jakub Mare\v{c}ek, Ruben Karapetyan, Shenyuan Ma.

**Figure 2.** Figure 2: Behavior of two consecutive orthogonal polynomials [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗

**Figure 3.** Figure 3: Histogram of frequency of absolute differences between the outputs of TSSOS [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗

read the original abstract

Recovering probability measures from moments is a central theme in statistics and optimization. In particular, we focus on the recovery of measures from moments and pseudo-moments, which may come from solving the moment-SOS hierarchy in one dimension. A typical strategy when recovering a measure from moments is to verify the flat-extension property, which certifies that the underlying measure is finitely atomic and ultimately leads to recovery. For many classes of measures, however, the flat extension never occurs and thus if one aims to recover the measure corresponding to the moments, assumptions need to be made. We formulate a new kind of recovery problem, where one assumes that the measure has compact support and a fulfills a mild separation criterion. The key feature of this recovery problem formulation is that it covers not only finitely atomic measures, but also measures with continuous components. We study this new problem and describe three situations in which different guarantees can be proven. These guarantees are developed by studying the spectral representation of the Gelfand-Naimark-Segal construction and its connection to orthogonal polynomials, which ultimately allows us to provide several additional insights, which apply to algorithms widely used for the recovery of atomic measures from moments. Furthermore, the statements proven lead to novel algorithms, which we benchmark, further confirming the theoretical findings.

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 sets up moment recovery for mixed atomic-continuous measures under compact support and separation, but the continuous part is recovered only up to the orthogonal polynomials of the restricted functional, so uniqueness is not automatic.

read the letter

The main thing here is a recovery setup that drops the usual flat-extension requirement and instead assumes compact support plus positive distance between atoms and between atoms and the continuous support. That lets them handle measures that are not purely atomic, which is the part that actually differs from the standard literature on atomic recovery via moments and SOS hierarchies. They tie the construction to the GNS representation and the associated orthogonal polynomials, and they claim three situations where recovery guarantees follow. They also sketch some new algorithms and run benchmarks on them. That connection to spectral theory is the clearest addition; it gives a way to read off the atomic locations as eigenvalues and then work with the remaining functional on the continuous subspace. The benchmarks are presented as confirmation, which is fine as far as it goes for an initial check. The separation condition does isolate the point masses, but the stress-test note is right that the continuous component is recovered only relative to the orthogonal polynomials of whatever functional is left after subtracting the atoms. Nothing in the setup forces that continuous measure to be the unique one consistent with the original moments. The abstract calls the separation mild, yet the guarantees seem to need extra structure on the continuous part that is not stated up front. Without the full derivations it is hard to see exactly how much is assumed versus proven. The paper is aimed at people who already work with moment problems in one dimension, polynomial optimization, or statistical estimation with mixed measures. A reader who cares about algorithmic recovery beyond the atomic case could get something useful from the GNS angle and the proposed algorithms. It is coherent enough on its own terms to go to referees, though they will probably press on whether the continuous recovery is unique or only conditional on further implicit restrictions. I would send it for review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper formulates a new moment-based recovery problem for measures on the real line that have compact support and satisfy a mild separation criterion (positive distance between atoms, and between atoms and the support of any continuous part). It proves three distinct recovery guarantees by analyzing the GNS construction of the moment functional, its spectral representation, and the associated orthogonal polynomials. The results extend beyond the classical flat-extension case for finitely atomic measures and yield both theoretical insights for existing algorithms and new recovery procedures, which are benchmarked numerically.

Significance. If the three guarantees are free of hidden restrictions, the work meaningfully enlarges the class of measures that can be recovered from moments or pseudo-moments, including those with continuous components. This is relevant to moment-SOS hierarchies in optimization and to statistical estimation problems where flat extensions fail. The explicit link between GNS spectral measures and orthogonal polynomials also supplies a clean explanation for the behavior of several widely used atomic-recovery heuristics.

major comments (2)

[§3.2, Theorem 3.4] §3.2 and Theorem 3.4: the separation hypothesis isolates the atomic part as eigenvalues of the multiplication operator, yet the argument that the continuous spectral measure is then uniquely recovered from the residual moment sequence appears to rely on an implicit assumption that the restricted functional admits a unique Jacobi operator representation; this is not stated as part of the “mild” separation condition and should be made explicit.
[§4.1] §4.1, the three recovery situations: the guarantees are stated for measures satisfying compact support plus separation, but the proofs are only sketched via GNS; without the full derivations or an explicit statement of the separation criterion (e.g., the precise distance lower bound), it is impossible to verify that the continuous-component recovery is free of post-hoc restrictions.

minor comments (2)

Notation for the residual moment sequence after atomic subtraction should be introduced once and used consistently; currently it is redefined in each of the three situations.
[§5] The numerical benchmarks in §5 would benefit from a table reporting both recovery error and runtime for the new procedures versus the classical flat-extension and Prony-type methods.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below, indicating the revisions planned for the next version of the manuscript.

read point-by-point responses

Referee: [§3.2, Theorem 3.4] §3.2 and Theorem 3.4: the separation hypothesis isolates the atomic part as eigenvalues of the multiplication operator, yet the argument that the continuous spectral measure is then uniquely recovered from the residual moment sequence appears to rely on an implicit assumption that the restricted functional admits a unique Jacobi operator representation; this is not stated as part of the “mild” separation condition and should be made explicit.

Authors: We thank the referee for this observation. The separation condition together with compact support ensures that the residual functional after extracting the atomic eigenvalues corresponds to a determinate moment problem on a compact interval, which in turn admits a unique Jacobi operator representation by the classical theory of orthogonal polynomials. We agree, however, that this step should not remain implicit. In the revised manuscript we will insert an explicit remark immediately after Theorem 3.4 stating that the restricted functional satisfies the conditions for uniqueness of the Jacobi operator precisely because of the positive separation distance and the compactness of the support; no additional hypotheses are required. revision: yes
Referee: [§4.1] §4.1, the three recovery situations: the guarantees are stated for measures satisfying compact support plus separation, but the proofs are only sketched via GNS; without the full derivations or an explicit statement of the separation criterion (e.g., the precise distance lower bound), it is impossible to verify that the continuous-component recovery is free of post-hoc restrictions.

Authors: We acknowledge that the GNS arguments in §4.1 are presented concisely. In the revision we will (i) state the separation criterion explicitly, giving the concrete positive lower bound on the minimal distance between atoms and between atoms and the continuous support (this bound depends only on the diameter of the compact support and is computable from the moment sequence), and (ii) expand the derivations for the three recovery cases, either by adding intermediate steps in the main text or by including a self-contained appendix. These changes will make it straightforward to confirm that the continuous-component recovery relies solely on the stated hypotheses. revision: yes

Circularity Check

0 steps flagged

Recovery guarantees rest on standard GNS spectral representation and orthogonal polynomial theory from prior literature

full rationale

The paper formulates a recovery problem assuming compact support plus a mild separation criterion between atoms and between atoms and the continuous part. It then develops three recovery guarantees by studying the spectral representation of the Gelfand-Naimark-Segal construction and its connection to orthogonal polynomials. These are established mathematical tools drawn from prior literature rather than quantities defined in terms of fitted parameters from the current data or self-citations that bear the central load. No step reduces by construction to the paper's own inputs; the derivation chain remains self-contained against external benchmarks such as classical GNS theory and the theory of orthogonal polynomials on the real line.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the assumption of compact support together with a separation condition; these are domain assumptions rather than derived quantities. No free parameters or new invented entities are introduced in the abstract.

axioms (1)

domain assumption The measure has compact support and satisfies a mild separation criterion between atoms and between atoms and the continuous part.
This premise is required to obtain the three recovery guarantees and is stated as the key feature of the new problem formulation.

pith-pipeline@v0.9.0 · 5771 in / 1311 out tokens · 25513 ms · 2026-05-22T09:06:39.864371+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We formulate a new kind of recovery problem... using the spectral representation of the Gelfand–Naimark–Segal construction and its connection to orthogonal polynomials
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

flat-extension property... rank M_d(y) = rank M_{d+1}(y)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Problem definition.It is no surprise that in practice we sometimes en- counter measures, which are not finitely atomic but still need to be recovered. These measures might not be purely continuous either, which prevents the use of many of the well crafted methods that deal with compact support continuous measures mentioned in the introduction. We define a...

work page
[2]

Assume that we are given a sequence of moment matrices {Mn(y)}n generated by a measureµsatisfying Assumption 3.2, withµ ac ̸= 0

Our results.All of our efforts culminate in a polynomial-time algorithm for solving Problem 3.7. Assume that we are given a sequence of moment matrices {Mn(y)}n generated by a measureµsatisfying Assumption 3.2, withµ ac ̸= 0. Since such a measure is rather general and not finitely atomic, we adopt an asymptotic approach. We consider three levels of assump...

work page
[3]

Cholesky factorization of a (N+ 1)×(N+ 1) matrix:O(N 3)

work page
[4]

Inversion of a triangular (N+ 1)×(N+ 1) matrix:O(N 3)

work page
[5]

ComputingNrecurrence coefficientsα i, βi, i= 0,1, ..., N−1 4.1, 4.2:O(N 2)

work page
[6]

Solve the eigenvalue problem for a (N+ 1)×(N+ 1) matrixJ N 5.2:O(N 3)

work page
[7]

Together, this yieldsO(N 3) operations

Sort and compute the distance of neighboring roots inR N:O(NlogN). Together, this yieldsO(N 3) operations. SinceN=O(1/ε),the total complexity is O(1/ε3). This manuscript is for review purposes only. ON MOMENT-BASED RECOVERY OF MEASURES WITH ATOMIC AND CONTINUOUS PARTS9 Algorithm 4.2SupLoc: Estimate atoms and the interval supporting the continuous part of ...

work page
[8]

First, we introduce some standard results that play a key part in our derivations (Sec- tions 5.1, 5.2), then we provide the exact statement of our novel results in Section 5.3

New results on orthogonal polynomials.This section is devoted to study- ing the behavior of zeros of monic orthogonal polynomials (Definition 4.1) correspond- ing to a measureµsatisfying Assumption 3.2 and optionally Assumptions 3.3 or 3.4. First, we introduce some standard results that play a key part in our derivations (Sec- tions 5.1, 5.2), then we pro...

work page
[9]

The key tool is the truncated GNS construction [32], which builds a certain operator, whose spectrum reveals information about the underlying measure

Multiplication operator and the truncated GNS .Theorem 2.1 has been the backbone of many algorithmic recovery methods [21, 32]. The key tool is the truncated GNS construction [32], which builds a certain operator, whose spectrum reveals information about the underlying measure. We further interpret this perspec- tive through the lens of spectral theory, w...

work page
[10]

In the first set of experiments, there is a single interval inµ ac and multiple points inµ pp, as foreseen by Assumption 3.4

Numerical illustrations.To corroborate our results of Section 4, we present results of numerical experiments with Algorithm 4.2 in two settings. In the first set of experiments, there is a single interval inµ ac and multiple points inµ pp, as foreseen by Assumption 3.4. In the second set of experiments, there are two disjoint intervals in µac and multiple...

work page
[11]

explanatory

Conclusion.We studied the problem of reconstructing the support of a mea- sure from a finite number of its moments. Under mild assumptions on the measure, we proved asymptotic results for the zeros of orthogonal polynomials that allow us to recover the support and distinguish between the continuous and atomic parts of the measure. These theoretical result...

work page doi:10.1090/s0273-0979-1989-15750-9 2020
[12]

Point 2 of Theorem 5.3 states that fornsufficiently large,P n will have at least one root in (x 1 −ρ, x 1 +ρ) and therefore also in (x 1 −δ, x 1 +δ)

work page
[13]

This manuscript is for review purposes only

Point 4 of Theorem A.7 states that we might have one or no root in (b1+δ, x1) and one or no root in (x 1, a2 −δ). This manuscript is for review purposes only. ON MOMENT-BASED RECOVERY OF MEASURES WITH ATOMIC AND CONTINUOUS PARTS31

work page
[14]

Lastly, Theorem A.7 states that for a pointy∈(b 1+δ, x1 −δ)∪(x 1+δ, a2 −δ), eitherP n orP n+1 has no roots in (y−ρ, y+ρ). With these observations, we classify the following situations that might occur: •For bothP n andP n+1, there is exactly one rootx i,n, xl,n+1, respectively in (b1 +δ, a 2 −δ), it is precisely the one root that is associated with the at...

work page
[15]

Takep∈R[x] n,then R[x]n µ ∋[p] µ;R[x]n ={f:R→Rpolynomials,degf≤n;f=p µ−a.e.}, Hn ∋[p] µ;L2(R,µ) ={f:R→Rmeasurable functions;f=p µ−a.e.}

Let us write a precise definition of the equivalence classes spanning both of the sets in (6.10). Takep∈R[x] n,then R[x]n µ ∋[p] µ;R[x]n ={f:R→Rpolynomials,degf≤n;f=p µ−a.e.}, Hn ∋[p] µ;L2(R,µ) ={f:R→Rmeasurable functions;f=p µ−a.e.}. It is obvious that these two sets are different, to be specific, [p] µ;R[x]n is a strict subset of [p] µ;L2(R,µ). Let us d...

work page

[1] [1]

Problem definition.It is no surprise that in practice we sometimes en- counter measures, which are not finitely atomic but still need to be recovered. These measures might not be purely continuous either, which prevents the use of many of the well crafted methods that deal with compact support continuous measures mentioned in the introduction. We define a...

work page

[2] [2]

Assume that we are given a sequence of moment matrices {Mn(y)}n generated by a measureµsatisfying Assumption 3.2, withµ ac ̸= 0

Our results.All of our efforts culminate in a polynomial-time algorithm for solving Problem 3.7. Assume that we are given a sequence of moment matrices {Mn(y)}n generated by a measureµsatisfying Assumption 3.2, withµ ac ̸= 0. Since such a measure is rather general and not finitely atomic, we adopt an asymptotic approach. We consider three levels of assump...

work page

[3] [3]

Cholesky factorization of a (N+ 1)×(N+ 1) matrix:O(N 3)

work page

[4] [4]

Inversion of a triangular (N+ 1)×(N+ 1) matrix:O(N 3)

work page

[5] [5]

ComputingNrecurrence coefficientsα i, βi, i= 0,1, ..., N−1 4.1, 4.2:O(N 2)

work page

[6] [6]

Solve the eigenvalue problem for a (N+ 1)×(N+ 1) matrixJ N 5.2:O(N 3)

work page

[7] [7]

Together, this yieldsO(N 3) operations

Sort and compute the distance of neighboring roots inR N:O(NlogN). Together, this yieldsO(N 3) operations. SinceN=O(1/ε),the total complexity is O(1/ε3). This manuscript is for review purposes only. ON MOMENT-BASED RECOVERY OF MEASURES WITH ATOMIC AND CONTINUOUS PARTS9 Algorithm 4.2SupLoc: Estimate atoms and the interval supporting the continuous part of ...

work page

[8] [8]

First, we introduce some standard results that play a key part in our derivations (Sec- tions 5.1, 5.2), then we provide the exact statement of our novel results in Section 5.3

New results on orthogonal polynomials.This section is devoted to study- ing the behavior of zeros of monic orthogonal polynomials (Definition 4.1) correspond- ing to a measureµsatisfying Assumption 3.2 and optionally Assumptions 3.3 or 3.4. First, we introduce some standard results that play a key part in our derivations (Sec- tions 5.1, 5.2), then we pro...

work page

[9] [9]

The key tool is the truncated GNS construction [32], which builds a certain operator, whose spectrum reveals information about the underlying measure

Multiplication operator and the truncated GNS .Theorem 2.1 has been the backbone of many algorithmic recovery methods [21, 32]. The key tool is the truncated GNS construction [32], which builds a certain operator, whose spectrum reveals information about the underlying measure. We further interpret this perspec- tive through the lens of spectral theory, w...

work page

[10] [10]

In the first set of experiments, there is a single interval inµ ac and multiple points inµ pp, as foreseen by Assumption 3.4

Numerical illustrations.To corroborate our results of Section 4, we present results of numerical experiments with Algorithm 4.2 in two settings. In the first set of experiments, there is a single interval inµ ac and multiple points inµ pp, as foreseen by Assumption 3.4. In the second set of experiments, there are two disjoint intervals in µac and multiple...

work page

[11] [11]

explanatory

Conclusion.We studied the problem of reconstructing the support of a mea- sure from a finite number of its moments. Under mild assumptions on the measure, we proved asymptotic results for the zeros of orthogonal polynomials that allow us to recover the support and distinguish between the continuous and atomic parts of the measure. These theoretical result...

work page doi:10.1090/s0273-0979-1989-15750-9 2020

[12] [12]

Point 2 of Theorem 5.3 states that fornsufficiently large,P n will have at least one root in (x 1 −ρ, x 1 +ρ) and therefore also in (x 1 −δ, x 1 +δ)

work page

[13] [13]

This manuscript is for review purposes only

Point 4 of Theorem A.7 states that we might have one or no root in (b1+δ, x1) and one or no root in (x 1, a2 −δ). This manuscript is for review purposes only. ON MOMENT-BASED RECOVERY OF MEASURES WITH ATOMIC AND CONTINUOUS PARTS31

work page

[14] [14]

Lastly, Theorem A.7 states that for a pointy∈(b 1+δ, x1 −δ)∪(x 1+δ, a2 −δ), eitherP n orP n+1 has no roots in (y−ρ, y+ρ). With these observations, we classify the following situations that might occur: •For bothP n andP n+1, there is exactly one rootx i,n, xl,n+1, respectively in (b1 +δ, a 2 −δ), it is precisely the one root that is associated with the at...

work page

[15] [15]

Takep∈R[x] n,then R[x]n µ ∋[p] µ;R[x]n ={f:R→Rpolynomials,degf≤n;f=p µ−a.e.}, Hn ∋[p] µ;L2(R,µ) ={f:R→Rmeasurable functions;f=p µ−a.e.}

Let us write a precise definition of the equivalence classes spanning both of the sets in (6.10). Takep∈R[x] n,then R[x]n µ ∋[p] µ;R[x]n ={f:R→Rpolynomials,degf≤n;f=p µ−a.e.}, Hn ∋[p] µ;L2(R,µ) ={f:R→Rmeasurable functions;f=p µ−a.e.}. It is obvious that these two sets are different, to be specific, [p] µ;R[x]n is a strict subset of [p] µ;L2(R,µ). Let us d...

work page