Rigidity of the Moment Hierarchy

• Rigidity of the moment hierarchy is a stabilization phenomenon in moment-based polynomial optimization where semidefinite relaxations achieve finite convergence and flat truncation.
• Kernel representations, such as the Christoffel–Darboux sections, enforce an exact reconstruction of minimizers through structured signed densities and rank stabilization.
• Quantitative rigidity links convergence rates to geometric error bounds, ensuring that local convexity and second-order conditions secure finite and exact recovery in structured settings.

As an Editor’s term, rigidity of the moment hierarchy denotes a family of stabilization phenomena in moment-based polynomial optimization in which semidefinite relaxations cease to act merely as asymptotic outer approximations and instead become exact, finitely atomic, or otherwise highly constrained at finite order. In the literature on the Moment-SOS and MoM/SoS hierarchies, this rigidity appears through finite convergence of relaxation values, flat truncation of moment matrices, exact reconstruction of minimizers, kernel-determined optimal signed densities, and convergence rates dictated by geometric error bounds of the feasible set (Baldi et al., 2020). A complementary interpretation identifies exact relaxations with Christoffel–Darboux kernel sections centered at minimizers (Lasserre, 2020), while more recent work shows that hidden convexity, zero-dimensional minimizer support, local optimality conditions, and Łojasiewicz exponents can each force the hierarchy into such stabilized regimes (Ðurašinović et al., 27 Feb 2026).

1. Foundational setup and principal notions of exactness

The standard setting is a polynomial optimization problem

$$f^*=\min_{x\in\mathbf{\Omega}} f(x),\qquad \mathbf{\Omega}:=\{x\in\mathbb{R}^d:\ g_j(x)\ge 0,\ j=1,\dots,m\},$$

or, equivalently, optimization on a compact basic semialgebraic set $S(\mathbf g)$. In the Moment-SOS formulation, one works with truncated moment functionals or moment sequences subject to positivity of moment and localizing matrices. In one representative formulation, the $n$-th relaxation is

$$\rho_n=\min\{\phi(f):\ \phi(1)=1,\ M_{n-d_j}(g_j\,\phi)\succeq 0,\ j=0,\dots,m+1\},$$

with $g_0:=1$, $d_j=\lceil \deg(g_j)/2\rceil$, and an added redundant ball constraint $g_{m+1}(x)=1-\|x\|^2\ge 0$ to guarantee Archimedeanity; under the Archimedean property, $\rho_n\uparrow f^*$ as $n\to\infty$ (Ðurašinović et al., 27 Feb 2026).

On the dual-primal MoM/SoS side, the hierarchy is organized by the quadratic module

$$Q(\mathbf g)=\Sigma^2+\Sigma^2 g_1+\cdots+\Sigma^2 g_s$$

and its dual cones

$S(\mathbf g)$0

The order-$S(\mathbf g)$1 SOS and MoM relaxations satisfy

$S(\mathbf g)$2

This distinction is important: SoS exactness means the polynomial certificate $S(\mathbf g)$3, whereas MoM exactness means that optimal truncated linear functionals are represented by positive measures supported on the feasible set, which is the stronger statement for recovery of minimizers (Baldi et al., 2020).

Within this framework, “rigidity” refers to more than monotone convergence. It concerns circumstances in which the hierarchy is forced into a sharply structured regime: the optimal value stabilizes after finitely many levels, the moment matrices exhibit rank stabilization, or the optimal dual object takes a uniquely constrained analytic form.

Mode of rigidity	Hallmark	Representative source
Finite convergence	$S(\mathbf g)$4 or $S(\mathbf g)$5 at finite order	(Ðurašinović et al., 27 Feb 2026)
Kernel rigidity	$S(\mathbf g)$6	(Lasserre, 2020)
Flat truncation	rank stabilization of moment matrices	(Baldi et al., 2020)
Quantitative rigidity	$S(\mathbf g)$7	(Tran et al., 1 Jul 2025)

These forms are complementary rather than competing. Some describe exact optimal values, others describe the structure of optimal moment data, and others quantify how rapidly pseudo-moments are forced toward true moment sequences.

2. Kernel representation and Christoffel–Darboux rigidity

A major reinterpretation of the hierarchy replaces abstract truncated moments by coefficients in an orthonormal polynomial basis of $S(\mathbf g)$8, where $S(\mathbf g)$9 is an arbitrary finite Borel reference measure whose support is exactly the compact feasible set $n$0. If $n$1 is an orthonormal polynomial basis and $n$2 is a feasible moment sequence, then the dual of the $n$3-th Moment-SOS relaxation can be viewed as minimizing

$n$4

over polynomial densities

$n$5

with coefficient vector satisfying

$n$6

The resulting object is generally a signed density $n$7, not a positive one (Lasserre, 2020).

This perspective yields one of the sharpest forms of rigidity in the literature. If the $n$8-th relaxation is exact and an optimal solution $n$9 has a representing measure supported at a global minimizer $$\rho_n=\min\{\phi(f):\ \phi(1)=1,\ M_{n-d_j}(g_j\,\phi)\succeq 0,\ j=0,\dots,m+1\},$$0, then the coefficients are forced to satisfy

$$\rho_n=\min\{\phi(f):\ \phi(1)=1,\ M_{n-d_j}(g_j\,\phi)\succeq 0,\ j=0,\dots,m+1\},$$1

so that

$$\rho_n=\min\{\phi(f):\ \phi(1)=1,\ M_{n-d_j}(g_j\,\phi)\succeq 0,\ j=0,\dots,m+1\},$$2

With the Christoffel–Darboux kernel

$$\rho_n=\min\{\phi(f):\ \phi(1)=1,\ M_{n-d_j}(g_j\,\phi)\succeq 0,\ j=0,\dots,m+1\},$$3

this becomes

$$\rho_n=\min\{\phi(f):\ \phi(1)=1,\ M_{n-d_j}(g_j\,\phi)\succeq 0,\ j=0,\dots,m+1\},$$4

At the minimizer,

$$\rho_n=\min\{\phi(f):\ \phi(1)=1,\ M_{n-d_j}(g_j\,\phi)\succeq 0,\ j=0,\dots,m+1\},$$5

and $$\rho_n=\min\{\phi(f):\ \phi(1)=1,\ M_{n-d_j}(g_j\,\phi)\succeq 0,\ j=0,\dots,m+1\},$$6 is the Christoffel function evaluated at $$\rho_n=\min\{\phi(f):\ \phi(1)=1,\ M_{n-d_j}(g_j\,\phi)\succeq 0,\ j=0,\dots,m+1\},$$7 (Lasserre, 2020).

The structural reason is the reproducing property

$$\rho_n=\min\{\phi(f):\ \phi(1)=1,\ M_{n-d_j}(g_j\,\phi)\succeq 0,\ j=0,\dots,m+1\},$$8

Hence the signed density $$\rho_n=\min\{\phi(f):\ \phi(1)=1,\ M_{n-d_j}(g_j\,\phi)\succeq 0,\ j=0,\dots,m+1\},$$9 mimics the Dirac measure $g_0:=1$0 on all polynomials of degree at most $g_0:=1$1. This sharply distinguishes the lower-bound Moment-SOS hierarchy from upper-bound hierarchies based on positive SOS densities: positivity is too restrictive to reproduce a Dirac mass exactly at finite degree, whereas signed kernel sections can do so on the finite-dimensional polynomial space relevant to the relaxation. The paper explicitly remarks that finite convergence “generically takes place,” and in such cases the optimizer is no longer arbitrary but constrained to this kernel form (Lasserre, 2020).

3. Flat truncation, real radicals, and algebraic stabilization

A second major rigidity mechanism is algebraic. In the MoM hierarchy, one studies moment matrices

$g_0:=1$2

and their kernels

$g_0:=1$3

Flat truncation is the rank condition

$g_0:=1$4

where $g_0:=1$5. This is the practical certificate that a truncated moment sequence comes from a finite atomic measure and that minimizers can be extracted effectively from the moment data (Baldi et al., 2020).

The underlying algebra is encoded by the support ideal

$g_0:=1$6

of a quadratic module $g_0:=1$7, and especially its real radical $g_0:=1$8. For a finitely generated quadratic module $g_0:=1$9, the closure theorem

$d_j=\lceil \deg(g_j)/2\rceil$0

shows that the dual MoM hierarchy behaves as if the primal SoS hierarchy had been enlarged by generators of the real radical of the support (Baldi et al., 2020).

The central flat-truncation characterization is that

$d_j=\lceil \deg(g_j)/2\rceil$1

is equivalent to the existence of a relaxation order for which a generic optimal truncated functional has flat truncation, under MoM finite convergence. This zero-dimensionality criterion identifies when the hierarchy has become rigid enough to reconstruct the minimizer set. In that regime, one obtains finitely atomic representations

$d_j=\lceil \deg(g_j)/2\rceil$2

and, for generic optimal functionals at sufficiently high order,

$d_j=\lceil \deg(g_j)/2\rceil$3

The distinction from mere SoS exactness is decisive: SoS exactness produces a polynomial certificate for the optimal value, whereas MoM exactness together with flat truncation yields both the value and the minimizers (Baldi et al., 2020).

The same paper also proves that Boundary Hessian Conditions imply SoS exactness, MoM exactness, and flat truncation for all sufficiently large orders. This situates rigidity within a broader local-to-global principle: local second-order structure at minimizers can force global stabilization of the hierarchy.

4. Hidden convexity and finite convergence without explicit convex reformulation

A more recent line of work shows that the hierarchy can become exact because of convexity that is present only on the feasible set, not in the ambient space. The setting is a compact convex feasible region

$d_j=\lceil \deg(g_j)/2\rceil$4

defined by SOS-concave polynomials, together with a polynomial objective $d_j=\lceil \deg(g_j)/2\rceil$5 that is globally nonconvex on $d_j=\lceil \deg(g_j)/2\rceil$6 but convex on $d_j=\lceil \deg(g_j)/2\rceil$7. The paper describes this as hidden or local convexity (Ðurašinović et al., 27 Feb 2026).

The algebraic certificate for this local convexity is

$d_j=\lceil \deg(g_j)/2\rceil$8

for suitable matrix polynomials $d_j=\lceil \deg(g_j)/2\rceil$9. This is a Putinar-like certificate showing that $g_{m+1}(x)=1-\|x\|^2\ge 0$0 is SOS-convex on $g_{m+1}(x)=1-\|x\|^2\ge 0$1, and it is weaker than global SOS-convexity on all of $g_{m+1}(x)=1-\|x\|^2\ge 0$2. If $g_{m+1}(x)=1-\|x\|^2\ge 0$3 is strongly convex on $g_{m+1}(x)=1-\|x\|^2\ge 0$4 and the quadratic module is Archimedean, then Putinar’s matrix Positivstellensatz yields precisely such a representation (Ðurašinović et al., 27 Feb 2026).

The decisive technical step is a Jensen-type inequality for feasible truncated linear functionals. If

$g_{m+1}(x)=1-\|x\|^2\ge 0$5

then, under the paper’s assumptions, one has

$g_{m+1}(x)=1-\|x\|^2\ge 0$6

When $g_{m+1}(x)=1-\|x\|^2\ge 0$7 is strongly convex on $g_{m+1}(x)=1-\|x\|^2\ge 0$8, this holds for sufficiently large $g_{m+1}(x)=1-\|x\|^2\ge 0$9. When all constraints are at most quadratic and $\rho_n\uparrow f^*$0 is SOS-convex on $\rho_n\uparrow f^*$1, it holds as soon as

$\rho_n\uparrow f^*$2

From this, the hierarchy becomes exact: $\rho_n\uparrow f^*$3 and

$\rho_n\uparrow f^*$4

is a certified global minimizer (Ðurašinović et al., 27 Feb 2026).

This result is notable because the hierarchy is not explicitly informed that the instance is convex on $\rho_n\uparrow f^*$5. The relaxations nonetheless exploit the hidden structure and become exact at a finite step. A plausible implication is that rigidity of the hierarchy need not come from a user-supplied convex reformulation; it can emerge from the interaction between localizing constraints and latent convexity certificates already present in the data of the instance.

5. Geometry-driven rates and quantitative rigidity

Not all rigidity is finite-order exactness. Another strand studies how rapidly the relaxed truncated moment cones approach the true moment cone. For a compact basic semialgebraic set

$\rho_n\uparrow f^*$6

the optimization problem

$\rho_n\uparrow f^*$7

can be reformulated over the set $\rho_n\uparrow f^*$8 of $\rho_n\uparrow f^*$9-truncated moment sequences of probability measures supported on $n\to\infty$0. The hierarchy replaces $n\to\infty$1 by an outer spectrahedral approximation, and its quality is measured by the Hausdorff distance

$n\to\infty$2

with analogous quantities for $n\to\infty$3 and $n\to\infty$4 (Tran et al., 1 Jul 2025).

Tchakaloff’s theorem is the key finite-dimensional bridge. Every $n\to\infty$5 can be represented as

$n\to\infty$6

with at most $n\to\infty$7 support points and positive weights summing to one. This permits quantitative comparison between pseudo-moment sequences and genuine moment sequences by projecting support points onto the feasible domain. The paper derives the basic estimate

$n\to\infty$8

so any rate for the Hausdorff distance transfers directly to a rate for the hierarchy’s error (Tran et al., 1 Jul 2025).

The central principle is that the convergence rate is governed by a Łojasiewicz exponent $n\to\infty$9 for the domain, via an error bound of the form

$$Q(\mathbf g)=\Sigma^2+\Sigma^2 g_1+\cdots+\Sigma^2 g_s$$0

on a suitable compact ambient set. Under the relevant assumptions,

$$Q(\mathbf g)=\Sigma^2+\Sigma^2 g_1+\cdots+\Sigma^2 g_s$$1

and similarly for the associated moment-SOS error. The paper gives several special cases: $$Q(\mathbf g)=\Sigma^2+\Sigma^2 g_1+\cdots+\Sigma^2 g_s$$2 for polytopes and domains satisfying the constraint qualification condition, $$Q(\mathbf g)=\Sigma^2+\Sigma^2 g_1+\cdots+\Sigma^2 g_s$$3 for domains satisfying the Polyak–Łojasiewicz condition or defined by locally strongly convex polynomials, and $$Q(\mathbf g)=\Sigma^2+\Sigma^2 g_1+\cdots+\Sigma^2 g_s$$4 for general polynomials over a sphere; the abstract also recalls previously established $$Q(\mathbf g)=\Sigma^2+\Sigma^2 g_1+\cdots+\Sigma^2 g_s$$5 behavior for the unit ball, hypercube, and standard simplex (Tran et al., 1 Jul 2025).

This is a quantitative form of rigidity. The hierarchy’s approximation behavior is not arbitrary: it is structurally dictated by how strongly the defining constraints control distance to the feasible set. Simpler domains with sharper error bounds force faster collapse of pseudo-moments toward genuine moments.

6. Matrix constraints and products of spheres

Rigidity phenomena persist in more structured optimization models. For polynomial matrix optimization

$$Q(\mathbf g)=\Sigma^2+\Sigma^2 g_1+\cdots+\Sigma^2 g_s$$6

with feasible set

$$Q(\mathbf g)=\Sigma^2+\Sigma^2 g_1+\cdots+\Sigma^2 g_s$$7

the matrix Moment-SOS hierarchy uses the quadratic module

$$Q(\mathbf g)=\Sigma^2+\Sigma^2 g_1+\cdots+\Sigma^2 g_s$$8

If $$Q(\mathbf g)=\Sigma^2+\Sigma^2 g_1+\cdots+\Sigma^2 g_s$$9 is archimedean and the nondegeneracy condition, strict complementarity condition, and second order sufficient condition hold at every minimizer, then the hierarchy has finite convergence: there exists $S(\mathbf g)$00 such that $S(\mathbf g)$01 for all $S(\mathbf g)$02. Under the same assumptions, every minimizer of the moment relaxation has a flat truncation for sufficiently large order, and the support $S(\mathbf g)$03 is zero dimensional (Huang et al., 2024).

The matrix paper also gives a low-order convex case. If $S(\mathbf g)$04 and $S(\mathbf g)$05 are SOS-convex, then the lowest-order relaxation is already exact: $S(\mathbf g)$06 If, in addition, there exists a strictly feasible point $S(\mathbf g)$07 with $S(\mathbf g)$08, then

$S(\mathbf g)$09

This shows that rigidity can appear either through generic second-order optimality conditions or through explicit SOS-convexity of the data (Huang et al., 2024).

A different structured setting is minimization of a multihomogeneous polynomial over a product of spheres

$S(\mathbf g)$10

For a generic multihomogeneous objective, the moment-SOS hierarchy has finite convergence and high-order moment optimizers satisfy flat truncation. The mechanism is a chain

$S(\mathbf g)$11

In this setting, LICQ is automatic because the gradients of the sphere constraints are linearly independent at every feasible point, and SCC is vacuous because the problem has no inequality constraints. The genericity statement is with respect to Lebesgue measure zero subsets in the coefficient space of multihomogeneous polynomials (Halaseh et al., 11 Dec 2025).

The product-of-spheres result is especially relevant for tensor optimization. The best rank-one approximation problem for a tensor can be rewritten as optimization of a multihomogeneous polynomial over a product of spheres, so finite convergence of the hierarchy yields exact computation after finitely many SDP relaxations for a generic tensor, together with flat-truncation-based extraction of minimizers or maximizers (Halaseh et al., 11 Dec 2025).

7. Adjacent notions, limits, and common misconceptions

The phrase “rigidity” has a distinct meaning in classical moment theory. For Hamburger and Stieltjes moment sequences, rigidity concerns whether changing only finitely many entries can preserve the moment property. The decisive dichotomy is between determinate and indeterminate sequences. The paper proves that a moment sequence allows all sufficiently small finite perturbations precisely when it is indeterminate, while determinate sequences are rigid in a manner quantified by the finite index of determinacy. It also proves that if all sufficiently small variations of a single entry preserve the moment property, then the sequence must be indeterminate (Dyachenko, 2017). This is a different use of rigidity from finite convergence or flat truncation in SDP hierarchies, though both concern how strongly moment constraints restrict admissible data.

A further terminological adjacency appears in rigidity theory on the geometric moment curve

$S(\mathbf g)$12

For points on this curve, the bar-and-joint rigidity matroid, the hyperconnectivity matroid, and the $S(\mathbf g)$13-cofactor rigidity matroid on a planar conic coincide (Ruiz et al., 2021). This suggests a terminological rather than methodological connection to moment hierarchies: the shared word “moment” refers there to the moment curve, not to moment sequences or moment-SOS relaxations.

Several misconceptions are corrected by the hierarchy literature itself. Finite convergence is not automatic: in polynomial matrix optimization, examples show failure of finite convergence when any one of nondegeneracy, strict complementarity, or second order sufficiency is dropped (Huang et al., 2024). Exactness of the optimal value is also not the whole story: SoS exactness does not by itself recover minimizers, whereas MoM exactness plus flat truncation does (Baldi et al., 2020). Conversely, the lower-bound hierarchy need not use positive densities; exact finite-order behavior may require signed polynomial densities, and in the Christoffel–Darboux picture the exact optimizer is a kernel section rather than an SOS density (Lasserre, 2020). Finally, hidden convexity need not be certified in advance by a tailored reformulation: a general-purpose Moment-SOS hierarchy can become exact at finite order even when convexity is only local to the feasible set and not visible globally in $S(\mathbf g)$14 (Ðurašinović et al., 27 Feb 2026).

Taken together, these results show that rigidity of the moment hierarchy is not a single theorem but a unifying descriptor for several stabilization mechanisms. Some are algebraic, through real radicals and zero-dimensional supports; some are analytic, through reproducing kernels and signed densities; some are geometric, through convexity, optimality conditions, and Łojasiewicz exponents. What they share is the same endpoint: after enough structure has been exposed, the hierarchy is no longer free to approximate loosely, but is forced into an exact or sharply constrained form.