Pseudomoment Cone and Hankel Spectrahedra

• Pseudomoment Cone is the dual of the SOS cone, comprising linear functionals that are nonnegative on squares of homogeneous polynomials.
• It is realized as a Hankel spectrahedral cone whose facial geometry governs moment relaxations, truncated sequences, and atomic rank-one decompositions.
• Key methodologies such as Gram representation and the RayDecomp algorithm enable unique atomic recovery and offer insights into convex algebraic geometry.

The homogeneous pseudo-moment cone $\Sigma_{n,2d}^*$ is the dual, in the sense of convex cones of linear functionals, of the sum-of-squares cone $\Sigma_{n,2d}$ of degree-$2d$ forms in $n$ variables. In matrix form, it is realized as a Hankel spectrahedral cone, and its facial geometry governs moment/SOS relaxations, the structure of truncated moment sequences, and the identifiability of atomic rank-one decompositions of moment matrices (Kang et al., 7 May 2026). In SOS and moment hierarchies it also appears as a spectrahedral shadow, so questions about its extremal structure interact with the general geometry of linear images of the PSD cone, including possible non-closedness and associated boundary pathologies (Jiang et al., 2020).

1. Definition and dual realizations

Let $H_{n,d}$ denote the real vector space of homogeneous polynomials of degree $d$ in $n$ variables. The cone of SOS forms of degree $2d$ is

$$\Sigma_{n,2d}:=\{f\in H_{n,2d}\mid f=\sum_i q_i^2,\ q_i\in H_{n,d}\},$$

and the cone of nonnegative forms is

$$P_{n,2d}:=\{f\in H_{n,2d}\mid f(x)\ge 0,\ \forall x\in\mathbb{R}^n\}.$$

Their dual cones of linear functionals $\Sigma_{n,2d}$0 are

$\Sigma_{n,2d}$1

and

$\Sigma_{n,2d}$2

The dual $\Sigma_{n,2d}$3 is called the homogeneous pseudo-moment cone (Kang et al., 7 May 2026).

By construction, $\Sigma_{n,2d}$4 and therefore $\Sigma_{n,2d}$5. This already separates the pseudo-moment cone from the cone coming from genuine measures: pseudo-moments are required to be nonnegative on squares, but not every such functional is induced by a representing measure (Kang et al., 7 May 2026).

A monomial basis is central to the standard matrix realization. For multi-indices $\Sigma_{n,2d}$6 with $\Sigma_{n,2d}$7, the degree-$\Sigma_{n,2d}$8 monomial vector is

$\Sigma_{n,2d}$9

The inhomogeneous vector of monomials up to degree $2d$0 is

$2d$1

and homogenization gives $2d$2 with $2d$3 (Kang et al., 7 May 2026).

The SOS cone has a Gram representation:

$2d$4

so

$2d$5

Dually, a linear functional $2d$6 is encoded by a homogeneous degree-$2d$7 pseudo-moment sequence $2d$8, with $2d$9, and the associated homogeneous moment matrix is

$n$0

Then

$n$1

hence

$n$2

(Kang et al., 7 May 2026).

2. Hankel spectrahedra, representable moments, and atoms

The matrix realization of the pseudo-moment cone is

$n$3

where

$n$4

Equivalently,

$n$5

Thus the pseudo-moment cone is a spectrahedral cone obtained by intersecting the PSD cone with a Hankel linear subspace (Kang et al., 7 May 2026).

By contrast, the representable moment-matrix cone is

$n$6

and one has

$n$7

A common misconception is that the pseudo-moment cone and the representable moment cone coincide. The inclusion above shows that representable moments form only a subcone of the full pseudo-moment cone (Kang et al., 7 May 2026).

Atoms arise from point evaluations. For any $n$8, the functional $n$9 belongs to $H_{n,d}$0, and

$H_{n,d}$1

a rank-one atomic moment matrix. Weighted atoms are finite conic sums

$H_{n,d}$2

Every rank-one extreme ray of $H_{n,d}$3 is generated by a scaled point evaluation $H_{n,d}$4, $H_{n,d}$5 (Kang et al., 7 May 2026).

This rank-one description is only part of the global extremal structure. Higher-rank extreme rays do exist globally in $H_{n,d}$6, including rank $H_{n,d}$7 for $H_{n,d}$8 and rank $H_{n,d}$9 for $d$0 (Kang et al., 7 May 2026). The significance of the recent local theory is not the absence of such rays globally, but the fact that they are excluded from certain minimal faces around generically generated atomic moment matrices.

3. Facial geometry and simplicial regularizability

The facial geometry of $d$1 is inherited from the PSD cone and then refined by Hankel constraints. If

$d$2

with $d$3 full column rank and $d$4, then

$d$5

Intersecting with the Hankel subspace yields

$d$6

(Kang et al., 7 May 2026).

The central geometric result of "Simplicial Regularizability of the Pseudo-Moment Cone and Carathéodory-Type Atomic Decomposition of Moment Matrices" establishes a local simplicial structure in the inhomogeneous-via-homogenization model (Kang et al., 7 May 2026). Let

$d$7

where $d$8. If the number of atoms $d$9 satisfies

$n$0

then for fixed $n$1 this upper bound scales as $n$2, and for generically chosen $n$3,

$n$4

and the set $n$5 is linearly independent (Kang et al., 7 May 2026).

The theorem is a local facial statement. Its interpretation is that the minimal face containing

$n$6

is a simplicial cone generated exactly by the planted rank-one atoms. Near such generically generated $n$7, the pseudo-moment cone locally regularizes to a cone isomorphic to $n$8, enabling unique atomic recovery (Kang et al., 7 May 2026).

This local regularity does not contradict the existence of higher-rank extreme rays. The same source states explicitly that higher-rank extreme rays exist globally, but they do not appear in the minimal face of $n$9 under the stated genericity and $2d$0 bound (Kang et al., 7 May 2026). This distinction is crucial for interpreting the result: it concerns a generic atomic regime, not the entirety of the cone.

4. Carathéodory-type extreme-ray decomposition

The same work develops a general decomposition procedure for spectrahedral cones, denoted RayDecomp (Kang et al., 7 May 2026). Given a nonempty spectrahedral cone $2d$1 and a nonzero point $2d$2, the algorithm returns coefficients and extreme rays

$2d$3

with each $2d$4 generating an extreme ray of $2d$5.

Its procedure is:

1. Initialize $2d$6, $2d$7.
2. While $2d$8:
   1. $2d$9.
   2. Draw $$\Sigma_{n,2d}:=\{f\in H_{n,2d}\mid f=\sum_i q_i^2,\ q_i\in H_{n,d}\},$$0 from any absolutely continuous distribution on $$\Sigma_{n,2d}:=\{f\in H_{n,2d}\mid f=\sum_i q_i^2,\ q_i\in H_{n,d}\},$$1.
   3. Solve

   $$\Sigma_{n,2d}:=\{f\in H_{n,2d}\mid f=\sum_i q_i^2,\ q_i\in H_{n,d}\},$$2

   and set $$\Sigma_{n,2d}:=\{f\in H_{n,2d}\mid f=\sum_i q_i^2,\ q_i\in H_{n,d}\},$$3 to the minimizer. 4. Compute

   $$\Sigma_{n,2d}:=\{f\in H_{n,2d}\mid f=\sum_i q_i^2,\ q_i\in H_{n,d}\},$$4 5. Update $$\Sigma_{n,2d}:=\{f\in H_{n,2d}\mid f=\sum_i q_i^2,\ q_i\in H_{n,d}\},$$5.

3. Return the decomposition (Kang et al., 7 May 2026).

If each SDP is solved exactly, then each step is well-defined, and with probability $$\Sigma_{n,2d}:=\{f\in H_{n,2d}\mid f=\sum_i q_i^2,\ q_i\in H_{n,d}\},$$6 the algorithm terminates in finitely many steps and returns an extreme-ray decomposition of $$\Sigma_{n,2d}:=\{f\in H_{n,2d}\mid f=\sum_i q_i^2,\ q_i\in H_{n,d}\},$$7 (Kang et al., 7 May 2026). The probabilistic mechanism is geometric: minimizing a random linear functional over the normalized face almost surely exposes a generator of an extreme ray.

When specialized to the pseudo-moment cone in the generic simplicial regime, the algorithm provably recovers the planted atoms and weights uniquely up to permutation. If

$$\Sigma_{n,2d}:=\{f\in H_{n,2d}\mid f=\sum_i q_i^2,\ q_i\in H_{n,d}\},$$8

with $$\Sigma_{n,2d}:=\{f\in H_{n,2d}\mid f=\sum_i q_i^2,\ q_i\in H_{n,d}\},$$9 in the theorem’s bound and $$P_{n,2d}:=\{f\in H_{n,2d}\mid f(x)\ge 0,\ \forall x\in\mathbb{R}^n\}.$$0 generic, then with probability $$P_{n,2d}:=\{f\in H_{n,2d}\mid f(x)\ge 0,\ \forall x\in\mathbb{R}^n\}.$$1

$$P_{n,2d}:=\{f\in H_{n,2d}\mid f(x)\ge 0,\ \forall x\in\mathbb{R}^n\}.$$2

(Kang et al., 7 May 2026).

For a rank-one output $$P_{n,2d}:=\{f\in H_{n,2d}\mid f(x)\ge 0,\ \forall x\in\mathbb{R}^n\}.$$3, if $$P_{n,2d}:=\{f\in H_{n,2d}\mid f(x)\ge 0,\ \forall x\in\mathbb{R}^n\}.$$4 is its unique nonzero eigenvalue and $$P_{n,2d}:=\{f\in H_{n,2d}\mid f(x)\ge 0,\ \forall x\in\mathbb{R}^n\}.$$5 is the associated unit eigenvector, then

$$P_{n,2d}:=\{f\in H_{n,2d}\mid f(x)\ge 0,\ \forall x\in\mathbb{R}^n\}.$$6

and $$P_{n,2d}:=\{f\in H_{n,2d}\mid f(x)\ge 0,\ \forall x\in\mathbb{R}^n\}.$$7 is read off from the degree-$$P_{n,2d}:=\{f\in H_{n,2d}\mid f(x)\ge 0,\ \forall x\in\mathbb{R}^n\}.$$8 entries of $$P_{n,2d}:=\{f\in H_{n,2d}\mid f(x)\ge 0,\ \forall x\in\mathbb{R}^n\}.$$9. The planted weight recovers as

$\Sigma_{n,2d}$00

(Kang et al., 7 May 2026).

Compared to flatness-based extraction on degree-$\Sigma_{n,2d}$01 truncated moment matrices, which can certify at most $\Sigma_{n,2d}$02 atoms from degree-$\Sigma_{n,2d}$03 data, this method recovers up to $\Sigma_{n,2d}$04 atoms without solving higher-degree relaxations (Kang et al., 7 May 2026). This suggests a different route to identifiability: not via higher-order flat extension, but via local simplicial facial geometry.

5. Numerical stabilization, complexity, and empirical behavior

A finite-precision robust implementation, RayDecompRestart, is described for the pseudo-moment setting (Kang et al., 7 May 2026). It includes facial reduction at each iteration: the algorithm restricts to the PSD face determined by the numerical range space of $\Sigma_{n,2d}$05, truncates small eigenvalues, removes redundant linear constraints via pivoted QR, and solves the reduced SDP in the face. The stated purpose is to restore strict feasibility and numerical stability.

The implementation also replaces the one-dimensional feasibility search by the closed-form step size

$\Sigma_{n,2d}$06

uses alternating projections onto the linear subspace $\Sigma_{n,2d}$07 and onto $\Sigma_{n,2d}$08, and employs a restart mechanism that detects SDP failures, reduces rank-thresholds, reprojects, and restarts (Kang et al., 7 May 2026).

For $\Sigma_{n,2d}$09, the ambient matrix size is

$\Sigma_{n,2d}$10

for fixed $\Sigma_{n,2d}$11. One inner-loop iteration costs, in the worst case,

$\Sigma_{n,2d}$12

time and

$\Sigma_{n,2d}$13

memory, where $\Sigma_{n,2d}$14 is the numerical rank of the input $\Sigma_{n,2d}$15. The total cost over $\Sigma_{n,2d}$16 iterations is

$\Sigma_{n,2d}$17

In the worst case $\Sigma_{n,2d}$18, this becomes

$\Sigma_{n,2d}$19

time and

$\Sigma_{n,2d}$20

memory (Kang et al., 7 May 2026).

The reported numerical behavior is twofold. First, a stabilized implementation using the MOSEK SDP solver, facial reduction, and alternating projections exhibits strong recovery across a range of $\Sigma_{n,2d}$21, with near-perfect recovery below a data-driven phase-transition threshold in $\Sigma_{n,2d}$22, often substantially larger than the conservative theoretical bound (Kang et al., 7 May 2026). Second, outside the guaranteed regime, the algorithm returns non-unique extreme rays, and empirically high-rank extreme rays appear once $\Sigma_{n,2d}$23 crosses the phase-transition threshold; one cited example is rank-$\Sigma_{n,2d}$24 rays for $\Sigma_{n,2d}$25 in the homogenized model (Kang et al., 7 May 2026). This suggests that the same procedure can function as a practical sampler of high-rank extreme rays, although that role is empirical rather than part of the proved guarantee.

6. Spectrahedral shadows, closedness, and broader convex-algebraic context

Pseudo-moment cones also arise as linear images of PSD cones. In SOS and moment hierarchies one works with moment or pseudo-moment matrices $\Sigma_{n,2d}$26 subject to linear constraints, and the truncated moment sequences extracted from $\Sigma_{n,2d}$27 form a spectrahedral shadow

$\Sigma_{n,2d}$28

for an appropriate subspace $\Sigma_{n,2d}$29 (Jiang et al., 2020). Consequently, the geometry of pseudo-moment cones is linked to the general theory of projections of the PSD cone.

"Bad Projections of the PSD Cone" studies when such linear images fail to be closed (Jiang et al., 2020). For a subspace $\Sigma_{n,2d}$30 with associated map $\Sigma_{n,2d}$31, the image cone

$\Sigma_{n,2d}$32

need not be closed, even though $\Sigma_{n,2d}$33 itself is closed. The closure satisfies

$\Sigma_{n,2d}$34

(Jiang et al., 2020). In the context of pseudo-moment cones, this means that some boundary points of the closure may not be attainable by actual PSD moment matrices after projection.

The paper provides several equivalent diagnostics for badness, including Pataki’s block characterization, an intrinsic criterion in terms of spectrahedral ranks and ideals, and a normal-cycle criterion based on complementarity (Jiang et al., 2020). It also identifies the Zariski closure of the bad locus as a hypersurface in the Grassmannian whose irreducible components are coisotropic hypersurfaces of symmetric determinantal varieties (Jiang et al., 2020). These results apply verbatim to pseudomoment cones viewed as spectrahedral shadows.

The practical implications are explicit. Non-closed pseudomoment cones lead to weak infeasibility and failure of strong duality in relaxations, resulting in duality gaps or numerical instability (Jiang et al., 2020). This is a different phenomenon from the local simplicial regularizability established in the Hankel spectrahedral realization of $\Sigma_{n,2d}$35 (Kang et al., 7 May 2026). A plausible implication is that the pseudo-moment cone should be understood through two complementary geometric lenses: as an intersection $\Sigma_{n,2d}$36, where local facial structure can become simplicial around generic atomic points, and as a spectrahedral shadow, where projection-induced non-closedness can create global boundary pathologies.

Within SOS optimization and moment hierarchies, these two perspectives address different technical issues. The simplicial regularizability result explains identifiability and decomposition in a generic $\Sigma_{n,2d}$37 atomic regime (Kang et al., 7 May 2026). The bad-projection framework explains when projected pseudomoment sets may violate Slater-type regularity and require facial reduction or extended dual constructions (Jiang et al., 2020). Taken together, they place the pseudomoment cone at the intersection of convex algebraic geometry, spectrahedral facial theory, and algorithmic moment decomposition.