[Paper Review] A Gramian Description of the Degree 4 Generalized Elliptope

Afonso S. Bandeira, Dmitriy Kunisky|arXiv (Cornell University)|Dec 30, 2018

Advanced Optimization Algorithms Research59 references4 citations

TL;DR

This paper provides a complete Gramian characterization of the degree 4 generalized elliptope, a tighter convex relaxation of the cut polytope in combinatorial optimization. It establishes a necessary and sufficient condition for a Gram matrix to belong to this set, proves a tight rank inequality between degree 2 and degree 4 pseudomoment matrices, and shows that the only shared extreme points between the elliptope and the degree 4 generalized elliptope are the cut matrices, thus eliminating spurious extreme points.

ABSTRACT

One of the most widely studied convex relaxations in combinatorial optimization is the relaxation of the cut polytope $\mathscr C^N$ to the elliptope $\mathscr E^N$, which corresponds to the degree 2 sum-of-squares (SOS) relaxation of optimizing a quadratic form over the hypercube $\{\pm 1\}^N$. We study the extension of this classical idea to degree 4 SOS, which gives an intermediate relaxation we call the degree 4 generalized elliptope $\mathscr E_4^N$. Our main result is a necessary and sufficient condition for the Gram matrix of a collection of vectors to belong to $\mathscr E_4^N$. Consequences include a tight rank inequality between degree 2 and degree 4 pseudomoment matrices, and a guarantee that the only extreme points of $\mathscr E^N$ also in $\mathscr E_4^N$ are the cut matrices; that is, $\mathscr E^N$ and $\mathscr E_4^N$ share no "spurious" extreme point. For Gram matrices of equiangular tight frames, we give a simple criterion for membership in $\mathscr{E}_4^N$. This yields new inequalities satisfied in $\mathscr{E}_4^N$ but not $\mathscr{E}^N$ whose structure is related to the Schläfli graph and which cannot be obtained as linear combinations of triangle inequalities. We also give a new proof of the restriction to degree 4 of a result of Laurent showing that $\mathscr{E}_4^N$ does not satisfy certain cut polytope inequalities capturing parity constraints. Though limited to this special case, our proof of the positive semidefiniteness of Laurent's pseudomoment matrix is short and elementary. Our techniques also suggest that membership in $\mathscr{E}_4^N$ is closely related to the partial transpose operation on block matrices, which has previously played an important role in the study of quantum entanglement. To illustrate, we present a correspondence between certain entangled bipartite quantum states and the matrices of $\mathscr{E}_4^N\setminus\mathscr{C}^N$.

Motivation & Objective

• To develop a complete algebraic description of the degree 4 generalized elliptope, a higher-degree sum-of-squares relaxation of the cut polytope.
• To identify the necessary and sufficient conditions under which a Gram matrix belongs to this relaxation.
• To prove that the only extreme points shared between the standard elliptope and the degree 4 generalized elliptope are the cut matrices, eliminating 'spurious' extreme points.
• To derive new inequalities—related to the Schläfli graph—that are valid in the degree 4 relaxation but not in the degree 2 case, and cannot be expressed as linear combinations of triangle inequalities.
• To provide a short, elementary proof of a result by Laurent on the failure of certain parity constraints in the degree 4 relaxation, using a novel pseudomoment witness construction.

Proposed method

• The authors introduce a pseudomoment witness construction to characterize membership in the degree 4 generalized elliptope via the Gram matrix of a vector system.
• They establish a duality between Gram matrices and pseudomoment matrices, showing that a matrix belongs to the degree 4 generalized elliptope if and only if a certain witness matrix is positive semidefinite.
• The method relies on a structural decomposition of symmetric matrices using projections onto subspaces of symmetric tensors, particularly $ V_{\mathsf{sym}} $ and $ V_{\mathsf{sym}}' $, to analyze the constraints on pseudomoment extensions.
• For equiangular tight frames, the authors derive a simplified criterion for membership in the degree 4 generalized elliptope, leveraging the frame's symmetry and coherence properties.
• They construct a specific positive semidefinite matrix $ \bm{A} \succeq \bm{0} $ with explicit coefficients tied to the Schläfli graph, which certifies new inequalities valid in $ \mathscr{E}_4^N $ but not in $ \mathscr{E}^N $.
• The proof of positive semidefiniteness is verified computationally using SageMath, though the authors note the lack of a conceptual explanation as an open problem.

Experimental results

Research questions

• RQ1 What is the complete algebraic condition for a Gram matrix to belong to the degree 4 generalized elliptope?
• RQ2 Are there new, non-trivial inequalities valid in the degree 4 relaxation that are not implied by triangle inequalities or the degree 2 relaxation?
• RQ3 Do the degree 4 generalized elliptope and the standard elliptope share any extreme points other than the cut matrices?
• RQ4 Can the pseudomoment matrix of a maximal equiangular tight frame be shown to be positive semidefinite via an elementary construction?
• RQ5 Is there a conceptual explanation for the positive semidefiniteness of the witness matrix $ \bm{A} $ used to derive Schläfli-type inequalities?

Key findings

• The paper provides a necessary and sufficient condition for a symmetric matrix to be in the degree 4 generalized elliptope, expressed through a witness matrix construction based on Gram vectors.
• It proves that the only extreme points common to both $ \mathscr{E}^N $ and $ \mathscr{E}_4^N $ are the cut matrices, implying no 'spurious' extreme points are shared.
• For equiangular tight frames, a simple criterion for membership in $ \mathscr{E}_4^N $ is derived, enabling the construction of new inequalities not implied by triangle inequalities.
• The authors construct a positive semidefinite matrix $ \bm{A} \in \mathbb{R}^{N^2 \times N^2} $ with explicit coefficients involving $ \gamma_1 = 1/126 $, $ \gamma_2 = 1/36 $, $ \kappa_1 = 2/9 $, and $ \kappa_2 = 1/28 $, which certifies a new class of Schläfli-type inequalities.
• The paper gives a short, elementary proof of a result by Laurent showing that certain parity constraints do not hold in $ \mathscr{E}_4^N $, using a novel pseudomoment witness instead of advanced tools from association schemes.
• The authors establish a correspondence between matrices in $ \mathscr{E}_4^N \setminus \mathscr{C}^N $ and entangled quantum states, linking the relaxation to the partial transpose operation in quantum information theory.