Matrix Extreme Points and Free extreme points of Free spectrahedra

TL;DR

The paper addresses whether matrix extreme points of real free spectrahedra can differ from free extreme points and proves three main results: (i) matrix extreme points can indeed fail to be free extreme, (ii) when the defining size satisfies $d=2$, matrix and free extreme points coincide, and (iii) practical, exact and numerical methods (including a novel Nullspace Purification technique) can construct and certify non-free matrix extreme points. The authors develop exact-arithmetic constructions for $g=3$ and $g=4$ cases, provide a detailed numerical dilation algorithm to generate extreme points with small kernels, and demonstrate through extensive experiments that non-free matrix extreme points are not rare, especially when the Arveson rank-nullity count exceeds the Matrix extreme count. They also show that the Free Carathéodory expansion algorithm can effectively express points as matrix convex combinations of free extreme points, with improved reliability thanks to Nullspace Purification. Overall, the work advances both the theory and computational practice of extreme points in free spectrahedra, with implications for noncommutative optimization and related applications in control and quantum information. The paper highlights the interplay between projective maps, dilation theory, and numerical stability in understanding the geometry of matrix convex sets.

Abstract

A spectrahedron is a convex set defined by a linear matrix inequality, i.e., the set of all $x \in \mathbb{R}^g$ such that \[ L_A(x) = I + A_1 x_1 + A_2 x_2 + \dots + A_g x_g \succeq 0 \] for some symmetric matrices $A_1,\ldots,A_g$. This can be extended to matrix spaces by taking $X$ to be a tuple of real symmetric matrices of any size and using the Kronecker product $$L_A(X) = I_n \otimes I_d + A_1 \otimes X_1 + A_2 \otimes X_2 + \dots + A_g \otimes X_g.$$ The solution set of $L_A (X) \succeq 0$ is called a \textit{free spectrahedron}. Free spectrahedra are important in systems engineering, operator algebras, and the theory of matrix convex sets. Matrix and free extreme points of free spectrahedra are of particular interest. While many authors have studied matrix and free extreme points of free spectrahedra, it has until now been unknown if these two types of extreme points are actually different. The results of this paper fall into three categories: theoretical, algorithmic, and experimental. Firstly, we prove the existence of matrix extreme points of free spectrahedra that are not free extreme. This is done by producing exact examples of matrix extreme points that are not free extreme. We also show that if the $A_i$ are $2 \times 2$ matrices, then matrix and free extreme points coincide. Secondly, we detail methods for constructing matrix extreme points of free spectrahedra that are not free extreme, both exactly and numerically. We also show how a recent result due to Kriel (Complex Anal.~Oper.~Theory 2019) can be used to efficiently test whether a point is matrix extreme. Thirdly, we provide evidence that a substantial number of matrix extreme points of free spectrahedra are not free extreme. Numerical work in another direction shows how to effectively write a given tuple in a free spectrahedron as a matrix convex combination of its free extreme points.

Figures (13)

• Figure 1: $g=2$. Histogram of $\mu$ data, viewed from two different angles. The plot indicates frequency of $(\mu,d)$ occurrences.
• Figure 2: This is a cross section of the 3D histogram for $g=2$ given in \ref{['fig:g2Hist90Off']}. In particular, this histogram shows the cross section for $d=2$ and $n_0 = 5.$ The horizontal axis is $0< \mu < 1$ and the vertical axis is the frequency with which a value of $\mu$ was observed. See \ref{['fig:g2HistGrid']} for the remaining cross sections.
• Figure 3: $g=2$. This figure shows a grid of histograms. These are cross sections of the 3D histogram \ref{['fig:g2Hist90Off']}. The rows correspond to $n_0$, low $n_0=2$ at the top thru $n_0=4$ at the bottom, and the columns are $d=2$ on the left thru $d=5$ on the right.
• Figure 4: $g=3$. Histogram of $\mu$ data, viewed from two angles.
• Figure 5: $g=4$. Histogram of $\mu$ data.

Theorems & Definitions (36)

• Lemma 1.1
• proof
• Remark 1.2
• Theorem 1.3: EH19
• Proposition 2.1: EHKM18
• Theorem 2.2: K+
• proof
• Corollary 2.3
• proof
• Proposition 2.4