Random Spectrahedra

Paul Breiding, Khazhgali Kozhasov, and Antonio Lerario

Contact the author: Please use for correspondence this email.
Submission date: 23. Nov. 2017
Pages: 24
published in: SIAM journal on optimization, 29 (2019) 4, p. 2608-2624 
DOI number (of the published article): 10.1137/18M1208812
Bibtex
Download full preprint: PDF (1325 kB)

Abstract:
Spectrahedral cones are linear sections of the cone of positive semidefinite symmetric matrices. We study statistical properties of random spectrahedral cones (intersected with the sphere)

<center class="math-display"> <img src="/fileadmin/preprint_img/2017/tex_2300a0x.png" alt="&amp;#x1D4AE; &amp;#x2113;,n = {(x0,...,x&amp;#x2113;) &amp;isin; S&amp;#x2113; | x0I + x1R1 + &amp;sdot;&amp;sdot;&amp;sdot;+ x&amp;#x2113;R&amp;#x2113; &amp;#x227B; 0} " class="math-display"></center> where R₁,…,R_ℓ are independent GOE(n)-distributed matrices rescaled by (2nℓ)^−1∕2.

We relate the expectation of the volume of 𝒮 _ℓ,n with some statistics of the smallest eigenvalue of a GOE(n) matrix, by providing explicit formulas for this quantity. These formulas imply that as ℓ,n →∞ on average 𝒮 _ℓ,n keeps a positive fraction of the volume of the sphere S^ℓ (the exact constant is Φ(−1) ≈ 0.1587, where Φ is the cumulative distribution function of a standard gaussian variable).

For ℓ = 2 spectrahedra are generically smooth, but already when ℓ = 3 singular points on their boundaries appear with positive probability. We relate the average number 𝔼σ_n of singular points on the boundary of a three-dimensional spectrahedron 𝒮 _3,n to the volume of the set of symmetric matrices whose two smallest eigenvalues coincide. In the case of quartic spectrahedra (n = 4) we show that 𝔼σ₄ = 6 −. Moreover, we prove that the average number 𝔼ρ_n of singular points on the random symmetroid surface

<center class="math-display"> <img src="/fileadmin/preprint_img/2017/tex_2300a2x.png" alt=" 3 &amp;Sigma;3,n = {(x0,x1,x2,x3) &amp;isin; S | det(x0I + x1R1 +x2R2 + x3R3) = 0}, " class="math-display"></center> equals n(n − 1). This quantity is related to the volume of the set of symmetric matrices with repeated eigenvalues.