The real positive definite completion problem for a simple cycle

Abstract

We consider the question of whether a real partial positive denite matrix (in which the specied o-diagonal entries consist of a full n cycle) has a positive denite completion. This lies in contrast to the previously studied chordal case. We give t w o solutions. In one, we describe about n 2 independent conditions on angles associated with a normalization of the data that are necessary and sucient. The second is more computational and allows presentation of all positive denite completions, as well as answering the existence question.

Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry

“…The dual cone C G consists of all partial matrices T ∈ R E with entries in positions (i, j) ∈ E, which can be extended to a full positive definite matrix. So, maximum likelihood estimation in Gaussian graphical models corresponds to the classical positive definite matrix completion problem (Barrett et al 1993;Grone et al 1984;Barrett et al 1996;Laurent 2001b). In this section we investigate our three guiding questions, first for chordal graphs, next for the chordless m-cycle C m , then for all graphs with five or less vertices, and finally for the m-wheel W m .…”

“…, x m ) such that (25) can be filled up to a positive definite matrix. The 2 × 2-minors of (25) Barrett et al (1993) gave a beautiful polyhedral description of the spectrahedron C ′ m . The idea is to replace each x i by its arc-cosine, that is, to substitute x i = cos(φ i ) into (25).…”

“…We call Γ The polyhedral characterization of C m given in Barrett et al (1993) translates into the following theorem. and we then use the following relation to write (28) as an algebraic expression in x 1 , .…”

“…Similarly, vanishing of Γ ′ m characterizes partial matrices (25) that can be completed to rank ≤ 2. Independently of the work of Barrett et al (1993), a solution to the problem of characterizing the cone C m appeared in the same year in the statistics literature, namely by Buhl (1993). For statisticians, the cone C m is the set of partial sample covariance matrices on the m-cycle for which the MLE exists.…”

“…If rank(S) < m then it can happen that the fiber is empty, in which case the MLE does not exist for (L, S). Work of Buhl (1993) and Barrett et al (1993) addresses this issue for graphical models; see Section 4 below. Our motivating statistical problem is to identify conditions on the pair (L, S) that ensure the existence of the MLE.…”