On the Maximum Entropy Completion of Circulant Covariance Matrices

Francesca Carli, Tryphon Georgiou
This paper deals with the positive-definite completion of partially specified (block-) circulant covariance matrices. In the absence of any constraint other than positivity, the maximal-determinant completion of a partially specified covari-ance matrix (i.e., the so-called maximum entropy completion) was shown by Dempster to have an inverse with zero-values at all locations where the original matrix was unspecified-this will be referred to as the Dempster property. In earlier work, Carli etal.
more » ... 2] showed that even under the constraint of a covariance being block-circulant, as long as the unspecified elements are in a single band, the maximum entropy completion has the Dempster property. The purpose of the present paper is to prove that circulant, block-circulant, or Hermitian constraints do not interfere with the Dempster property of the maximum entropy completion. I.e., regardless of which elements are specified, the completion has the Dempster property. This fact is a direct consequence of the invariance of the determinant to the group of transformations that leave circulant, block-circulant, or Hermitian matrices invariant. A description of the set of all positive extensions is discussed and connections between this set and the factorization of certain polynomials in many variables, facilitated by the circulant structure, is highlighted.