  We study exact matrix completion from partially available data with hidden
connectivity patterns. Exact matrix completion was shown to be possible
recently by Cosse and Demanet in 2021 with Lasserre's relaxation using the
trace of the variable matrix as the objective function with given data
structured in a chain format. In this study, we introduce a structure for the
objective function so that the resulting sum-of-squares (SOS) relaxation, the
dual of Lasserre's SDP relaxation, produces a rank-($N$-1) solution, where $N$
denotes the size of variable matrix in the SOS relaxation. Specifically, the
arrowhead structure is employed for the coefficient matrix of the objective
function. We show that a matrix can be exactly completed through the SOS
relaxation when the connectivity of given data is not explicitly displayed or
follows a chain format. The theoretical exactness is proved using the rank of
the Gram matrix for the SOS relaxation. We also present numerical algorithms
designed to find the coefficient matrix in the SOS relaxation. Numerical
experiments illustrate the validity of the proposed algorithm.
