We propose to use non-negative matrix factorization (NMF) to estimate the unknown pairwise distances and reconstruct a distance matrix for microphone array position calibration. We develop new multiplicative update rules for NMF with incomplete input matrix that take into account the symmetry of the distance matrix. Additionally, we develop a convex matrix completion method which is related to an $l_2$-regularized symmetric NMF. Thorough experiments demonstrate that the proposed methods lead to substantial improvement over the state-of-the-art techniques in a wide range of signal-to-noise and unknown-distance ratios. The convex symmetric matrix completion method was found to be the most robust method with less computational cost.