Theoretical Analysis of Euclidean Distance Matrix Completion for Ad hoc Microphone Array Calibration

We consider the problem of ad~hoc microphone array calibration where the distance matrix consisted of all microphones pairwise distances have entries missing corresponding to distances greater than $d_{\text{max}}$. Furthermore, the known entries are noisy modeled through additive independent random variables with strictly sub-Gaussian distribution, $\textsc{S}\textsc{ub}(c^2(d))$ with a bounded constant dependent on the distance $d$ between the microphone pairs. In this report, we exploit matrix completion approach to recover the full distance matrix. We derive the theoretical guarantees of microphone calibration performance which demonstrates that the error of calibrating a network of $N$ microphones using matrix completion decreases as $\mathcal{O}(N^{-1/2})$.