Relative impulse responses between microphones are usually long and dense due to the reverberant acoustic environment. Estimating them from short and noisy recordings poses a long-standing challenge of audio signal processing. In this paper we apply a novel strategy based on ideas of Compressed Sensing. Relative transfer function (RTF) corresponding to the relative impulse response can often be estimated accurately from noisy data but only for certain frequencies. This means that often only an incomplete measurement of the RTF is available. A complete RTF estimate can be obtained through finding its sparsest representation in the time-domain: that is, through computing the sparsest among the corresponding relative impulse responses. Based on this approach, we propose to estimate the RTF from noisy data in three steps. First, the RTF is estimated using any conventional method such as the non-stationarity-based estimator by Gannot et al. or through Blind Source Separation. Second, frequencies are determined for which the RTF estimate appears to be accurate. Third, the RTF is reconstructed through solving a weighted $\ell_1$ convex program, which we propose to solve via a computationally efficient variant of the SpaRSA (Sparse Reconstruction by Separable Approximation) algorithm. An extensive experimental study with real-world recordings has been conducted. It has been shown that the proposed method is capable of improving many conventional estimators used as the first step in most situations.