We propose a novel deep learning-based channel estimation technique for high-dimensional communication signals that does not require any training. Our method is broadly applicable to channel estimation for multicarrier signals with any number of antennas, and has low enough complexity to be used in a mobile station. The proposed deep channel estimator can outperform LS estimation with nearly the same complexity, and approach MMSE estimation performance to within 1 dB without knowing the second order statistics. The only complexity increase with respect to LS estimator lies in fitting the parameters of a deep neural network (DNN) periodically on the order of the channel coherence time. We empirically show that the main benefit of this method accrues from the ability of this specially designed DNN to exploit correlations in the time-frequency grid. The proposed estimator can also reduce the number of pilot tones needed in an OFDM time-frequency grid, e.g. in an LTE scenario by 98% (68%) when the channel coherence time interval is 73ms (4.5ms).