We present a method to remove the effects of sensor-specific noise in multiple-channel recordings such as magnetoencephalography (MEG) or electroencephalography (EEG). The method assumes that every source of interest is picked up by more than one sensor, as is the case with systems with spatially dense sensors. To reduce noise, each sensor signal is projected on the subspace spanned by its neighbors and replaced by its projection. In this process, components specific to the sensor (typically wide-band noise and/or 'glitches') are eliminated, while sources of interest are retained. Evaluation with real and simulated MEG signals shows that the method removes sensor-specific noise effectively, without removing or distorting signals of interest. It complements existing noise-reduction methods that target environmental or physiological noise.