In this paper, we propose a novel regression-based algorithm for suppressing the residual echo present in the output of an acoustic echo canceller (AEC). We learn a functional relationship between the magnitudes of many frames of the speaker signal and the magnitude of the echo residual, per subband. We estimate and track the parameters of this function using adaptive algorithms (e.g. NLMS). We show that this approach can be interpreted as a rank-1 approximation to a more general regression model, and can address shortcomings of the earlier approaches based on correlation analysis. Preliminary results using linear regression on magnitudes of real audio signals in both mono and stereo situations demonstrate an average of 7 dB of echo suppression over the AEC output signal under a wide variety of conditions without near-end signal distortion. The framework is general enough to promise even further reductions in the future.