We consider the multichannel blind deconvolution problem where we observe theoutput of multiple channels that are all excited with the same unknown input.From these observations, we wish to estimate the impulse responses of each ofthe channels. We show that this problem is well-posed if the channels follow abilinear model where the ensemble of channel responses is modeled as lying in alow-dimensional subspace but with each channel modulated by an independentgain. Under this model, we show how the channel estimates can be found byminimizing a quadratic functional over a non-convex set. We analyze two methods for solving this non-convex program, and provideperformance guarantees for each. The first is a method of alternatingeigenvectors that breaks the program down into a series of eigenvalue problems.The second is a truncated power iteration, which can roughly be interpreted asa method for finding the largest eigenvector of a symmetric matrix with theadditional constraint that it adheres to our bilinear model. As with mostnon-convex optimization algorithms, the performance of both of these algorithmsis highly dependent on having a good starting point. We show how such astarting point can be constructed from the channel measurements. Our performance guarantees are non-asymptotic, and provide a sufficientcondition on the number of samples observed per channel in order to guaranteechannel estimates of a certain accuracy. Our analysis uses a model with a"generic" subspace that is drawn at random, and we show the performance boundshold with high probability. Mathematically, the key estimates are derived byquantifying how well the eigenvectors of certain random matrices approximatethe eigenvectors of their mean. We also present a series of numerical results demonstrating that theempirical performance is consistent with the presented theory.
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