We study the question of reconstructing two signals f and g from their convolution y = f * g. This problem, known as blind deconvolution, pervades many areas of science and technology, including astronomy, medical imaging, optics, and wireless communications. A key challenge of this intricate non-convex optimization problem is that it might exhibit many local minima. We present an efficient numerical algorithm that is guaranteed to recover the exact solution, when the number of measurements is (up to log-factors) slightly larger than the information-theoretical minimum, and under reasonable conditions on f and g. The proposed regularized gradient descent algorithm converges at a geometric rate and is provably robust in the presence of noise. To the best of our knowledge, our algorithm is the first blind deconvolution algorithm that is numerically efficient, robust against noise, and comes with rigorous recovery guarantees under certain subspace conditions. Moreover, numerical experiments do not only provide empirical verification of our theory, but they also demonstrate that our method yields excellent performance even in situations beyond our theoretical framework.

Matrices of low rank can be uniquely determined from fewer linear measurements, or entries, than the total number of entries in the matrix. Moreover, there is a growing literature of computationally e cient algorithms which can recover a low rank matrix from such limited information, typically referred to as matrix completion. We introduce a particularly simple yet highly e cient alternating projection algorithm which uses an adaptive stepsize calculated to be exact for a restricted subspace. This method is proven to have near optimal order recovery guarantees from dense measurement masks, and is observed to have average case performance superior in some respects to other matrix completion algorithms for both dense measurement masks and from entry measurements. In particular, this proposed algorithm is able to recover matrices from extremely close to the minimum number of measurements necessary.

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We introduce the Conjugate Gradient Iterative Hard Thresholding (CGIHT) family of algorithms for the efficient solution of constrained underdetermined linear systems of equations arising in compressed sensing, row sparse approximation, and matrix completion. CGIHT is designed to balance the low per iteration complexity of simple hard thresholding algorithms with the fast asymptotic convergence rate of employing the conjugate gradient method. We establish provable recovery guarantees and stability to noise for variants of CGIHT with sufficient conditions in terms of the restricted isometry constants of the sensing operators. Extensive empirical performance comparisons establish significant computational advantages for CGIHT both in terms of the size of problems which can be accurately approximated and in terms of overall computation time.

We establish theoretical recovery guarantees of a family of Riemannian optimization algorithms for low rank matrix recovery, which is about recovering an m × n rank r matrix from p < mn number of linear measurements. The algorithms are first interpreted as iterative hard thresholding algorithms with subspace projections. Based on this connection, we show that provided the restricted isometry constant R 3r of the sensing operator is less than C κ / √ r, the Riemannian gradient descent algorithm and a restarted variant of the Riemannian conjugate gradient algorithm are guaranteed to converge linearly to the underlying rank r matrix if they are initialized by one step hard thresholding. Empirical evaluation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements necessary.