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Blind deconvolution and demixing is the problem of reconstructing convolved signals and kernels from the sum of their convolutions. This problem arises in many applications, such as blind MIMO. This work presents a separable approach to blind deconvolution and demixing via convex optimization. Unlike previous works, our formulation allows separation into smaller optimization problems, which significantly improves complexity. We develop recovery guarantees, which comply with those of the original non-separable problem, and demonstrate the method performance under several normalization constraints. [ABSTRACT FROM AUTHOR]

We consider simultaneous blind deconvolution of $r$ source signals from their noisy superposition, a problem also referred to blind demixing and deconvolution. This signal processing problem occurs in the context of the Internet of Things where a massive number of sensors sporadically communicate only short messages over unknown channels. We show that robust recovery of message and channel vectors can be achieved via convex optimization when random linear encoding using i.i.d. complex Gaussian matrices is used at the devices and the number of required measurements at the receiver scales with the degrees of freedom of the overall estimation problem. Since the scaling is linear in $r$ our result significantly improves over recent works. [ABSTRACT FROM AUTHOR]

We study the problem of demixing a pair of sparse signals from noisy, nonlinear observations of their superposition. Mathematically, we consider a nonlinear signal observation model, y_i = g(a_i^Tx) + e_i, \ i=1,\ldots,m, where x = \Phi w+\Psi z denotes the superposition signal, \Phi and \Psi, and w, z\in \mathbb {R}^n are sparse coefficient vectors of the constituent signals, and e_i represents the noise. Moreover, g represents a nonlinear link function, and a_i\in \mathbb {R}^n is the ith row of the measurement matrix A\in \mathbb R^m\times n. Problems of this nature arise in several applications ranging from astronomy, computer vision, and machine learning. In this paper, we make some concrete algorithmic progress for the above demixing problem. Specifically, we consider two scenarios: first, the case when the demixing procedure has no knowledge of the link function, and second is the case when the demixing algorithm has perfect knowledge of the link function. In both cases, we provide fast algorithms for recovery of the constituents w and $z$ from the observations. Moreover, we support these algorithms with a rigorous theoretical analysis and derive (nearly) tight upper bounds on the sample complexity of the proposed algorithms for achieving stable recovery of the component signals. We also provide a range of numerical simulations to illustrate the performance of the proposed algorithms on both real and synthetic signals and images. [ABSTRACT FROM PUBLISHER]

Suppose that we have r sensors and each one intends to send a function {g}_{i} (e.g., a signal or an image) to a receiver common to all r sensors. During transmission, each gi gets convolved with a function {f}i . The receiver records the function y , given by the sum of all these convolved signals. When and under which conditions is it possible to recover the individual signals gi and the blurring functions fi from just one received signal y ? This challenging problem, which intertwines blind deconvolution with blind demixing, appears in a variety of applications, such as audio processing, image processing, neuroscience, spectroscopy, and astronomy. It is also expected to play a central role in connection with the future Internet-of-Things. We will prove that under reasonable and practical assumptions, it is possible to solve this, otherwise, highly ill-posed problem and recover the $r$ transmitted functions gi and the impulse responses fi in a robust, reliable, and efficient manner, from just one single received function y by solving a semidefinite program. We derive explicit bounds on the number of measurements needed for successful recovery and prove that our method is robust in the presence of noise. Our theory is actually suboptimal, since numerical experiments demonstrate that, quite remarkably, recovery is still possible if the number of measurements is close to the number of degrees of freedom. [ABSTRACT FROM PUBLISHER]