arXiv subjects

Shuyang Ling

Publications and source records attributed to Shuyang Ling.

23 records · Page 2Linked to original sources

arXiv2017

Regularized Gradient Descent: A Nonconvex Recipe for Fast Joint Blind Deconvolution and Demixing

We study the question of extracting a sequence of functions $\{\boldsymbol{f}_i, \boldsymbol{g}_i\}_{i=1}^s$ from observing only the sum of their convolutions, i.e., from $\boldsymbol{y} = \sum_{i=1}^s \boldsymbol{f}_i\ast \boldsymbol{g}_i$. While convex optimization techniques are able to solve this joint blind deconvolution-demixing problem provably and robustly under certain conditions, for medium-size or large-size problems we need computationally faster methods without sacrificing the benefits of mathematical rigor that come with convex methods. In this paper, we present a non-convex algorithm which guarantees exact recovery under conditions that are competitive with convex optimization methods, with the additional advantage of being computationally much more efficient. Our two-step algorithm converges to the global minimum linearly and is also robust in the presence of additive noise. While the derived performance bounds are suboptimal in terms of the information-theoretic limit, numerical simulations show remarkable performance even if the number of measurements is close to the number of degrees of freedom. We discuss an application of the proposed framework in wireless communications in connection with the Internet-of-Things.

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arXiv2017

Self-Calibration and Bilinear Inverse Problems via Linear Least Squares

Whenever we use devices to take measurements, calibration is indispensable. While the purpose of calibration is to reduce bias and uncertainty in the measurements, it can be quite difficult, expensive, and sometimes even impossible to implement. We study a challenging problem called \emph{self-calibration}, i.e., the task of designing an algorithm for devices so that the algorithm is able to perform calibration automatically. More precisely, we consider the setup $\boldsymbol{y} = \mathcal{A}(\boldsymbol{d}) \boldsymbol{x} + \boldsymbolε$ where only partial information about the sensing matrix $\mathcal{A}(\boldsymbol{d})$ is known and where $\mathcal{A}(\boldsymbol{d})$ linearly depends on $\boldsymbol{d}$. The goal is to estimate the calibration parameter $\boldsymbol{d}$ (resolve the uncertainty in the sensing process) and the signal/object of interests $\boldsymbol{x}$ simultaneously. For three different models of practical relevance, we show how such a \emph{bilinear} inverse problem, including blind deconvolution as an important example, can be solved via a simple \emph{linear least squares} approach. As a consequence, the proposed algorithms are numerically extremely efficient, thus potentially allowing for real-time deployment. We also present a variation of the least squares approach, which leads to a~\emph{spectral method}, where the solution to the bilinear inverse problem can be found by computing the singular vector associated with the smallest singular value of a certain matrix derived from the bilinear system. Explicit theoretical guarantees and stability theory are derived for both techniques; and the number of sampling complexity is nearly optimal (up to a poly-log factor). Applications in imaging sciences and signal processing are discussed and numerical simulations are presented to demonstrate the effectiveness and efficiency of our approach.

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arXiv2017

Blind Deconvolution Meets Blind Demixing: Algorithms and Performance Bounds

Suppose that we have $r$ sensors and each one intends to send a function $\boldsymbol{g}_i$ (e.g.\ a signal or an image) to a receiver common to all $r$ sensors. During transmission, each $\boldsymbol{g}_i$ gets convolved with a function $\boldsymbol{f}_i$. The receiver records the function $\boldsymbol{y}$, given by the sum of all these convolved signals. When and under which conditions is it possible to recover the individual signals $\boldsymbol{g}_i$ and the blurring functions $\boldsymbol{f}_i$ from just one received signal $\boldsymbol{y}$? 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 $\boldsymbol{g}_i$ and the impulse responses $\boldsymbol{f}_i$ in a robust, reliable, and efficient manner from just one single received function $\boldsymbol{y}$ 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 sub-optimal, 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.

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arXiv2016

Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex Optimization

We study the question of reconstructing two signals $f$ and $g$ from their convolution $y = f\ast g$. This problem, known as {\em 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.

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arXiv2015

Self-Calibration and Biconvex Compressive Sensing

The design of high-precision sensing devises becomes ever more difficult and expensive. At the same time, the need for precise calibration of these devices (ranging from tiny sensors to space telescopes) manifests itself as a major roadblock in many scientific and technological endeavors. To achieve optimal performance of advanced high-performance sensors one must carefully calibrate them, which is often difficult or even impossible to do in practice. In this work we bring together three seemingly unrelated concepts, namely Self-Calibration, Compressive Sensing, and Biconvex Optimization. The idea behind self-calibration is to equip a hardware device with a smart algorithm that can compensate automatically for the lack of calibration. We show how several self-calibration problems can be treated efficiently within the framework of biconvex compressive sensing via a new method called SparseLift. More specifically, we consider a linear system of equations y = DAx, where both x and the diagonal matrix D (which models the calibration error) are unknown. By "lifting" this biconvex inverse problem we arrive at a convex optimization problem. By exploiting sparsity in the signal model, we derive explicit theoretical guarantees under which both x and D can be recovered exactly, robustly, and numerically efficiently via linear programming. Applications in array calibration and wireless communications are discussed and numerical simulations are presented, confirming and complementing our theoretical analysis.

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