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Register a new userConvex Sparse Blind Deconvolution
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Added by
Qingyun Sun
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In the blind deconvolution problem, we observe the convolution of an unknown filter and unknown signal and attempt to reconstruct the filter and signal. The problem seems impossible in general, since there are seemingly many more unknowns than knowns . Nevertheless, this problem arises in many application fields; and empirically, some of these fields have had success using heuristic methods -- even economically very important ones, in wireless communications and oil exploration. Todays fashionable heuristic formulations pose non-convex optimization problems which are then attacked heuristically as well. The fact that blind deconvolution can be solved under some repeatable and naturally-occurring circumstances poses a theoretical puzzle. To bridge the gulf between reported successes and theorys limited understanding, we exhibit a convex optimization problem that -- assuming signal sparsity -- can convert a crude approximation to the true filter into a high-accuracy recovery of the true filter. Our proposed formulation is based on L1 minimization of inverse filter outputs. We give sharp guarantees on performance of the minimizer assuming sparsity of signal, showing that our proposal precisely recovers the true inverse filter, up to shift and rescaling. There is a sparsity/initial accuracy tradeoff: the less accurate the initial approximation, the greater we rely on sparsity to enable exact recovery. To our knowledge this is the first reported tradeoff of this kind. We consider it surprising that this tradeoff is independent of dimension. We also develop finite-$N$ guarantees, for highly accurate reconstruction under $Ngeq O(k log(k) )$ with high probability. We further show stable approximation when the true inverse filter is infinitely long and extend our guarantees to the case where the observations are contaminated by stochastic or adversarial noise.
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Multichannel blind deconvolution is the problem of recovering an unknown signal $f$ and multiple unknown channels $x_i$ from their circular convolution $y_i=x_i circledast f$ ($i=1,2,dots,N$). We consider the case where the $x_i$s are sparse, and convolution with $f$ is invertible. Our nonconvex optimization formulation solves for a filter $h$ on the unit sphere that produces sparse output $y_icircledast h$. Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of $f$ up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of $f$ and $x_i$ using a simple manifold gradient descent (MGD) algorithm. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods.
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The Internet of Things and specifically the Tactile Internet give rise to significant challenges for notions of security. In this work, we introduce a novel concept for secure massive access. The core of our approach is a fast and low-complexity blind deconvolution algorithm exploring a bi-linear and hierarchical compressed sensing framework. We show that blind deconvolution has two appealing features: 1) There is no need to coordinate the pilot signals, so even in the case of collisions in user activity, the information messages can be resolved. 2) Since all the individual channels are recovered in parallel, and by assumed channel reciprocity, the measured channel entropy serves as a common secret and is used as an encryption key for each user. We will outline the basic concepts underlying the approach and describe the blind deconvolution algorithm in detail. Eventually, simulations demonstrate the ability of the algorithm to recover both channel and message. They also exhibit the inherent trade-offs of the scheme between economical recovery and secret capacity.
We revisit the Blind Deconvolution problem with a focus on understanding its robustness and convergence properties. Provable robustness to noise and other perturbations is receiving recent interest in vision, from obtaining immunity to adversarial attacks to assessing and describing failure modes of algorithms in mission critical applications. Further, many blind deconvolution methods based on deep architectures internally make use of or optimize the basic formulation, so a clearer understanding of how this sub-module behaves, when it can be solved, and what noise injection it can tolerate is a first order requirement. We derive new insights into the theoretical underpinnings of blind deconvolution. The algorithm that emerges has nice convergence guarantees and is provably robust in a sense we formalize in the paper. Interestingly, these technical results play out very well in practice, where on standard datasets our algorithm yields results competitive with or superior to the state of the art. Keywords: blind deconvolution, robust continuous optimization
The task of finding a sparse signal decomposition in an overcomplete dictionary is made more complicated when the signal undergoes an unknown modulation (or convolution in the complementary Fourier domain). Such simultaneous sparse recovery and blind demodulation problems appear in many applications including medical imaging, super resolution, self-calibration, etc. In this paper, we consider a more general sparse recovery and blind demodulation problem in which each atom comprising the signal undergoes a distinct modulation process. Under the assumption that the modulating waveforms live in a known common subspace, we employ the lifting technique and recast this problem as the recovery of a column-wise sparse matrix from structured linear measurements. In this framework, we accomplish sparse recovery and blind demodulation simultaneously by minimizing the induced atomic norm, which in this problem corresponds to the block $ell_1$ norm minimization. For perfect recovery in the noiseless case, we derive near optimal sample complexity bounds for Gaussian and random Fourier overcomplete dictionaries. We also provide bounds on recovering the column-wise sparse matrix in the noisy case. Numerical simulations illustrate and support our theoretical results.
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