From Blind deconvolution to Blind Super-Resolution through convex programming

Abstract

This paper discusses the recovery of an unknown signal x ∈ R L through the result of its convolution with an unknown filter h ∈ R L . This problem, also known as blind deconvolution, has been studied extensively by the signal processing and applied mathematics communities, leading to a diversity of proofs and algorithms based on various assumptions on the filter and its input. Sparsity of this filter, or in contrast, non vanishing of its Fourier transform are instances of such assumptions. The main result of this paper shows that blind deconvolution can be solved through nuclear norm relaxation in the case of a fully unknown channel, as soon as this channel is probed through a few N µ 2 m K 1/2 input signals xn = Cnmn, n = 1, . . . , N, that are living in known K-dimensional subspaces Cn of R L . This result holds with high probability on the genericity of the subspaces Cn as soon as L K 3/2 and N K 1/2 up to log factors. Our proof system relies on the construction of a certificate of optimality for the underlying convex program. This certificate expands as a Neumann series and is shown to satisfy the conditions for the recovery of the matrix encoding the unknowns by controlling the terms in this series. We apply specific concentration bounds to the first two terms in the series, in order to reduce the sample complexities, and bound the remaining terms through a more general argument. The first term in the series is bounded through the subexponential Bernstein inequality. The second term is decomposed into two contributions, each corresponding to a fourth order gaussian chaos. The first contribution, containing the univariate fourth order monomials in the random vectors defining the subspaces Cn, is bounded through a matrix version of the Rosenthal-Pinelis inequality. The second contribution, containing the cross terms, is bounded by using a decoupling argument for U-Statistics. An incidental consequence of the result of this paper, following from the lack of assumptions on the filter, is that nuclear norm relaxation can be extended from blind deconvolution to blind super-resolution, as soon as the unknown ideal low pass filter has a sufficiently large support compared to the ambient dimension L. Numerical experiments supporting the theory as well as its application to blind super-resolution are provided.Acknowledgement. AC was supported by the FNRS, FSMP, BAEF and Francqui Foundations. AC thanks MIT Math, Harvard IACS and The University of Chicago, for hosting him during this work. AC is grateful to Laurent Demanet and Ali Ahmed for interesting discussions as well as Joel Tropp for pointing out the matrix version of the Rosenthal-Pinelis inequality.

Blind Gain and Phase Calibration via Sparse Spectral Methods

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“…The lifting technique has been applied on a wide range of blind deconvolution and sensor calibration models [4,1,9,7,44], among which [7,44] are mostly related with our model. In [7], Chi considered a slightly different single-snapshot model:…”

“…Let n 0 := g 2 f which can be estimated from R y based on the following lemma (see Appendix B for the proof): Lemma 2. Let R y be defined in (9). Then…”