Almost lossless analog compression without phase information

Riegler, Erwin; Taubock, Georg

doi:10.1109/isit.2015.7282605

Cited by 7 publications

(4 citation statements)

References 19 publications

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“…where we used (17) together with dim B (S ) dim B (S) thanks to S ⊆ S. Since lim j→∞ log(1/δ j ) = ∞, the convergence of the left-hand side (LHS) of (23) to a finite negative number implies that lim j→∞ log M S (δ j )δ k j = −∞ and hence lim j→∞ M S (δ j )δ k j = 0. Taking the limit j → ∞ in (20)- (22) implies that the LHS in (20) equals zero, which concludes the proof.…”

Section: Probabilistic Uncertainty Relationsupporting

confidence: 51%

“…4.3], but use a new proof technique that is more direct. Finally, we note that the probabilistic uncertainty relation developed here is a quite general tool and has been applied to establish information-theoretic limits of matrix completion [19] and of phase retrieval [20].…”

Section: Probabilistic Uncertainty Relationmentioning

confidence: 98%

“…where (20) follows from a union bound over the covering balls B n (x i , δ j ) of S and in (21) we set…”

Section: Probabilistic Uncertainty Relationmentioning

confidence: 99%

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Almost Lossless Analog Signal Separation and Probabilistic Uncertainty Relations

Stotz

Riegler

Agustsson

et al. 2017

IEEE Trans. Inform. Theory

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We propose an information-theoretic framework for analog signal separation. Specifically, we consider the problem of recovering two analog signals, modeled as general random vectors, from the noiseless sum of linear measurements of the signals. Our framework is inspired by the groundbreaking work of Wu and Verdú (2010) on analog compression and encompasses, inter alia, inpainting, declipping, super-resolution, the recovery of signals corrupted by impulse noise, and the separation of (e.g., audio or video) signals into two distinct components. The main results we report are general achievability bounds for the compression rate, i.e., the number of measurements relative to the dimension of the ambient space the signals live in, under either measurability or Hölder continuity imposed on the separator. Furthermore, we find a matching converse for sources of mixed discrete-continuous distribution. For measurable separators our proofs are based on a new probabilistic uncertainty relation which shows that the intersection of generic subspaces with general sets of sufficiently small Minkowski dimension is empty. Hölder continuous separators are dealt with by introducing the concept of regularized probabilistic uncertainty relations. The probabilistic uncertainty relations we develop are inspired by embedding results in dynamical systems theory due to Sauer et al. (1991) and-conceptually-parallel classical Donoho-Stark and Elad-Bruckstein uncertainty principles at the heart of compressed sensing theory. Operationally, the new uncertainty relations take the theory of sparse signal separation beyond traditional sparsity-as measured in terms of the number of non-zero entries-to the more general notion of low description complexity as quantified by Minkowski dimension. Finally, our approach also allows to significantly strengthen key results in Wu and Verdú (2010).

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Section: Probabilistic Uncertainty Relationsupporting

confidence: 51%

Section: Probabilistic Uncertainty Relationmentioning

confidence: 98%

See 1 more Smart Citation

Almost Lossless Analog Signal Separation and Probabilistic Uncertainty Relations

Stotz

Riegler

Agustsson

et al. 2017

IEEE Trans. Inform. Theory

Self Cite

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“…In order to rigorously express the MI between the observations and the SOI, we adopt a Bayesian framework for the phase retrieval setup, similar to the approach in [28]. Computing the MI between the observations and the SOI is a difficult task.…”

Section: Introductionmentioning

confidence: 99%

Measurement Matrix Design for Phase Retrieval Based on Mutual Information

Shlezinger

Dabora

Eldar

2018

IEEE Trans. Signal Process.

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In phase retrieval problems, a signal of interest (SOI) is reconstructed based on the magnitude of a linear transformation of the SOI observed with additive noise. The linear transform is typically referred to as a measurement matrix. Many works on phase retrieval assume that the measurement matrix is a random Gaussian matrix, which, in the noiseless scenario with sufficiently many measurements, guarantees invertability of the transformation between the SOI and the observations, up to an inherent phase ambiguity. However, in many practical applications, the measurement matrix corresponds to an underlying physical setup, and is therefore deterministic, possibly with structural constraints. In this work we study the design of deterministic measurement matrices, based on maximizing the mutual information between the SOI and the observations. We characterize necessary conditions for the optimality of a measurement matrix, and analytically obtain the optimal matrix in the low signal-to-noise ratio regime. Practical methods for designing general measurement matrices and masked Fourier measurements are proposed. Simulation tests demonstrate the performance gain achieved by the suggested techniques compared to random Gaussian measurements for various phase recovery algorithms.

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