The Generalized Lasso With Non-Linear Observations

Plan, Yaniv; Vershynin, Roman

doi:10.1109/tit.2016.2517008

Cited by 154 publications

(217 citation statements)

References 41 publications

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“…To our knowledge, this corollary is new. It recovers previous results that only apply to a single, fixed λ, as in [20,11]. It is known to be nearly minimax optimal for many constraint sets of interest and for stochastic noise term z, in which case z 2 would be replaced by its expected value [21].…”

Section: 7supporting

confidence: 78%

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A Simple Tool for Bounding the Deviation of Random Matrices on Geometric Sets

Liaw

Mehrabian

Plan

et al. 2017

Lecture Notes in Mathematics

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Abstract. Let A be an isotropic, sub-gaussian m × n matrix. We prove that the process Zx := Ax 2 − √ m x 2 has sub-gaussian increments, that is, Zx − Zy ψ 2 ≤ C x − y 2 for any x, y ∈ R n . Using this, we show that for any bounded set T ⊆ R n , the deviation of Ax 2 around its mean is uniformly bounded by the Gaussian complexity of T . We also prove a local version of this theorem, which allows for unbounded sets. These theorems have various applications, some of which are reviewed in this paper. In particular, we give a new result regarding model selection in the constrained linear model.

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Section: 7supporting

confidence: 78%

“…Predating, but especially following, the works in compressed sensing, there have also been several works which tackle the general case, giving results for arbitrary T [11,25,17,16,1,3,20,21]. The deviation inequalities of this paper allow for a general treatment as well.…”

Section: 6mentioning

confidence: 91%

A Simple Tool for Bounding the Deviation of Random Matrices on Geometric Sets

Liaw

Mehrabian

Plan

et al. 2017

Lecture Notes in Mathematics

Self Cite

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“…Perhaps surprisingly, Plan and Versyynin [PV16] demonstrated that, even with quantized measurements, the G-Lasso achieves good recovery performance when the measurement vectors are Gaussian 4 . An appealing feature of their theoretical result is that, similar to (2), their error bounds are simple to state and clearly isolate the effect of the specific quantization scheme, on one hand, and of the problem geometry, on the other hand.…”

Section: This Gives Rise To the Following Natural Questionmentioning

confidence: 99%

The Generalized Lasso for Sub-Gaussian Measurements With Dithered Quantization

Thrampoulidis

Rawat

2020

IEEE Trans. Inform. Theory

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In the problem of structured signal recovery from high-dimensional linear observations, it is commonly assumed that full-precision measurements are available. Under this assumption, the recovery performance of the popular Generalized Lasso (G-Lasso) is by now well-established. In this paper, we extend these types of results to the practically relevant settings with quantized measurements. We study two extremes of the quantization schemes, namely, uniform and one-bit quantization; the former imposes no limit on the number of quantization bits, while the second only allows for one bit. In the presence of a uniform dithering signal and when measurement vectors are sub-gaussian, we show that the same algorithm (i.e., the G-Lasso) has favorable recovery guarantees for both uniform and one-bit quantization schemes. Our theoretical results, shed light on the appropriate choice of the range of values of the dithering signal and accurately capture the error dependence on the problem parameters. For example, our error analysis shows that the G-Lasso with one-bit uniformly dithered measurements leads to only a logarithmic rate loss compared to the fullprecision measurements.

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“…Our works naturally relates to the literature of one-bit compressed sensing (CS) [5]. The vast majority of performance guarantees for one-bit CS are order-wise in nature, e.g., [18], [22], [21], [23]. To the best of our knowledge, the only existing sharp results are presented in [29] for Gaussian measurement vectors.…”

Section: Prior Workmentioning

confidence: 99%

Sharp Guarantees for Solving Random Equations with One-Bit Information

Taheri

Pedarsani

Thrampoulidis

2019

2019 57th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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We study the performance of a wide class of convex optimization-based estimators for recovering a signal from corrupted one-bit measurements in high-dimensions. Our general result predicts sharply the performance of such estimators in the linear asymptotic regime when the measurement vectors have entries IID Gaussian. This includes, as a special case, the previously studied least-squares estimator and various novel results for other popular estimators such as least-absolute deviations, hinge-loss and logistic-loss. Importantly, we exploit the fact that our analysis holds for generic convex loss functions to prove a bound on the best achievable performance across the entire class of estimators. Numerical simulations corroborate our theoretical findings and suggest they are accurate even for relatively small problem dimensions.

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The Generalized Lasso With Non-Linear Observations

Cited by 154 publications

References 41 publications

A Simple Tool for Bounding the Deviation of Random Matrices on Geometric Sets

A Simple Tool for Bounding the Deviation of Random Matrices on Geometric Sets

The Generalized Lasso for Sub-Gaussian Measurements With Dithered Quantization

Sharp Guarantees for Solving Random Equations with One-Bit Information

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