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- Emmanuel J. Candès, Yaniv Plan
- Proceedings of the IEEE
- 2010

On the heels of compressed sensing, a remarkable new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest form, the problem is to recover a matrix from a small sample… (More)

- Emmanuel J. Candès, Yaniv Plan
- IEEE Transactions on Information Theory
- 2011

This paper introduces a simple and very general theory of compressive sensing. In this theory, the sensing mechanism simply selects sensing vectors independently at random from a probability distribution <i>F</i>; it includes all standard models-e.g., Gaussian, frequency measurements-discussed in the literature, but also provides a framework for new… (More)

- Yaniv Plan, Roman Vershynin
- IEEE Transactions on Information Theory
- 2013

This paper develops theoretical results regarding noisy 1-bit compressed sensing and sparse binomial regression. We demonstrate that a single convex program gives an accurate estimate of the signal, or coefficient vector, for both of these models. We show that an -sparse signal in can be accurately estimated from m = O(s log(n/s)) single-bit measurements… (More)

- Mark A. Davenport, Yaniv Plan, Ewout van den Berg, Mary Wootters
- ArXiv
- 2012

In this paper we develop a theory of matrix completion for the extreme case of noisy 1-bit observations. Instead of observing a subset of the real-valued entries of a matrix M , we obtain a small number of binary (1-bit) measurements generated according to a probability distribution determined by the realvalued entries of M . The central question we ask is… (More)

- Yaniv Plan, Roman Vershynin
- ArXiv
- 2011

We give the first computationally tractable and almost optimal solution to the problem of one-bit compressed sensing, showing how to accurately recover an s-sparse vector x ∈ R from the signs of O(s log(n/s)) random linear measurements of x. The recovery is achieved by a simple linear program. This result extends to approximately sparse vectors x. Our… (More)

- Emmanuel J. Candès, Yaniv Plan
- ArXiv
- 2009

This paper presents several novel theoretical results regarding the recovery of a low-rank matrix from just a few measurements consisting of linear combinations of the matrix entries. We show that properly constrained nuclear-normminimization stably recovers a low-rank matrix from a constant number of noisy measurements per degree of freedom; this seems to… (More)

- Emmanuel J. Candès, Yaniv Plan
- IEEE Transactions on Information Theory
- 2011

This paper presents several novel theoretical results regarding the recovery of a low-rank matrix from just a few measurements consisting of linear combinations of the matrix entries. We show that properly constrained nuclear-norm minimization stably recovers a low-rank matrix from a constant number of noisy measurements per degree of freedom; this seems to… (More)

Consider measuring a vector x ∈ R through the inner product with several measurement vectors, a1, a2, . . . , am. It is common in both signal processing and statistics to assume the linear response model yi = 〈ai, x〉+ εi, where εi is a noise term. However, in practice the precise relationship between the signal x and the observations yi may not follow the… (More)

Testing for the significance of a subset of regression coefficients in a linear model, a staple of statistical analysis, goes back at least to the work of Fisher who introduced the analysis of variance (ANOVA). We study this problem under the assumption that the coefficient vector is sparse, a common situation in modern high-dimensional settings. Suppose we… (More)

- Yaniv Plan, Roman Vershynin
- IEEE Transactions on Information Theory
- 2016

We study the problem of signal estimation from non-linear observations when the signal belongs to a low-dimensional set buried in a high-dimensional space. A rough heuristic often used in practice postulates that the non-linear observations may be treated as noisy linear observations, and thus, the signal may be estimated using the generalized Lasso. This… (More)