Improved Matrix Uncertainty Selector

By Mathieu Rosenbaum, Alexandre B. Tsybakov. 2011

71 citations recorded by the source corpus.

We consider the regression model with observation error in the design: y = Xµ ⁄ + »; Z = X + ¥: Here the random vector y 2R n and the random n£p matrix Z are observed, the n £ p matrix X is unknown, ¥ is an n £ p random noise matrix, » 2 R n is a random noise vector, and µ ⁄ is a vector of unknown parameters to be estimated. We consider the setting where the dimension p can be much larger than the sample size n and µ ⁄ is sparse. Because of the presence of the noise matrix ¥, the commonly used Lasso and Dantzig selector are unstable. An alternative procedure called the Matrix Uncertainty (MU) selector has been proposed in Rosenbaum and Tsybakov (2010) in order to account for the noise. The properties of the MU selector have been studied in Rosenbaum and Tsybakov (2010) for sparse µ ⁄ under the assumption that the noise matrix ¥ is deterministic and its values are small. In this paper, we propose a modiflcation of the MU selector when ¥ is a random matrix with zero-mean entries having the variances that can be estimated. This is, for example, the case in the model where the entries of X are missing at random. We show both theoretically and numerically that, under these conditions, the new estimator called the Compensated MU selector achieves better accuracy of estimation than the original MU selector.

Subfields: Computational Mechanics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/657756cbefa50a5b5e909d3bf04d9ee0848ee7cd

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In Computational Mechanics

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Citations in the corpus

Measurement error in Lasso: impact and likelihood bias correction (2012, background): “An improved MU selector is presented in Rosenbaum and Tsybakov (2013).”
High-dimensional regression with noisy and missing data: Provable guarantees with non-convexity (2011, result): “simulations and further discussions of the consequences of Corollary1. (b)We may also compare the results in (a) with bounds from past work on high-dimensional sparse regression with noisy covariates [18]. In this work, Rosenbaum and Tsybakov derive similar concentration bounds on sub-Gaussian matri”
Robust Sparse Regression under Adversarial Corruption (2013, background): “Note that the same is true for standard Dantzig selector and Lasso, and their modified versions in (Rosenbaum & Tsybakov, 2011; Loh & Wainwright, 2012).”
Robust High Dimensional Sparse Regression and Matching Pursuit (2013, background): “Another line of work has developed approaches to handle stochastic noise or small bounded noise in the covariates [17], [26], [27], [23], [9].”