Detection boundary in sparse regression

By Yu. I. Ingster, Alexandre B. Tsybakov, Nicolas Verzelen. 2010

131 citations recorded by the source corpus.

We study the problem of detection of a p-dimensional sparse vector of parameters in the linear regression model with Gaussian noise. We establish the detection boundary, i.e., the necessary and sufficient conditions for the possibility of successful detection as both the sample size n and the dimension p tend to the infinity. Testing procedures that achieve this boundary are also exhibited. Our results encompass the high-dimensional setting (p>> n). The main message is that, under some conditions, the detection boundary phenomenon that has been proved for the Gaussian sequence model, extends to high-dimensional linear regression. Finally, we establish the detection boundaries when the variance of the noise is unknown. Interestingly, the detection boundaries sometimes depend on the knowledge of the variance in a high-dimensional setting.

Subfields: Computational Mechanics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5005a726e3b6b22d5c266776f3e2b65a242cd851

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Citations in the corpus

Adaptive Global Testing for Functional Linear Models (2014, background): “Such a framework (also known as the detection boundary problem) has been studied in highdimensional and nonparametric testing problems such as Ingster (1982), Donoho and Jin (2004), Ingster et al. (2010), Ingster et al. (2012), Arias-Castro et al. (2011).”
Hypothesis Testing For Densities and High-Dimensional Multinomials: Sharp Local Minimax Rates (2017, background): “Finally, it would also be interesting to further explore the extent to which the local-minimax perspective can lead to a better understanding of composite-null inference problems [1, 2, 5, 8, 10, 23, 30].”
Finding and Leveraging Structure in Learning Problems (2012, background): “[99] and Arias-Castro [8] studied detection of sparse vectors from compressive measurements.”
Minimax risks for sparse regressions: Ultra-high-dimensional phenomenons (2010, unknown): “[34] have recently provided the asymptotic minimax separation distance with exact constant for known variance when the design satisfies very specific conditions.”
Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity (2015, methodology): “The technique has also been used more recently in [15, 19] for detection and confidence ball in sparse linear regression.”
Statistical significance in high-dimensional linear models (2012, unknown): “For the extreme case with G n = { 1 ,... Ingster ,p n } , we are in the setting of detection of the global hypotheses, see for example , Tsybakov and Verzelen (2010) for characterizing the detection boundary in case of independent covariables.”