Restricted Eigenvalue Conditions on Subgaussian Random Matrices

By Shuheng Zhou. 2009

86 citations recorded by the source corpus.

It is natural to ask: what kinds of matrices satisfy the Restr icted Eigenvalue (RE) condition? In this paper, we associate the RE condition (Bickel-Ritov-Tsybakov 09) with the complexity of a subset of the sphere in R p , where p is the dimensionality of the data, and show that a class of random matrices with independent rows, but not necessarily independent columns, satisfy the RE condition, when the sample size is above a certain lower bound. Here we explicitly introduce an additional covariance structure to the class of random matrices that we have known by now that satisfy the Restricted Isometry Property

Subfields: Applied Mathematics, Computational Theory and Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b570697525b947451d8ba0933c9cbde5f13286b7

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In Applied Mathematics

In Computational Theory and Mathematics

In Statistics and Probability

Citations in the corpus

Confidence intervals for causal effects with invalid instruments by using two‐stage hard thresholding with voting (2016, unknown): “6 in Zhou (2009), if n ≥ 1/θ2×c′M3 1 max { 12(2 + γ0) (2)M1s log (5ep/4s) , 9 log p } , then with probability at least 1 − 2 exp ( −cθ2n/M3 1 ) , for all δ such that there exists |J0| ≤ 4s and ∥δJc 0∥1 ≤ γ0∥δJ0∥1, we have ∥Zδ∥2/( √ n∥Σ 1 2 δ∥2) ≥ 1 − θ.”
Non-negative least squares for high-dimensional linear models: consistency and sparse recovery without regularization (2012, background): “ R(α,S). The compatibility condition is slightly weaker than the correspondingrestricted eigenvalue condition [3] which applies to random sub-Gaussian matrices for s suﬃciently small in a broad sense [55]. 8 Theorem4. Assume that y= Xβ∗ +ε, where β∗ 0 has support S, |S| = s, εhas i.i.d. sub-Gaussia”
Restricted Eigenvalue Properties for Correlated Gaussian Designs (2010, methodology): “Finally, in work that followed up on the initial posting of this work (Raskutti et al., 2009), a paper by Zhou (2009) provides an extension of Theorem 1 to the case of correlated random matrices with sub-Gaussian entries.”
A unified framework for high-dimensional analysis of $M$-estimators with decomposable regularizers (2009, methodology): “Related analysis by Zhou [83] extends these types of guarantees to the case of sub-Gaussian designs.”