Restricted Eigenvalue Conditions on Subgaussian Random Matrices - Mathematics > Statistics TheoryReport as inadecuate




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Abstract: It is natural to ask: what kinds of matrices satisfy the RestrictedEigenvalue RE condition? In this paper, we associate the RE conditionBickel-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 ofrandom 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 ofrandom matrices that we have known by now that satisfy the Restricted IsometryProperty as defined in Candes and Tao 05 and hence the RE condition, in orderto compose a broader class of random matrices for which the RE condition holds.In this case, tools from geometric functional analysis in characterizing theintrinsic low-dimensional structures associated with the RE condition has beencrucial in analyzing the sample complexity and understanding its statisticalimplications for high dimensional data.



Author: Shuheng Zhou

Source: https://arxiv.org/







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