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Given i.i.d. observations of a random vector X ∈ R, we study the problem of estimating both its covariance matrix Σ, and its inverse covariance or concentration matrix Θ = (Σ). When X is multivariate Gaussian, the non-zero structure of Θ is specified by the graph of an associated Gaussian Markov random field; and a popular estimator for such sparse Θ is the… (More)

- Garvesh Raskutti, Martin J. Wainwright, Bin Yu
- Journal of Machine Learning Research
- 2010

Methods based on l1-relaxation, such as basis pursuit and the Lasso, are very popular for sparse regression in high dimensions. The conditions for success of these methods are now well-understood: (1) exact recovery in the noiseless setting is possible if and only if the design matrix X satisfies the restricted nullspace property, and (2) the squared… (More)

Given i.i.d. observations of a random vector X ∈ R, we study the problem of estimating both its covariance matrix Σ∗, and its inverse covariance or concentration matrix Θ∗ = (Σ). We estimate Θ∗ by minimizing an l1-penalized log-determinant Bregman divergence; in the multivariate Gaussian case, this approach corresponds to l1-penalized maximum likelihood,… (More)

- Garvesh Raskutti, Martin J. Wainwright, Bin Yu
- Journal of Machine Learning Research
- 2012

Sparse additive models are families of d-variate functions that have the additive decomposition f∗ = ∑ j∈S f ∗ j , where S is a unknown subset of cardinality s d. We consider the case where each component function f∗ j lies in a reproducing kernel Hilbert space, and analyze a simple kernel-based convex program for estimating the unknown function f∗. Working… (More)

- Garvesh Raskutti, Martin J. Wainwright, Bin Yu
- IEEE Transactions on Information Theory
- 2011

Consider the high-dimensional linear regression model <i>y</i> = <i>X</i> β<sup>*</sup> + <i>w</i>, where <i>y</i> ∈ \BBR<i>n</i> is an observation vector, <i>X</i> ∈ \BBR<i>n</i> × <i>d</i> is a design matrix with <i>d</i> >; <i>n</i>, β<sup>*</sup> ∈ \BBR<i>d</i> is an unknown regression vector, and <i>w</i> ~… (More)

- Garvesh Raskutti, Martin J. Wainwright, Bin Yu
- IEEE Trans. Information Theory
- 2011

Consider the standard linear regression model Y = Xβ+w, where Y ∈ R is an observation vector, X ∈ R is a design matrix, β ∈ R is the unknown regression vector, and w ∼ N (0,σI) is additive Gaussian noise. This paper studies the minimax rates of convergence for estimation of β for #p-losses and in the #2-prediction loss, assuming that β belongs to an #q-ball… (More)

Consider the high-dimensional linear regression model y = Xβ∗ +w, where y ∈ R is an observation vector, X ∈ R is a design matrix with d > n, the quantity β∗ ∈ R is an unknown regression vector, and w ∼ N (0, σI) is additive Gaussian noise. This paper studies the minimax rates of convergence for estimating β∗ in either l2-loss and l2-prediction loss,… (More)

We consider the problem of estimating the graph structure associated with a Gaussian Markov random field (GMRF) from i.i.d. samples. We study the performance of study the performance of the l1-regularized maximum likelihood estimator in the high-dimensional setting, where the number of nodes in the graph p, the number of edges in the graph s and the maximum… (More)

Many algorithms for inferring causality rely heavily on the faithfulness assumption. The main justification for imposing this assumption is that the set of unfaithful distributions has Lebesgue measure zero, since it can be seen as a collection of hypersurfaces in a hypercube. However, due to sampling error the faithfulness condition alone is not sufficient… (More)

- Garvesh Raskutti, Martin J. Wainwright, Bin Yu
- ArXiv
- 2009

Consider the standard linear regression model Y = Xβ+w, where Y ∈ R is an observation vector, X ∈ R is a design matrix, β ∈ R is the unknown regression vector, and w ∼ N (0, σI) is additive Gaussian noise. This paper studies the minimax rates of convergence for estimation of β for lp-losses and in the l2-prediction loss, assuming that β belongs to an… (More)