Philippe Rigollet

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In high-dimensional linear regression, the goal pursued here is to estimate an unknown regression function using linear combinations of a suitable set of covariates. One of the key assumptions for the success of any statistical procedure in this setup is to assume that the linear combination is sparse in some sense, for example, that it involves only few(More)
We perform a finite sample analysis of the detection levels for sparse principal components of a high-dimensional covariance matrix. Our minimax optimal test is based on a sparse eigenvalue statistic. Alas, computing this test is known to be NP-complete in general, and we describe a computationally efficient alternative test using convex relaxations. Our(More)
In the context of sparse principal component detection, we bring evidence towards the existence of a statistical price to pay for computational efficiency. We measure the performance of a test by the smallest signal strength that it can detect and we propose a computationally efficient method based on semidefinite programming. We also prove that the(More)
We consider semi-supervised classification when part of the available data is unlabeled. These unlabeled data can be useful for the classification problem when we make an assumption relating the behavior of the regression function to that of the marginal distribution. Seeger (2000) proposed the well-known cluster assumption as a reasonable one. We propose a(More)
In a regression setup with deterministic design, we study the pure aggregation problem and introduce a natural extension from the Gaussian distribution to distributions in the exponential family. While this extension bears strong connections with generalized linear models, it does not require identifiability of the parameter or even that the model on the(More)