that is quantitatively close to an isometry when restricted to certain subspaces. They showed that it is sufficient for either exact or optimally approximate recovery of sufficiently sparse solutions. Their proofs, which involved the theory of convex duality, were markedly simplified in collaboration with Romberg, to use only linear algebra and elementary ideas of harmonic analysis. These ideas and results were later improved by Candes. Candes and Tao also considered relaxations of the sparsity condition, such as power-law decay of coefficients. They complemented these results by drawing on a large corpus of past results in random matrix theory to show that, according to the Gaussian ensemble, a large number of matrices satisfy the restricted isometry property. In 2009, Candes and Benjamin Recht considered an analogous problem for recovering a matrix from knowledge of only a few of its entries and the information that the matrix is of low rank. They formulated the problem in terms of convex optimisation, studying minimisation of the nuclear norm. Candes and Tao, in 2010, developed further results and techniques for the same problem. Improved results were later found by Recht. Similar problems and results have also been considered by a number of other authors. In 2007, Candes and Tao introduced a novel statistical estimator for linear regression, which they called the "Dantzig selector." They proved a number of results on its success as an estimator and model selector, roughly in parallel to their earlier work on compressed sensing. A number of other authors have since studied the Dantzig selector, comparing it to similar objects such as the statistical lasso introduced in the 1990s. Trevor Hastie, Robert Tibshirani, and Jerome H. Friedman conclude that it is "somewhat unsatisfactory" in a number of cases. Nonetheless it remains of significant interest in the statistical literature. Random matrices In the 1950s, Eugene Wigner initiated the study of random matrices and their eigenvalues. Wigner studied the case of hermitian and symmetric matrices, proving a "semicircle law" for their eigenvalues. In 2010, Tao and Van Vu made a major contribution to the study of non-symmetric random matrices. They showed that if is large and the entries of a matrix are selected randomly according to any fixed probability distribution of average 0 and standard deviation 1, then the eigenvalues of will tend to be uniformly scattered across the disk of radius around the origin; this can be made precise using the language