In this talk I will consider a general noisy linear regression setting Y = + \epsilon, that simultaneously describes the usual "vector" linear regression setting, and the "matrix" linear regression setting. I will consider the problem of inference in this model, i.e. estimation of the underlying parameter \theta, and associated uncertainty quantification.

PART I : Setting, applications, and low dimensional solution I will first present applications for both setting (vector and matrix regression) and in particular provide a short introduction to quantum tomography. I will also present the classical solution for inference in this model when it is low dimensional.

PART II : High dimensional estimation and local uncertainty quantification I will then consider the problem of doing inference on the parameter \theta when it is high dimensional, i.e. when the dimension of the parameter \theta is higher than the number of samples. An usual solution to the misspecification of this problem is to restrict the set of admissible parameters \theta to nested sub classes of growing complexity indexed by k. I will present the solutions that are usually proposed and some of their limits, in particular in the domains of computational efficiency and local (coordinate by coordinate) uncertainty quantification. I will present an approach that aims at tackling these limitations.

PART III : Global and adaptive uncertainty quantification in high dimension I will finally consider the problem of global and adaptive uncertainty quantification for \theta. The estimation error on \theta depends on the complexity k of the class to which \theta belongs, and so should ideally the precision of the uncertainty quantification methods. The problem here is that k is unknown. I will present results that curiously highlight the difference between the vector and the matrix case.

This talk is based on the following joint works with Jens Eisert, David Gross, Arlene K.H. Kim and Richard Nickl: Alexandra Carpentier, Jens Eisert, David Gross and Richard Nickl. Uncertainty Quantification for Matrix Compressed Sensing and Quantum Tomography Problems. arXiv preprint arXiv:1504.03234, 2015. A. Carpentier and A.K.H. Kim. An iterative hard thresholding estimator for low rank matrix recovery with explicit limiting distribution. arXiv preprint arXiv:1502.04654, 2015. A. Carpentier and R. Nickl, 2015.