Title: Confidence sets for low rank inference problems, & Quantum computation

Abstract: Recovery rates for algorithms that estimate high-dimensional parameters are well studied. Uncertainty quantification and the valid construction of confidence sets ("error bars") for such algorithms are much less well understood. For high-dimensional *sparse* parameters one can derive information-theoretic barriers for the existence of confidence sets, unless one imposes signal strength conditions (Nickl & van de Geer, 2013, AoS). Perhaps surprisingly we show here that, in contrast, for certain *low rank* recovery problems fully `honest' confidence sets exist for the entire matrix parameter. We will describe these favourable situations, and what they depend upon: naturally the geometry of the loss function plays a crucial role, as well as the probabilistic properties of the sensing matrices used. A particularly important application will be given to quantum tomography problems arising in quantum computing, where our non-asymptotic confidence sets allow for the construction of an adaptive sequential algorithm that recovers an arbitrary quantum state by a minimal number of measurements, in a fully data-driven way. Interestingly this result is valid for the most important experimental ensembles arising from Pauli matrices only if one enforces the `quantum shape constraint' on the unknown matrix. This is joint work with  A. Carpentier, D. Gross, J. Eisert [arXiv:1504.03234]]