Dimensionality Reduction — Notes 3

where the inf is taken over all admissible sequences. We also let dX(T ) denote the diameter of T with respect to norm ‖·‖X . For the remainder of this section we make the definitions πrx = argminy∈Tr‖y − x‖X and ∆rx = πrx− πr−1x. Throughout this section we let ‖ · ‖ denote the `2→2 operator norm in the case of matrix arguments, and the `2 norm in the case of vector arguments. Krahmer, Mendelson, and Rauhut showed the following theorem [KMR14].