Euclidean Sections of with Sublinear Randomness and Error-Correction over the Reals

It is well-known that R has subspaces of dimension proportional to N on which the `1 and `2 norms are uniformly equivalent, but it is unknown how to construct them explicitly. We show that, for any δ > 0, such a subspace can be generated using only N random bits. This improves over previous constructions of Artstein-Avidan and Milman, and of Lovett and Sodin, which require O(N log N), and O(N) random bits, respectively. Such subspaces are known to also yield error-correcting codes over the reals and compressed sensing matrices. Our subspaces are defined by the kernel of a relatively sparse matrix (with at most N non-zero entries per row), and thus enable compressed sensing in near-linear O(N) time. As in the work of Guruswami, Lee, and Razborov, our construction is the continuous analog of a Tanner code, and makes use of expander graphs to impose a collection of local linear constraints on vectors in the subspace. Our analysis is able to achieve uniform equivalence of the `1 and `2 norms (independent of the dimension). It has parallels to iterative decoding of Tanner codes, and leads to an analogous near-linear time algorithm for error-correction over reals.