Expander codes over reals, Euclidean sections, and compressed sensing

Classical results from the 1970's state that w.h.p. a random subspace of TV-dimensional Euclidean space of proportional (linear in TV) dimension is “well-spread” in the sense that vectors in the subspace have their ¿2 mass smoothly spread over a linear number of coordinates. Such well-spread subspaces are intimately connected to low distortion embeddings, compressed sensing matrices, and error-correction over reals. We describe a construction inspired by expander/Tanner codes that can be used to produce well-spread subspaces of O(TV) dimension using sub-linear randomness (or in sub-exponential time). These results were presented in our paper [10]. We also discuss the connection of our subspaces to compressed sensing, and describe a near-linear time iterative recovery algorithm for compressible signals.