We develop an algorithm for estimating the values of a vector x in R^n over a support S of size k from a randomized sparse binary linear sketch Ax of size O(k). Given Ax and S, we can recover x' with ||x' - x_S||_2 <= eps ||x - x_S||_2 with probability at least 1 - k^{-Ω(1)}. The recovery takes O(k) time. While interesting in its own right, this primitive also has a number of applications. For example, we can: 1. Improve the linear k-sparse recovery of heavy hitters in Zipfian distributions with O(k log n) space from a (1+eps) approximation to a (1 + o(1)) approximation, giving the first such approximation in O(k log n) space when k <= O(n^{1-eps}). 2. Recover block-sparse vectors with O(k) space and a (1+eps) approximation. Previous algorithms required either omega(k) space or omega(1) approximation.