We present the first sublinear memory sketch that can be queried to find the nearest neighbors in a dataset. Our online sketching algorithm compresses an N element dataset to a sketch of size O(N^b log^3 N) in O(N^(b+1) log^3 N) time, where b < 1. This sketch can correctly report the nearest neighbors of any query that satisfies a stability condition parameterized by b. We achieve sublinear memory performance on stable queries by combining recent advances in locality sensitive hash (LSH)-based estimators, online kernel density estimation, and compressed sensing. Our theoretical results shed new light on the memory-accuracy tradeoff for nearest neighbor search, and our sketch, which consists entirely of short integer arrays, has a variety of attractive features in practice. We provide rigorous theoretical guarantees and present a thorough evaluation of the memory-recall tradeoff of our method on a friend recommendation task in social media networks, including the Google plus graph. We find that RACE provides orders of magnitude better compression than the random projection based alternative while retaining the ability to report the nearest neighbors of practical queries. We expect that our theory will lead to new insight on geometric sketching problems while our methods will enable new possibilities in high-speed data mining and IoT settings.