Linear sketches turn an n-dimensional input into a concise lower-dimensional representation via a linear transformation. In almost any realistic setting a linear sketch faces the possibility that its inputs are correlated with previous evaluations of the sketch. Known techniques no longer guarantee the correctness of the output in the presence of such correlations. We therefore ask: are linear sketches inherently non-robust to adaptively chosen inputs? We give a strong affirmative answer to this question. Specifically, we show that no linear sketch approximates the Euclidean norm of its input to within an arbitrary multiplicative approximation factor on a polynomial number of adaptively chosen inputs. The result remains true even if the dimension of the sketch is d = n - o(n) and the sketch is given unbounded computation time. Our result is based on an algorithm with running time polynomial in d that adaptively finds a distribution over inputs on which the sketch is incorrect with constant probability. Our result implies several corollaries for related problems including lp-norm estimation and l2/l2 compressed sensing in the presence of adaptively chosen inputs.

Joint work with Moritz Hardt.