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Linear sketches are powerful algorithmic tools that turn an n-dimensionalinput into a concise lower-dimensional representation via a lineartransformation. Such sketches have seen a wide range of applications includingnorm estimation over data streams, compressed sensing, and distributedcomputing. In almost any realistic setting, however, a linear sketch faces thepossibility that its inputs are correlated with previous evaluations of thesketch. Known techniques no longer guarantee the correctness of the output inthe presence of such correlations. We therefore ask: Are linear sketchesinherently non-robust to adaptively chosen inputs? We give a strong affirmativeanswer to this question. Specifically, we show that no linear sketchapproximates the Euclidean norm of its input to within an arbitrarymultiplicative approximation factor on a polynomial number of adaptively choseninputs. 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 onan algorithm with running time polynomial in d that adaptively finds adistribution over inputs on which the sketch is incorrect with constantprobability. Our result implies several corollaries for related problemsincluding lp-norm estimation and compressed sensing. Notably, we resolve anopen problem in compressed sensing regarding the feasibility of l2/l2-recoveryguarantees in the presence of computationally bounded adversaries.

How Robust are Linear Sketches to Adaptive Inputs?