Given an n ×d dimensional dataset A, a projection query specifies a subset C [d] of columns which yields a new n ×|C| array. We study the space complexity of computing data analysis functions over such subspaces, including heavy hitters and norms, when the subspaces are revealed only after observing the data. We show that this important class of problems is typically hard: for many problems, we show 2Ω(d) lower bounds. However, we present upper bounds which demonstrate space dependency better than 2d. That is, for c,c' in(0,1) and a parameter N=2d an Nc-approximation can be obtained in space min(Nc',n), showing that it is possible to improve on the naïve approach of keeping information for all 2d subsets of d columns. Our results are based on careful constructions of instances using coding theory and novel combinatorial reductions that exhibit such space-approximation tradeoffs.
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