Tight Bounds for the Subspace Sketch Problem with Applications
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In the subspace sketch problem one is given an n× d matrix A with O((nd)) bit entries, and would like to compress it in an arbitrary way to build a small space data structure Q_p, so that for any given x ∈R^d, with probability at least 2/3, one has Q_p(x)=(1±ϵ)Ax_p, where p≥ 0, and where the randomness is over the construction of Q_p. The central question is: How many bits are necessary to store Q_p? This problem has applications to the communication of approximating the number of non-zeros in a matrix product, the size of coresets in projective clustering, the memory of streaming algorithms for regression in the row-update model, and embedding subspaces of L_p in functional analysis. A major open question is the dependence on the approximation factor ϵ. We show if p≥ 0 is not a positive even integer and d=Ω((1/ϵ)), then Ω̃(ϵ^-2d) bits are necessary. On the other hand, if p is a positive even integer, then there is an upper bound of O(d^p(nd)) bits independent of ϵ. Our results are optimal up to logarithmic factors, and show in particular that one cannot compress A to O(d) "directions" v_1,...,v_O(d), such that for any x, Ax_1 can be well-approximated from 〈 v_1,x〉,...,〈 v_O(d),x〉. Our lower bound rules out arbitrary functions of these inner products (and in fact arbitrary data structures built from A), and thus rules out the possibility of a singular value decomposition for ℓ_1 in a very strong sense. Indeed, as ϵ→ 0, for p = 1 the space complexity becomes arbitrarily large, while for p = 2 it is at most O(d^2 (nd)). As corollaries of our main lower bound, we obtain new lower bounds for a wide range of applications, including the above, which in many cases are optimal.
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