One-pass additive-error subset selection for ℓ_p subspace approximation
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We consider the problem of subset selection for ℓ_p subspace approximation, that is, to efficiently find a small subset of data points such that solving the problem optimally for this subset gives a good approximation to solving the problem optimally for the original input. Previously known subset selection algorithms based on volume sampling and adaptive sampling <cit.>, for the general case of p ∈ [1, ∞), require multiple passes over the data. In this paper, we give a one-pass subset selection with an additive approximation guarantee for ℓ_p subspace approximation, for any p ∈ [1, ∞). Earlier subset selection algorithms that give a one-pass multiplicative (1+ϵ) approximation work under the special cases. Cohen et al. <cit.> gives a one-pass subset section that offers multiplicative (1+ϵ) approximation guarantee for the special case of ℓ_2 subspace approximation. Mahabadi et al. <cit.> gives a one-pass noisy subset selection with (1+ϵ) approximation guarantee for ℓ_p subspace approximation when p ∈{1, 2}. Our subset selection algorithm gives a weaker, additive approximation guarantee, but it works for any p ∈ [1, ∞).
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