Recovering PCA from Hybrid-(ℓ_1,ℓ_2) Sparse Sampling of Data Elements
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This paper addresses how well we can recover a data matrix when only given a few of its elements. We present a randomized algorithm that element-wise sparsifies the data, retaining only a few its elements. Our new algorithm independently samples the data using sampling probabilities that depend on both the squares (ℓ_2 sampling) and absolute values (ℓ_1 sampling) of the entries. We prove that the hybrid algorithm recovers a near-PCA reconstruction of the data from a sublinear sample-size: hybrid-(ℓ_1,ℓ_2) inherits the ℓ_2-ability to sample the important elements as well as the regularization properties of ℓ_1 sampling, and gives strictly better performance than either ℓ_1 or ℓ_2 on their own. We also give a one-pass version of our algorithm and show experiments to corroborate the theory.
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