Tight Sensitivity Bounds For Smaller Coresets
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An ε-coreset for Least-Mean-Squares (LMS) of a matrix A∈R^n× d is a small weighted subset of its rows that approximates the sum of squared distances from its rows to every affine k-dimensional subspace of R^d, up to a factor of 1±ε. Such coresets are useful for hyper-parameter tuning and solving many least-mean-squares problems such as low-rank approximation (k-SVD), k-PCA, Lassso/Ridge/Linear regression and many more. Coresets are also useful for handling streaming, dynamic and distributed big data in parallel. With high probability, non-uniform sampling based on upper bounds on what is known as importance or sensitivity of each row in A yields a coreset. The size of the (sampled) coreset is then near-linear in the total sum of these sensitivity bounds. We provide algorithms that compute provably tight bounds for the sensitivity of each input row. It is based on two ingredients: (i) iterative algorithm that computes the exact sensitivity of each point up to arbitrary small precision for (non-affine) k-subspaces, and (ii) a general reduction of independent interest from computing sensitivity for the family of affine k-subspaces in R^d to (non-affine) (k+1)- subspaces in R^d+1. Experimental results on real-world datasets, including the English Wikipedia documents-term matrix, show that our bounds provide significantly smaller and data-dependent coresets also in practice. Full open source is also provided.
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