1. What are the contributions mentioned in the paper "Sampling based dimension reduction for subspace approximation" ?
The authors consider the problem of subspace approximation, i. e., they want to find a kdimensional linear subspace that minimizes the sum of p-th powers of distances to given points a1, a2,..., am ∈ R n, for p ≥ 1.. The authors also consider the corresponding projective clustering problem where instead of one k-dimensional subspace they want to find s of them such that the p-th root of the sum of of the p-th powers of distances from each ai to its nearest subspace is minimized.. The p = 2 case for subspace approximation ( also known as low-rank matrix approximation ) is well studied because a k-dimensional subspace that minimizes the sum of squared distances is spanned by the top k right singular vectors of a matrix A ∈ Rm×n ( with rows a1, a2,..., am ), and can be computed in time O ( min { mn2, m2n } ) using Singular Value Decomposition ( SVD ).. The authors show that one can get
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