Papers › Dimensionality Reduction for Sum-of-Distances Metric
Dimensionality Reduction for Sum-of-Distances Metric
Zhili Feng, Praneeth Kacham, David P. Woodruff
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We give a dimensionality reduction procedure to approximate the sum of distances of a given set of n points in Rᵈ to any "shape" that lies in a k-dimensional subspace. Here, by "shape" we mean any set of points in Rᵈ. Our algorithm takes an input in the form of an n ×d matrix A, where each row of A denotes a data point, and outputs a subspace P of dimension O(k³/ϵ⁶) such that the projections of each of the n points onto the subspace P and the distances of each of the points to the subspace P are sufficient to obtain an ϵ-approximation to the sum of distances to any arbitrary shape that lies in a k-dimensional subspace of Rᵈ. These include important problems such as k-median, k-subspace approximation, and (j,l) subspace clustering with j ·l ≤k. Dimensionality reduction reduces the data storage requirement to (n+d)k³/ϵ⁶ from nnz$(A). Here nnz(A)$ could potentially be as large as nd. Our algorithm runs in time nnz$(A)/\epsilon^2 + (n+d)poly(k/\epsilon), up to logarithmic factors. For dense matrices, where nnz(A) \approx nd$, we give a faster algorithm, that runs in time $nd + (n+d)poly(k/\epsilon)$ up to logarithmic factors. Our dimensionality reduction algorithm can also be used to obtain poly$(k/\epsilon)$ size coresets for k-median and (k,1)-subspace approximation problems in polynomial time.
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