New Subset Selection Algorithms for Low Rank Approximation: Offline and Online
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Subset selection for the rank k approximation of an n× d matrix A offers improvements in the interpretability of matrices, as well as a variety of computational savings. This problem is well-understood when the error measure is the Frobenius norm, with various tight algorithms known even in challenging models such as the online model, where an algorithm must select the column subset irrevocably when the columns arrive one by one. In contrast, for other matrix losses, optimal trade-offs between the subset size and approximation quality have not been settled, even in the offline setting. We give a number of results towards closing these gaps. In the offline setting, we achieve nearly optimal bicriteria algorithms in two settings. First, we remove a √(k) factor from a result of [SWZ19] when the loss function is any entrywise loss with an approximate triangle inequality and at least linear growth. Our result is tight for the ℓ_1 loss. We give a similar improvement for entrywise ℓ_p losses for p>2, improving a previous distortion of k^1-1/p to k^1/2-1/p. Our results come from a technique which replaces the use of a well-conditioned basis with a slightly larger spanning set for which any vector can be expressed as a linear combination with small Euclidean norm. We show that this technique also gives the first oblivious ℓ_p subspace embeddings for 1<p<2 with Õ(d^1/p) distortion, which is nearly optimal and closes a long line of work. In the online setting, we give the first online subset selection algorithm for ℓ_p subspace approximation and entrywise ℓ_p low rank approximation by implementing sensitivity sampling online, which is challenging due to the sequential nature of sensitivity sampling. Our main technique is an online algorithm for detecting when an approximately optimal subspace changes substantially.
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