Least-Mean-Squares Coresets for Infinite Streams

Consider a stream of <inline-formula><tex-math notation="LaTeX">$d$</tex-math></inline-formula>-dimensional rows (points in <inline-formula><tex-math notation="LaTeX">$\mathbb {R}^{d}$</tex-math></inline-formula>) arriving sequentially. An <inline-formula><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula>-coreset is a positively weighted subset that approximates their sum of squared distances to any linear subspace of <inline-formula><tex-math notation="LaTeX">$\mathbb{R}^{d}$</tex-math></inline-formula>, up to a <inline-formula><tex-math notation="LaTeX">$1 \pm \epsilon$</tex-math></inline-formula> factor. Unlike other data summarizations, such a coreset: (1) can be used to minimize faster any optimization function that uses this sum, such as regularized or constrained regression, (2) preserves input sparsity; (3) easily interpretable; (4) avoids numerical errors; (5) applies to problems with constraints on the input, such as subspaces that are spanned by few input points. Our main result is the first algorithm that returns such an <inline-formula><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula>-coreset using finite and constant memory during the streaming, i.e., independent of <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula>, the number of rows seen so far. The coreset consists of <inline-formula><tex-math notation="LaTeX">$O(d \log ^{2} d / \epsilon ^{2})$</tex-math></inline-formula> weighted rows, which is nearly optimal according to existing lower bounds of <inline-formula><tex-math notation="LaTeX">$\Omega (d / \epsilon ^{2})$</tex-math></inline-formula>. We support our findings with experiments on the Wikipedia dataset benchmarked against state-of-the-art algorithms.