Least-Mean-Squares Coresets for Infinite Streams

Consider a stream of
$d$
-dimensional rows (points in
$\mathbb {R}^{d}$
) arriving sequentially. An
$\epsilon$
-coreset is a positively weighted subset that approximates their sum of squared distances to any linear subspace of
$\mathbb {R}^{d}$
, up to a
$1 \pm \epsilon$
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
$\epsilon$
-coreset using finite and constant memory during the streaming, i.e., independent of
$n$
, the number of rows seen so far. The coreset consists of
$O(d \log ^{2}\;d / \epsilon ^{2})$
weighted rows, which is nearly optimal according to existing lower bounds of
$\Omega (d / \epsilon ^{2})$
. We support our findings with experiments on the Wikipedia dataset benchmarked against state-of-the-art algorithms.