Composable Core-sets for Determinant Maximization Problems via Spectral Spanners

We study a generalization of classical combinatorial graph spanners to the spectral setting. Given a set of vectors V subset of R-d, we say a set U subset of V is an alpha-spectral k-spanner, for k <= d, if for all v is an element of V there is a probability distribution (mu)v supported on U such that vv(T) <= k alpha center dot E-u similar to mu v uu(T), where for two matrices A, B is an element of R-dxd we write A <=(k) B iff the sum of the bottom d - k + 1 eigenvalues of B - A is nonnegative. In particular, A <= d B iff A <= B. We show that any set V has an (O) over tilde (k)-spectral spanner of size (O) over tilde (k) and this bound is almost optimal in the worst case. We use spectral spanners to study composable core-sets for spectral problems. We show that for many objective functions one can use a spectral spanner, independent of the underlying function, as a core-set and obtain almost optimal composable core-sets. For example, for the k-determinant maximization problem, we obtain an (O) over tilde (k)(k)-composable core-set, and we show that this is almost optimal in the worst case. Our algorithm is a spectral analogue of the classical greedy algorithm for finding (combinatorial) spanners in graphs. We expect that our spanners find many other applications in distributed or parallel models of computation.