Composable core-sets for determinant maximization problems via spectral spanners

Abstract

We study a generalization of classical combinatorial graph spanners to the spectral setting. Given a set of vectors V ⊆ ℝd, we say a set U ⊆ V is an α-spectral kspanner, for k ≤ d, if for all v ∈ V there is a probability distribution µv supported on U such that vv ≼k α· Eu∼µvuu, where for two matrices A,B ∈ ℝd×d 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 Õ(k)-spectral spanner of size Õ(k) and this bound is almost optimal in the worst case. We use spectral spanners to study composable coresets 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 Õ(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.