An Illuminating Algorithm for the Light Bulb Problem

The Light Bulb Problem is one of the most basic problems in data analysis. One is given as input n vectors in -1,1d, which are all independently and uniformly random, except for a planted pair of vectors with inner product at least ρ · d for some constant ρ > 0. The task is to find the planted pair. The most straightforward algorithm leads to a runtime of Omega(n2). Algorithms based on techniques like Locality-Sensitive Hashing achieve runtimes of n2 - O(ρ); as ρ gets small, these approach quadratic. Building on prior work, we give a new algorithm for this problem which runs in time O(n1.582 + nd), regardless of how small ρ is. This matches the best known runtime due to Karppa et al. Our algorithm combines techniques from previous work on the Light Bulb Problem with the so-called `polynomial method in algorithm design,' and has a simpler analysis than previous work. Our algorithm is also easily derandomized, leading to a deterministic algorithm for the Light Bulb Problem with the same runtime of O(n1.582 + nd), improving previous results.