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 $\{-1,1\}^d$, which are all independently and uniformly random, except for a planted pair of vectors with inner product at least $\rho \cdot d$ for some constant $\rho > 0$. The task is to find the planted pair. The most straightforward algorithm leads to a runtime of $\Omega(n^2)$. Algorithms based on techniques like Locality-Sensitive Hashing achieve runtimes of $n^{2 - O(\rho)}$; as $\rho$ gets small, these approach quadratic. Building on prior work, we give a new algorithm for this problem which runs in time $O(n^{1.582} + nd),$ regardless of how small $\rho$ 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 $O(n^{1.582} + nd),$ improving previous results.