## The xyz algorithm for fast interaction search in high-dimensional data

Gian-Andrea Thanei, Nicolai Meinshausen, Rajen D. Shah; 19(37):1−42, 2018.

### Abstract

When performing regression on a data set with $p$ variables, it is often of interest to go beyond using main linear effects and include interactions as products between individual variables. For small-scale problems, these interactions can be computed explicitly but this leads to a computational complexity of at least $\mathcal{O}(p^2)$ if done naively. This cost can be prohibitive if $p$ is very large.
We introduce a new randomised algorithm that is able to discover interactions with high probability and under mild conditions has a runtime that is subquadratic in $p$. We show that strong interactions can be discovered in almost linear time, whilst finding weaker interactions requires $\mathcal{O}(p^\alpha)$ operations for $1<\alpha<2$ depending on their strength. The underlying idea is to transform interaction search into a closest pair problem which can be solved efficiently in subquadratic time. The algorithm is called $xyz$ and is implemented in the language R. We demonstrate its efficiency for application to genome-wide association studies, where more than $10^{11}$ interactions can be screened in under $280$ seconds with a single-core $1.2$ GHz CPU.

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