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Detecting interaction effects among predictors on the response variable is acrucial step in various applications. In this paper, we first propose a simplemethod for sure screening interactions (SSI). Although its computationcomplexity is $O(p^2n)$, SSI works well for problems of moderate dimensionality(e.g., $p=10^3\sim10^4$), without the heredity assumption. To ultra-highdimensional problems (e.g., $p = 10^6$), motivated by discretization associatedBoolean representation and operations and the contingency table for discretevariables, we propose a fast algorithm, named "BOLT-SSI". The statisticaltheory has been established for SSI and BOLT-SSI, guaranteeing their surescreening property. The performance of SSI and BOLT-SSI are evaluated bycomprehensive simulation and real case studies. Numerical results demonstratethat SSI and BOLT-SSI can often outperform their competitors in terms ofcomputational efficiency and statistical accuracy. The proposed method can beapplied for fully detecting interactions with more than 300,000 predictors.Based on this study, we believe that there is a great need to rethink therelationship between statistical accuracy and computational efficiency. We haveshown that the computational performance of a statistical method can often begreatly improved by exploring the advantages of computational architecture witha tolerable loss of statistical accuracy.

BOLT-SSI: A Statistical Approach to Screening Interaction Effects for Ultra-High Dimensional Data