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Variable selection, also known as feature selection in machine learning,plays an important role in modeling high dimensional data and is key todata-driven scientific discoveries. We consider here the problem of detectinginfluential variables under the general index model, in which the response isdependent of predictors through an unknown function of one or more linearcombinations of them. Instead of building a predictive model of the responsegiven combinations of predictors, we model the conditional distribution ofpredictors given the response. This inverse modeling perspective motivates usto propose a stepwise procedure based on likelihood-ratio tests, which iseffective and computationally efficient in identifying important variableswithout specifying a parametric relationship between predictors and theresponse. For example, the proposed procedure is able to detect variables withpairwise, three-way or even higher-order interactions among $p$ predictors witha computational time of $O(p)$ instead of $O(p^k)$ (with $k$ being the highestorder of interactions). Its excellent empirical performance in comparison withexisting methods is demonstrated through simulation studies as well as realdata examples. Consistency of the variable selection procedure when both thenumber of predictors and the sample size go to infinity is established.

Variable selection for general index models via sliced inverse regression