and more specific regions”. Later papers fail to improve upon this heuristic justification. Practically, the advantage of synthetic sampling versus random over-sampling diminishes as the dimension of the feature space increases. In general, for each minority sample, we require at least $d$ synthetic samples to fully describe its neighborhood. This is often infeasible due to the sample size of the majority class and to computational complexity. Hence, it is typical to fix the number of synthetic samples regardless of the dimension of the feature space [@chawla2002smote], which may fail to “create large and less specific decision regions” when the dimension is high.

Motivated by these issues, we propose to directly penalize the Surface-to-Volume Ratio (SVR) of the decision set. A primary issue with imbalanced data is estimating a decision set consisting of small neighborhoods around each minority class sample. By penalizing SVR we favor regularly shaped decision sets much less subject to such over-fitting. With this motivation, we propose a new class of SVR-Tree algorithms.

The rest of the paper is organized as follows. Section 2 describes our methodology. Section 3 provides theory on estimation and feature selection consistency. Section 4 presents numerical studies for real datasets. Section 5 contains 