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Sparse shrunk additive models and sparse random feature models have beendeveloped separately as methods to learn low-order functions, where there arefew interactions between variables, but neither offers computationalefficiency. On the other hand, $\ell_2$-based shrunk additive models areefficient but do not offer feature selection as the resulting coefficientvectors are dense. Inspired by the success of the iterative magnitude pruningtechnique in finding lottery tickets of neural networks, we propose a newmethod -- Sparser Random Feature Models via IMP (ShRIMP) -- to efficiently fithigh-dimensional data with inherent low-dimensional structure in the form ofsparse variable dependencies. Our method can be viewed as a combined process toconstruct and find sparse lottery tickets for two-layer dense networks. Weexplain the observed benefit of SHRIMP through a refined analysis on thegeneralization error for thresholded Basis Pursuit and resulting bounds oneigenvalues.  From function approximation experiments on both synthetic data and real-worldbenchmark datasets, we show that SHRIMP obtains better than or competitive testaccuracy compared to state-of-art sparse feature and additive methods such asSRFE-S, SSAM, and SALSA. Meanwhile, SHRIMP performs feature selection with lowcomputational complexity and is robust to the pruning rate, indicating arobustness in the structure of the obtained subnetworks. We gain insight intothe lottery ticket hypothesis through SHRIMP by noting a correspondence betweenour model and weight/neuron subnetworks.

SHRIMP: Sparser Random Feature Models via Iterative Magnitude Pruning