Recent advances in deep learning highlight the need for personalized models that can learn from small samples, handle high-dimensional features, and remain interpretable. To address this, we propose the Sparse Deep Additive Model with Interactions (SDAMI), a framework that combines sparsity-driven feature selection with deep subnetworks for flexible function approximation. Central to SDAMI is the Effect Footprint principle, which posits that higher-order interactions leave detectable marginal traces on constituent variables, enabling their discovery without exhaustive search. SDAMI executes this principle through a three-stage strategy: (1) screening for footprint variables, (2) disentangling main effects from interactions via group lasso, and (3) modeling components with dedicated deep subnetworks. Theoretical analysis confirms that footprints vanish only under measure-zero symmetry conditions that are rare in practice, ensuring consistent interaction recovery. Extensive simulations demonstrate that SDAMI successfully identifies pure interactions that heredity-based baselines fundamentally miss, recovering complex effect structures with near-zero false positive rates. Together, these results position SDAMI as a principled framework for interpretable high-dimensional regression.

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