Separable Neural Networks: Approximation Theory, NTK Regime, and Preconditioned Gradient Descent

Separable neural networks (SepNNs) are emerging neural architectures that significantly reduce computational costs by factorizing a multivariate function into linear combinations of univariate functions, benefiting downstream applications such as implicit neural representations (INRs) and physics-informed neural networks (PINNs). However, fundamental theoretical analysis for SepNN, including detailed representation capacity and spectral bias characterization & alleviation, remains unexplored. This work makes three key contributions to theoretically understanding and improving SepNN. First, using Weierstrass-based approximation and universal approximation theory, we prove that SepNN can approximate any multivariate function with arbitrary precision, confirming its representation completeness. Second, we derive the neural tangent kernel (NTK) regimes for SepNN, showing that the NTK of infinite-width SepNN converges to a deterministic (or random) kernel under infinite (or fixed) decomposition rank, with corresponding convergence and spectral bias characterization. Third, we propose an efficient separable preconditioned gradient descent (SepPGD) for optimizing SepNN, which alleviates the spectral bias of SepNN by provably adjusting its NTK spectrum. The SepPGD enjoys an efficient $\mathcal{O}(nD)$ complexity for $n^D$ training samples, which is much more efficient than previous neural network PGD methods. Extensive experiments for kernel ridge regression, image and surface representation using INRs, and numerical PDEs using PINNs validate the efficiency of SepNN and the effectiveness of SepPGD for alleviating spectral bias.