Deep Learning with Learnable Product-Structured Activations

Modern neural architectures are fundamentally constrained by their reliance on fixed activation functions, limiting their ability to adapt representations to task-specific structure and efficiently capture high-order interactions. We introduce deep low-rank separated neural networks (LRNNs), a novel architecture generalizing MLPs that achieves enhanced expressivity by learning adaptive, factorized activation functions. LRNNs generalize the core principles underpinning continuous low-rank function decomposition to the setting of deep learning, constructing complex, high-dimensional neuron activations through a multiplicative composition of simpler, learnable univariate transformations. This product structure inherently captures multiplicative interactions and allows each LRNN neuron to learn highly flexible, data-dependent activation functions. We provide a detailed theoretical analysis that establishes the universal approximation property of LRNNs and reveals why they are capable of excellent empirical performance. Specifically, we show that LRNNs can mitigate the curse of dimensionality for functions with low-rank structure. Moreover, the learnable product-structured activations enable LRNNs to adaptively control their spectral bias, crucial for signal representation tasks. These theoretical insights are validated through extensive experiments where LRNNs achieve state-of-the-art performance across diverse domains including image and audio representation, numerical solution of PDEs, sparse-view CT reconstruction, and supervised learning tasks. Our results demonstrate that LRNNs provide a powerful and versatile building block with a distinct inductive bias for learning compact yet expressive representations.