Memory capacity of two layer neural networks with smooth activations

Determining the memory capacity of two-layer neural networks with m hidden neurons and input dimension d (i.e., md+m total trainable parameters), which refers to the largest size of general data the network can memorize, is a fundamental machine-learning question. For non-polynomial real analytic activation functions, such as sigmoids and smoothed rectified linear units (smoothed ReLUs), we establish a lower bound of md/2 and optimality up to a factor of approximately 2. Analogous prior results were limited to Heaviside and ReLU activations, with results for smooth activations suffering from logarithmic factors and requiring random data. To analyze the memory capacity, we examine the rank of the network's Jacobian by computing the rank of matrices involving both Hadamard powers and the Khati-Rao product. Our computation extends classical linear algebraic facts about the rank of Hadamard powers. Overall, our approach differs from previous works on memory capacity and holds promise for extending to deeper models and other architectures.