Understanding the Mathematical Foundation of Neural Tangent Kernel

This technical analysis explores the Neural Tangent Kernel (NTK), a theoretical framework that explains how neural networks optimize during gradient descent training. The work examines why overparameterized networks can achieve good generalization even when they contain far more parameters than training examples. The NTK theory provides rigorous mathematical justification for why sufficiently wide neural networks converge to global minima during training. The exposition covers essential mathematical preliminaries including Jacobian matrices, differential equations, the central limit theorem, and Taylor expansions before diving into kernel theory itself. At its core, NTK functions as a similarity metric revealing how parameter updates on one data point influence predictions across other examples. For networks with infinite width, the kernel becomes deterministic and independent of initialization, ensuring consistent learning dynamics across different starting conditions. The study establishes a deep connection between neural networks and Gaussian processes, demonstrating that network outputs behave like Gaussian distributions when hidden layers are sufficiently wide. This theoretical bridge provides formal guarantees about neural network training behavior.