捷径特征作为神经切线核的顶级本征函数：线性神经网络案例及更多

深度诱导的NTK：桥接过参数化神经网络与深度神经核

Depth-induced NTK: Bridging Over-parameterized Neural Networks and Deep Neural Kernels

本文提出基于捷径架构的深度诱导NTK核，突破传统无限宽假设，在深度趋于无穷时收敛于高斯过程，并通过理论与实验证明其能稳定核动力学、缓解退化，深化了对深度学习和缩放定律的理解。

Spectral Analysis of the Neural Tangent Kernel for Deep Residual Networks

Deep residual network architectures have been shown to achieve superior accuracy over classical feed-forward networks, yet their success is still not fully understood. Focusing on massively over-parameterized, fully connected residual networks with ReLU activation through their respective neural tangent kernels (ResNTK), we provide here a spectral analysis of these kernels. Specifically, we show that, much like NTK for fully connected networks (FC-NTK), for input distributed uniformly on the hypersphere $\mathbb{S}^{d-1}$, the eigenfunctions of ResNTK are the spherical harmonics and the eigenv…

Neural Spectrum Alignment: Empirical Study

Expressiveness and generalization of deep models was recently addressed via the connection between neural networks (NNs) and kernel learning, where first-order dynamics of NN during a gradient-descent (GD) optimization were related to gradient similarity kernel, also known as Neural Tangent Kernel (NTK). In the majority of works this kernel is considered to be time-invariant, with its properties being defined entirely by NN architecture and independent of the learning task at hand. In contrast, in this paper we empirically explore these properties along the optimization and show that in practi…

Implicit Bias of MSE Gradient Optimization in Underparameterized Neural Networks

We study the dynamics of a neural network in function space when optimizing the mean squared error via gradient flow. We show that in the underparameterized regime the network learns eigenfunctions of an integral operator $T_{K^\infty}$ determined by the Neural Tangent Kernel (NTK) at rates corresponding to their eigenvalues. For example, for uniformly distributed data on the sphere $S^{d - 1}$ and rotation invariant weight distributions, the eigenfunctions of $T_{K^\infty}$ are the spherical harmonics. Our results can be understood as describing a spectral bias in the underparameterized regim…

On the Similarity between the Laplace and Neural Tangent Kernels

Recent theoretical work has shown that massively overparameterized neural networks are equivalent to kernel regressors that use Neural Tangent Kernels(NTK). Experiments show that these kernel methods perform similarly to real neural networks. Here we show that NTK for fully connected networks is closely related to the standard Laplace kernel. We show theoretically that for normalized data on the hypersphere both kernels have the same eigenfunctions and their eigenvalues decay polynomially at the same rate, implying that their Reproducing Kernel Hilbert Spaces (RKHS) include the same sets of fu…

具有随机隐藏权重的简单 ReLU 网络的神经正切核与 Fisher 信息矩阵

Neural Tangent Kernels and Fisher Information Matrices for Simple ReLU Networks with Random Hidden Weights

本文研究随机隐藏权重两层 ReLU 网络的 Fisher 信息矩阵与 NTK，揭示二者的线性变换关系，给出 NTK 谱分解、主要特征函数形式及网络函数的近似公式。