Fast Neural Tangent Kernel Alignment, Norm and Effective Rank via Trace Estimation
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908651177902080 |
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| author | Hazelden, James |
| author_facet | Hazelden, James |
| contents | The Neural Tangent Kernel (NTK) characterizes how a model's state evolves over Gradient Descent. Computing the full NTK matrix is often infeasible, especially for recurrent architectures. Here, we introduce a matrix-free perspective, using trace estimation to rapidly analyze the empirical, finite-width NTK. This enables fast computation of the NTK's trace, Frobenius norm, effective rank, and alignment. We provide numerical recipes based on the Hutch++ trace estimator with provably fast convergence guarantees. In addition, we show that, due to the structure of the NTK, one can compute the trace using only forward- or reverse-mode automatic differentiation, not requiring both modes. We show these so-called one-sided estimators can outperform Hutch++ in the low-sample regime, especially when the gap between the model state and parameter count is large. In total, our results demonstrate that matrix-free randomized approaches can yield speedups of many orders of magnitude, leading to faster analysis and applications of the NTK. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10796 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fast Neural Tangent Kernel Alignment, Norm and Effective Rank via Trace Estimation Hazelden, James Machine Learning Artificial Intelligence Numerical Analysis G.1.3 The Neural Tangent Kernel (NTK) characterizes how a model's state evolves over Gradient Descent. Computing the full NTK matrix is often infeasible, especially for recurrent architectures. Here, we introduce a matrix-free perspective, using trace estimation to rapidly analyze the empirical, finite-width NTK. This enables fast computation of the NTK's trace, Frobenius norm, effective rank, and alignment. We provide numerical recipes based on the Hutch++ trace estimator with provably fast convergence guarantees. In addition, we show that, due to the structure of the NTK, one can compute the trace using only forward- or reverse-mode automatic differentiation, not requiring both modes. We show these so-called one-sided estimators can outperform Hutch++ in the low-sample regime, especially when the gap between the model state and parameter count is large. In total, our results demonstrate that matrix-free randomized approaches can yield speedups of many orders of magnitude, leading to faster analysis and applications of the NTK. |
| title | Fast Neural Tangent Kernel Alignment, Norm and Effective Rank via Trace Estimation |
| topic | Machine Learning Artificial Intelligence Numerical Analysis G.1.3 |
| url | https://arxiv.org/abs/2511.10796 |