Grassmannian Geometry and Global Convergence of Variable Projection for Neural Networks
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917234876612608 |
|---|---|
| author | Dus, Mathias |
| author_facet | Dus, Mathias |
| contents | Training deep neural networks and Physics-Informed Neural Networks (PINNs) often leads to ill-conditioned and stiff optimization problems. A key structural feature of these models is that they are linear in the output-layer parameters and nonlinear in the hiddenlayer parameters, yielding a separable nonlinear least-squares formulation. In this work, we study the classical variable projection (VarPro) method for such problems in the context of deep neural networks. We provide a geometric formulation on the Grassmannian and analyze the structure of critical points and convergence properties of the reduced problem. When the feature map is parametrized by a neural network, we show that these properties persist except in rank-deficient regimes, which we address via a regularized Grassmannian framework. Numerical experiments for regression and PINNs, including an efficient solver for the heat equation, illustrate the practical effectiveness of the approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22897 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Grassmannian Geometry and Global Convergence of Variable Projection for Neural Networks Dus, Mathias Optimization and Control Training deep neural networks and Physics-Informed Neural Networks (PINNs) often leads to ill-conditioned and stiff optimization problems. A key structural feature of these models is that they are linear in the output-layer parameters and nonlinear in the hiddenlayer parameters, yielding a separable nonlinear least-squares formulation. In this work, we study the classical variable projection (VarPro) method for such problems in the context of deep neural networks. We provide a geometric formulation on the Grassmannian and analyze the structure of critical points and convergence properties of the reduced problem. When the feature map is parametrized by a neural network, we show that these properties persist except in rank-deficient regimes, which we address via a regularized Grassmannian framework. Numerical experiments for regression and PINNs, including an efficient solver for the heat equation, illustrate the practical effectiveness of the approach. |
| title | Grassmannian Geometry and Global Convergence of Variable Projection for Neural Networks |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2601.22897 |