NOFE - Neural Operator Function Embedding
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arXiv
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| Main Authors: | , , , , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866913135756050432 |
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| author | Uebbing, Lars Joakimsen, Harald L. Chen, Siyan Leontidis, Georgios Wickstrøm, Kristoffer K. Kampffmeyer, Michael C. Lefèvre, Sébastien Salberg, Arnt-Børre Jenssen, Robert |
| author_facet | Uebbing, Lars Joakimsen, Harald L. Chen, Siyan Leontidis, Georgios Wickstrøm, Kristoffer K. Kampffmeyer, Michael C. Lefèvre, Sébastien Salberg, Arnt-Børre Jenssen, Robert |
| contents | Most dimensionality reduction methods treat data as discrete point clouds, ignoring the continuous domain structure inherent to many real-world processes. To bridge this gap, we introduce Neural Operator Function Embedding (NOFE), a domain-aware framework for continuous dimensionality reduction. NOFE learns function-to-function mappings via a Graph Kernel Operator, enabling mesh-free evaluation at arbitrary query locations independent of input discretization. We establish NOFE as approximation of sheaf-to-sheaf mappings, generalizing Sheaf Neural Networks to continuous domains. We evaluate NOFE across different datasets, comparing it against PCA, t-SNE, and UMAP. Our results demonstrate that NOFE significantly outperforms baselines in local structure preservation, achieving a local Stress of 0.111 compared to 0.398 for PCA, 0.773 for t-SNE, and 0.791 for UMAP for the ERA5 climate reanalysis dataset. NOFE also exhibits robust sampling independence, reducing the Patch Stitching Error by up to $20.0\times$ relative to UMAP (59.0 vs. 267.6 under regional normalization) and ensuring consistency across disjoint domain patches. While maintaining competitive global structure preservation (Stress-1: 0.379 vs. PCA's 0.268), NOFE resolves fine-grained structures and produces smooth, consistent embeddings that generalize across varying sample densities, addressing key limitations of discrete reduction methods. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_11970 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | NOFE - Neural Operator Function Embedding Uebbing, Lars Joakimsen, Harald L. Chen, Siyan Leontidis, Georgios Wickstrøm, Kristoffer K. Kampffmeyer, Michael C. Lefèvre, Sébastien Salberg, Arnt-Børre Jenssen, Robert Machine Learning Most dimensionality reduction methods treat data as discrete point clouds, ignoring the continuous domain structure inherent to many real-world processes. To bridge this gap, we introduce Neural Operator Function Embedding (NOFE), a domain-aware framework for continuous dimensionality reduction. NOFE learns function-to-function mappings via a Graph Kernel Operator, enabling mesh-free evaluation at arbitrary query locations independent of input discretization. We establish NOFE as approximation of sheaf-to-sheaf mappings, generalizing Sheaf Neural Networks to continuous domains. We evaluate NOFE across different datasets, comparing it against PCA, t-SNE, and UMAP. Our results demonstrate that NOFE significantly outperforms baselines in local structure preservation, achieving a local Stress of 0.111 compared to 0.398 for PCA, 0.773 for t-SNE, and 0.791 for UMAP for the ERA5 climate reanalysis dataset. NOFE also exhibits robust sampling independence, reducing the Patch Stitching Error by up to $20.0\times$ relative to UMAP (59.0 vs. 267.6 under regional normalization) and ensuring consistency across disjoint domain patches. While maintaining competitive global structure preservation (Stress-1: 0.379 vs. PCA's 0.268), NOFE resolves fine-grained structures and produces smooth, consistent embeddings that generalize across varying sample densities, addressing key limitations of discrete reduction methods. |
| title | NOFE - Neural Operator Function Embedding |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2605.11970 |