Spatial Adapter: Structured Spatial Decomposition and Closed-Form Covariance for Frozen Predictors
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| Format: | Preprint |
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2026
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| _version_ | 1866914556771565568 |
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| author | Wang, Wen-Ting Wu, Wei-Ying Huang, Hao-Yun Wang, Xuan-Chun |
| author_facet | Wang, Wen-Ting Wu, Wei-Ying Huang, Hao-Yun Wang, Xuan-Chun |
| contents | We present the Spatial Adapter, a parameter-efficient post-hoc layer that equips any frozen first-stage predictor with a structured spatial representation of its residual field and an induced closed-form spatial covariance. The adapter operates as a cascade second stage on residuals, jointly learning a spatially regularized orthonormal basis and per-sample scores via a tractable mini-batch ADMM procedure, without modifying any first-stage parameter. Because the first-stage parameters are frozen, the adapter does not retrain the backbone; its role is to supply a compressed distributional summary of the residual field. Smoothness, sparsity, and orthogonality together turn a generic low-rank factorization into an identifiable spatial representation whose induced residual covariance admits a closed-form low-rank-plus-noise estimator; the effective rank is determined data-adaptively by spectral thresholding, while the nominal rank K is an optimization-side upper bound only. This covariance enables kriging-style spatial prediction at unobserved locations, with plug-in uncertainty quantification as a secondary downstream use. Across synthetic data, Weather2K for spatial-holdout prediction, and GWHD patch grids as a basis-transferability diagnostic, the adapter recovers residual spatial structure when paired with frozen first stages from linear models to deep spatiotemporal and vision backbones; the added representation uses fewer than K(N+T) parameters alongside a compact residual-trend network. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_11394 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spatial Adapter: Structured Spatial Decomposition and Closed-Form Covariance for Frozen Predictors Wang, Wen-Ting Wu, Wei-Ying Huang, Hao-Yun Wang, Xuan-Chun Machine Learning Artificial Intelligence Applications Methodology 62M30, 62H25, 68T07, 90C26 We present the Spatial Adapter, a parameter-efficient post-hoc layer that equips any frozen first-stage predictor with a structured spatial representation of its residual field and an induced closed-form spatial covariance. The adapter operates as a cascade second stage on residuals, jointly learning a spatially regularized orthonormal basis and per-sample scores via a tractable mini-batch ADMM procedure, without modifying any first-stage parameter. Because the first-stage parameters are frozen, the adapter does not retrain the backbone; its role is to supply a compressed distributional summary of the residual field. Smoothness, sparsity, and orthogonality together turn a generic low-rank factorization into an identifiable spatial representation whose induced residual covariance admits a closed-form low-rank-plus-noise estimator; the effective rank is determined data-adaptively by spectral thresholding, while the nominal rank K is an optimization-side upper bound only. This covariance enables kriging-style spatial prediction at unobserved locations, with plug-in uncertainty quantification as a secondary downstream use. Across synthetic data, Weather2K for spatial-holdout prediction, and GWHD patch grids as a basis-transferability diagnostic, the adapter recovers residual spatial structure when paired with frozen first stages from linear models to deep spatiotemporal and vision backbones; the added representation uses fewer than K(N+T) parameters alongside a compact residual-trend network. |
| title | Spatial Adapter: Structured Spatial Decomposition and Closed-Form Covariance for Frozen Predictors |
| topic | Machine Learning Artificial Intelligence Applications Methodology 62M30, 62H25, 68T07, 90C26 |
| url | https://arxiv.org/abs/2605.11394 |