Spatial Adapter: Structured Spatial Decomposition and Closed-Form Covariance for Frozen Predictors

• URL: http://arxiv.org/abs/2605.11394v1
• Date: Tue, 12 May 2026 01:29:56 GMT
• Title: Spatial Adapter: Structured Spatial Decomposition and Closed-Form Covariance for Frozen Predictors
• Abstract summary: We present a parameter-efficient post-hoc layer that equips any frozen first-stage predictor with a structured spatial representation of its residual field.<n>The adapter operates as a cascade second stage on residuals, jointly learning a spatially regularized orthonormal basis and per-sample scores.<n>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.
• Score: 0.6643390346826856
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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