🤖 AI Summary
This work addresses the limitations of frozen backbone networks in residual field modeling—specifically, their inability to adequately capture spatial structure and quantify uncertainty—by introducing Spatial Adapter, a parameter-efficient post-processing layer. The method constructs an identifiable low-rank representation through a structured spatial decomposition that enforces smoothness, sparsity, and orthogonality, enabling closed-form derivation of spatial covariance. An effective rank is determined via a data-adaptive spectral thresholding scheme, facilitating kriging interpolation and uncertainty quantification. By jointly optimizing spatial orthogonal bases and sample-specific scores using mini-batch ADMM, and integrating low-rank-plus-noise covariance estimation with a compact trend network, the approach successfully recovers diverse residual spatial structures across various tasks—from linear to deep spatiotemporal or vision backbones—while maintaining a parameter budget below K(N+T).
📝 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.