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

arXiv: https://arxiv.org/abs/2605.11394 · HTML: https://mlsanity.com/paper/2605.11394/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 34 references that are in the map.
- First posted on arXiv 12 May 2026.

## Builds on (most cited first, 13 of 13)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Segment Anything](https://mlsanity.com/paper/2304.02643/) (2023)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [DoRA: Weight-Decomposed Low-Rank Adaptation](https://mlsanity.com/paper/2402.09353/) (2024)
- [Parameter-Efficient Fine-Tuning for Large Models: A Comprehensive Survey](https://mlsanity.com/paper/2403.14608/) (2024)
- [Bayesian Low-rank Adaptation for Large Language Models](https://mlsanity.com/paper/2308.13111/) (2023)
- [Global Wheat Head Detection (GWHD) dataset: a large and diverse dataset of high resolution RGB labelled images to develop and benchmark wheat head detection methods](https://mlsanity.com/paper/2005.02162/) (2020)
- [Statistical Deep Learning for Spatial and Spatio-Temporal Data](https://mlsanity.com/paper/2206.02218/) (2022)
- [Regularized Principal Component Analysis for Spatial Data](https://mlsanity.com/paper/1501.03221/) (2015)

## Abstract (arXiv)

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.