# Fermat Active Laplace Learning for Semi-Supervised Hyperspectral Image Classification

arXiv: https://arxiv.org/abs/2608.02483 · HTML: https://mlsanity.com/paper/2608.02483/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 46 references that are in the map.
- First posted on arXiv 3 Aug 2026.

## Builds on (most cited first, 17 of 17)

- [Self-supervised Learning in Remote Sensing: A Review](https://mlsanity.com/paper/2206.13188/) (2022)
- [Poisson Learning: Graph Based Semi-Supervised Learning At Very Low Label Rates](https://mlsanity.com/paper/2006.11184/) (2020)
- [A survey of active learning algorithms for supervised remote sensing image classification](https://mlsanity.com/paper/2104.07784/) (2021)
- [A Survey: Deep Learning for Hyperspectral Image Classification with Few Labeled Samples](https://mlsanity.com/paper/2112.01800/) (2021)
- [A Comprehensive Survey for Hyperspectral Image Classification: The Evolution from Conventional to Transformers and Mamba Models](https://mlsanity.com/paper/2404.14955/) (2024)
- [Path-Based Spectral Clustering: Guarantees, Robustness to Outliers, and Fast Algorithms](https://mlsanity.com/paper/1712.06206/) (2017)
- [Balancing Geometry and Density: Path Distances on High-Dimensional Data](https://mlsanity.com/paper/2012.09385/) (2020)
- [Approximate Leave-One-Out for Fast Parameter Tuning in High Dimensions](https://mlsanity.com/paper/1807.02694/) (2018)
- [Active Transfer Learning Network: A Unified Deep Joint Spectral-Spatial Feature Learning Model For Hyperspectral Image Classification](https://mlsanity.com/paper/1904.02454/) (2019)
- [Fermat Distances: Metric Approximation, Spectral Convergence, and Clustering Algorithms](https://mlsanity.com/paper/2307.05750/) (2023)
- [Power Weighted Shortest Paths for Clustering Euclidean Data](https://mlsanity.com/paper/1905.13345/) (2019)
- [Learning by Active Nonlinear Diffusion](https://mlsanity.com/paper/1905.12989/) (2019)
- [Intrinsic persistent homology via density-based metric learning](https://mlsanity.com/paper/2012.07621/) (2020)
- [Spatially regularized active diffusion learning for high-dimensional images](https://mlsanity.com/paper/1911.02155/) (2019)
- [Active Diffusion and VCA-Assisted Image Segmentation of Hyperspectral Images](https://mlsanity.com/paper/2204.06298/) (2022)
- [Poisson Reweighted Laplacian Uncertainty Sampling for Graph-based Active Learning](https://mlsanity.com/paper/2210.15786/) (2022)
- [High-dimensional Semi-supervised Classification via the Fermat Distance](https://mlsanity.com/paper/2604.23573/) (2026)

## Abstract (arXiv)

Two active learning algorithms for hyperspectral image (HSI) classification are proposed that combine density-aware Fermat distances with Poisson-reweighted harmonic label propagation. Our methods actively query points using an uncertainty-based acquisition function, extending Poisson ReWeighted Laplace Learning (PWLL). Our first algorithm, Fermat Active Laplace Learning (FALL), builds an affinity matrix using Fermat distances between all data points. Then, PWLL is run with a diagonal perturbation using the minimum-norm acquisition function. In contrast, Approximate FALL (A-FALL) computes Fermat distances between each data point and landmark pixels selected via farthest-point sampling and constructs the affinity matrix using landmark multidimensional scaling. After several query rounds, A-FALL selects the Fermat exponent p using a leave-one-out cross-validation variant. FALL and A-FALL leverage Fermat distances and subsequent harmonic label propagation to provide a density-aware estimation of the data manifold, improving labeling accuracy. Experiments on Salinas A and Pavia show the effectiveness of FALL and the scalability of A-FALL to large HSI scenes.