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Semi-supervised learning aims to learn prediction models from both labeled and unlabeled samples. There has been extensive research in this area. Among existing work, generative mixture models with Expectation-Maximization (EM) is a popular method due to clear statistical properties. However, existing literature on EM-based semi-supervised learning largely focuses on unstructured prediction, assuming that samples are independent and identically distributed. Studies on EM-based semi-supervised approach in structured prediction is limited. This paper aims to fill the gap through a comparative study between unstructured and structured methods in EM-based semi-supervised learning. Specifically, we compare their theoretical properties and find that both methods can be considered as a generalization of self-training with soft class assignment of unlabeled samples, but the structured method additionally considers structural constraint in soft class assignment. We conducted a case study on real-world flood mapping datasets to compare the two methods. Results show that structured EM is more robust to class confusion caused by noise and obstacles in features in the context of the flood mapping application.

Spatial classification with limited feature observations has been a challenging problem in machine learning. The problem exists in applications where only a subset of sensors are deployed at certain regions or partial responses are collected in field surveys. Existing research mostly focuses on addressing incomplete or missing data, e.g., data cleaning and imputation, classification models that allow for missing feature values, or modeling missing features as hidden variables and applying the EM algorithm. These methods, however, assume that incomplete feature observations only happen on a small subset of samples, and thus cannot solve problems where the vast majority of samples have missing feature observations. To address this issue, we propose a new approach that incorporates physics-aware structural constraints into the model representation. Our approach assumes that a spatial contextual feature is observed for all sample locations and establishes spatial structural constraint from the spatial contextual feature map. We design efficient algorithms for model parameter learning and class inference. Evaluations on real-world hydrological applications show that our approach significantly outperforms several baseline methods in classification accuracy, and the proposed solution is computationally efficient on a large data volume.

Spatial classification with limited observations is important in geographical applications where only a subset of sensors are deployed at certain spots or partial responses are collected in field surveys. For example, in observation-based flood inundation mapping, there is a need to map the full flood extent on geographic terrains based on earth imagery that partially covers a region. Existing research mostly focuses on addressing incomplete or missing data through data cleaning and imputation or modeling missing values as hidden variables in the EM algorithm. These methods, however, assume that missing feature observations are rare and thus are ineffective in problems whereby the vast majority of feature observations are missing. To address this issue, we recently proposed a new approach that incorporates physics-aware structural constraint into the model representation. We design efficient learning and inference algorithms. This paper extends our recent approach by allowing feature values of samples in each class to follow a multi-modal distribution. Evaluations on real-world flood mapping applications show that our approach significantly outperforms baseline methods in classification accuracy, and the multi-modal extension is more robust than our early single-modal version. Computational experiments show that the proposed solution is computationally efficient on large datasets.