Towards general-purpose representation learning of polygonal geometries

Abstract

Neural network representation learning for spatial data (e.g., points, polylines, polygons, and networks) is a common need for geographic artificial intelligence (GeoAI) problems. In recent years, many advancements have been made in representation learning for points, polylines, and networks, whereas little progress has been made for polygons, especially complex polygonal geometries. In this work, we focus on developing a general-purpose polygon encoding model, which can encode a polygonal geometry (with or without holes, single or multipolygons) into an embedding space. The result embeddings can be leveraged directly (or finetuned) for downstream tasks such as shape classification, spatial relation prediction, building pattern classification, cartographic building generalization, and so on. To achieve model generalizability guarantees, we identify a few desirable properties that the encoder should satisfy: loop origin invariance, trivial vertex invariance, part permutation invariance, and topology awareness. We explore two different designs for the encoder: one derives all representations in the spatial domain and can naturally capture local structures of polygons; the other leverages spectral domain representations and can easily capture global structures of polygons. For the spatial domain approach we propose ResNet1D, a 1D CNN-based polygon encoder, which uses circular padding to achieve loop origin invariance on simple polygons. For the spectral domain approach we develop NUFTspec based on Non-Uniform Fourier Transformation (NUFT), which naturally satisfies all the desired properties. We conduct experiments on two different tasks: 1) polygon shape classification based on the commonly used MNIST dataset; 2) polygon-based spatial relation prediction based on two new datasets (DBSR-46K and DBSR-cplx46K) constructed from OpenStreetMap and DBpedia. Our results show that NUFTspec and ResNet1D outperform multiple existing baselines with significant margins. While ResNet1D suffers from model performance degradation after shape-invariance geometry modifications, NUFTspec is very robust to these modifications due to the nature of the NUFT representation. NUFTspec is able to jointly consider all parts of a multipolygon and their spatial relations during prediction while ResNet1D can recognize the shape details which are sometimes important for classification. This result points to a promising research direction of combining spatial and spectral representations.

Appendix A

1.1 A.1 The Type Statistic of Polygonal Geometries in DBSR-cplx46K

Table 10.

Table 10 The place type statistic of geographic entities in DBSR-46K and DBSR-cplx46K dataset

1.2 A.2 Model Hyperparameter Tuning

We use grid search for hyperparameter tuning. For all polygon encoders on both tasks, we tune the learning rate lr over $\{0.02, 0.01, 0.005, 0.002, 0.001\}$, the polygon embedding dimension over $d \in \{256, 512, 1024\}$. As for all DDSL and NUFTspec-based models, we tune the frequency number $N_{wx} = \{16, 20, 24, 28, 32, 36, 40, 44\}$ for the shape classification task while $N_{wx} = \{16, 32, 64\}$ for the spatial relation prediction task. As for NUFTspec[gmf]-based models, we tune the $w_{min}=\{0.2, 0.4, 0.5, 0.8, 1.0\}$ and we tune $w_{max}$ around $N_{wx}/2$. For all PCA models, we vary $K_{PCA}$ such that the top $K_{PCA}$ PCA components can account for different data variance $\sum \nolimits _{PCA} = \{80\%, 85\%, 90\%, 95\%\}$. As for ResNet1D, we tune the KDelta point encoder’s neighbor size $2t \in \{0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}$ and tune the number of $\text {ResNet1D}_{cp}$ - $\mathcal {K} \in \{ 1, 2, 3\}$. For DDSL+LeNet5, we tune the hidden dimension of LeNet5 over $\{128, 256, 512, 1024\}$. As to NUFTspec-based models, DDSL+MLP, and DDSL+PCA+MLP, we tune the number of hidden layers h and the number of hidden dimension o in $MLP_{F}(\cdot )$ over $h = \{1, 2, 3\}$, $o = \{512, 1024\}$. We also try different NUFT spectral feature normalization method $\Psi (\cdot )$ such as no normalization, L2 normalization, and batch normalization. We find out no normalization usually leads to the best performance on all three datasets.

The best hyperparameter combinations for all models on MNIST-cplx70k are shown in Table 11. As for DBSR-46K and DBSR-cplx46K, each model’s best hyperparameter combinations are shown in Table 12.

Table 11 The best hyperparameter combinations for each model on MNIST-cplx70k dataset

Table 12 The best hyperparameter combinations for each model on DBSR-46K and DBSR-cplx46K dataset