Recent geometric deep learning works define convolution operations in local regions and have enjoyed remarkable success on non-Euclidean data, including graph and point clouds. However, the high-level geometric correlations between the input and its neighboring coordinates or features are not fully exploited, resulting in suboptimal segmentation performance. In this article, we propose a novel graph convolution architecture, which we term as Taylor Gaussian mixture model (GMM) network (TGNet), to efficiently learn expressive and compositional local geometric features from point clouds. The TGNet is composed of basic geometric units, TGConv, that conduct local convolution on irregular point sets and are parametrized by a family of filters. Specifically, these filters are defined as the products of the local point features and the neighboring geometric features extracted from local coordinates. These geometric features are expressed by Gaussian weighted Taylor kernels. Then, a parametric pooling layer aggregates TGConv features to generate new feature vectors for each point. TGNet employs TGConv on multiscale neighborhoods to extract coarse-to-fine semantic deep features while improving its scale invariance. Additionally, a conditional random field (CRF) is adopted within the output layer to further improve the segmentation results. Using three point cloud data sets, qualitative and quantitative experimental results demonstrate that the proposed method achieves 62.2% average accuracy on ScanNet, 57.8% and 68.17% mean intersection over union (mIoU) on Stanford Large-Scale 3D Indoor Spaces (S3DIS) and Paris-Lille-3D data sets, respectively.