SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels

By Matthias Fey, Jan Eric Lenssen, Frank Weichert, Heinrich Müller. 2017

511 citations recorded by the source corpus.

We present Spline-based Convolutional Neural Networks (SplineCNNs), a variant of deep neural networks for irregular structured and geometric input, e.g., graphs or meshes. Our main contribution is a novel convolution operator based on B-splines, that makes the computation time independent from the kernel size due to the local support property of the B-spline basis functions. As a result, we obtain a generalization of the traditional CNN convolution operator by using continuous kernel functions parametrized by a fixed number of trainable weights. In contrast to related approaches that filter in the spectral domain, the proposed method aggregates features purely in the spatial domain. In addition, SplineCNN allows entire end-to-end training of deep architectures, using only the geometric structure as input, instead of handcrafted feature descriptors. For validation, we apply our method on tasks from the fields of image graph classification, shape correspondence and graph node classification, and show that it outperforms or pars state-of-the-art approaches while being significantly faster and having favorable properties like domain-independence. Our source code is available on GitHub1.

Subfields: Artificial Intelligence, Computer Vision and Pattern Recognition, Graph Learning

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Citations in the corpus

Fast Graph Representation Learning with PyTorch Geometric (2019, methodology): “…GCN operator from Schlichtkrull et al. (2018), PointNet++ (Qi et al., 2017), PointCNN (Li et al., 2018), and the continuous kernel-based methods MPNN (Gilmer et al., 2017), MoNet (Monti et al., 2017), SplineCNN (Fey et al., 2018) and the edge convolution operator (EdgeCNN) from Wang et al. (2018b).”
Hierarchical Graph Representation Learning with Differentiable Pooling (2018, background): “Lastly, there are some recent works that learn hierarchical graph representations by combining GNNs with deterministic graph clustering algorithms [8, 13, 35], following a two-stage approach.”
Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks (2018, methodology): “The code was built upon the work of (Fey et al. 2018) and is provided at https://github. com/chrsmrrs/k-gnn .”
Just Jump: Dynamic Neighborhood Aggregation in Graph Neural Networks (2019, background): “Many different graph neural network variants have been proposed and significantly advanced the state-of-the-art in this field (Defferrard et al., 2016; Kipf & Welling, 2017; Monti et al., 2017; Gilmer et al., 2017; Hamilton et al., 2017; Veličković et al., 2018; Fey et al., 2018).”
Towards Sparse Hierarchical Graph Classifiers (2018, background): “…broadly fall into two categories: 1) aggregating node representations in a global pooling step after each [8] or after the ﬁnal message passing step [19, 5, 11], and 2) aggregating node representations into clusters which coarsen the graph in a hierarchical manner [4, 22, 6, 20, 24, 10, 21, 28, 1].”
Dynamic Graph CNN for Learning on Point Clouds (2018, background): “the shape into a domain with shift-invariant structure such as the sphere [Sinha et al. 2016], torus [Maron et al. 2017], plane [Ezuz et al. 2017], sparse network lattice [Su et al. 2018], or spline [Fey et al. 2018]. Finally, we should mention geometric generative models, which attempt to generaliz”
A Comprehensive Survey on Graph Neural Networks (2019, methodology): “Under the MoNet framework, several existing approaches for manifolds such as Geodesic CNN (GCNN) [90], Anisotropic CNN (ACNN) [91], Spline CNN [92], and for graphs such as GCN [22], DCNN [25] can be generalized as special instances of MoNet by constructing nonparametric weight functions.”
Distilling Knowledge From Graph Convolutional Networks (2020, background): “In recent years, graph convolutional network [9, 6, 21, 27, 10, 42] has been proved to be a powerful model for non-grid data, which is typically represented as a set of nodes with features along with a graph that represents the relationship among nodes.”
Geometric Graph Convolutional Neural Networks (2019, methodology): “Its performance is slightly better than SplineCNN, which reports state-of-the-art results on this task.”
Graph convolutional networks: a comprehensive review (2019, methodology): “SplineCNN [58] follows the same framework [i.”
Relational Representation Learning for Dynamic (Knowledge) Graphs: A Survey (2019, unknown): “Several approaches perform a deterministic hierarchical graph clustering step and combine the node representations in each cluster to learn hierarchical representations [59, 75, 204].”
FSPool: Learning Set Representations with Featurewise Sort Pooling (2019, methodology): “We used the torch-geometric library (Fey et al., 2018) to implement this model.”
Deep Closest Point: Learning Representations for Point Cloud Registration (2019, background): “Learning on graphs and point sets: A broad class of deep architectures for geometric data termed geometric deep learning [7] includes recent methods learning on graphs [51, 60, 12] and point clouds [33, 34, 50, 57].”
Graph Neural Networks With Convolutional ARMA Filters (2019, methodology): “Another approach followed in most GNNs consists of pre-computing coarsened versions of the graph using hierarchical clustering (Bruna et al., 2013; Defferrard et al., 2016; Monti et al., 2017; Fey et al., 2018).”
PRNet: Self-Supervised Learning for Partial-to-Partial Registration (2019, methodology): “Other works, including SPLATNet [30], SplineCNN [31], KPConv [32], and GWCNN [33], transform 3D shapes to regular grids for feature learning.”
Lagrangian Fluid Simulation with Continuous Convolutions (2020, methodology): “Training with SplineCNN Convs takes 3 to 4 days on a single GPU.”
Kervolutional Neural Networks (2019, methodology): “To process graph inputs, Spline-CNN [14] extends convolution by using continuous B-spline basis, which is parametrized by a constant number of trainable control values.”