RBF-GNN: Rational Basis Functions for Pseudo-Coordinate based Graph Convolutions

Paweł Batorski; Abtin Pourhadi; Paul Swoboda

cs.CVSep 29, 2026

RBF-GNN: Rational Basis Functions for Pseudo-Coordinate based Graph Convolutions

Authors: Paweł Batorski, Abtin Pourhadi, Paul Swoboda

Organizations: Heinrich Heine University Düsseldorf

Abstract

We propose RBF-GNN, a new pseudo-coordinate based graph neural network architecture that takes into account Euclidean, spherical or angular coordinates and uses them to induce a powerful spatial inductive bias. Similar in architecture to SplineCNN, we improve upon the latter by replacing the less efficient sparse-activation based B-splines whose number grows exponentially with dimension by rational Padé basis functions. For effective training we propose a spline-subspace initialization and a variance-preserving weight rescaling. Experimentally, we evaluate on a number of popular neural network architectures that use SplineCNNs. We replace only the SplineCNNs with RBF-GNN. We achieve improved results, including on semantic keypoint matching, shape matching, event based camera computer vision tasks. Code is available at https://github.com/pawelswoboda/RationalBasisCNN.

Figures & tables

Figure 2: Overview of RBF-GNN. Each graph edge $(j,i)$ carries a normalized pseudo-coordinate $\mathbf{u}_{ij}$ . The $K$ learnable safe rational basis functions are evaluated at $\mathbf{u}_{ij}$ and mix the weight matrices $\Theta_{p}$ into the edge-specific continuous kernel $\Theta(\mathbf{u}_{ij})$ . The resulting neighbor messages are aggregated to produce $\mathbf{x}^{\prime}_{i}$ ; the rational-basis coefficients and convolution weights are trained jointly.

Panel of Figure 3 — Figure 3: Construction of two dimensional pseudo-coordinates for $D{=}2$ . Left: The convolution at node $i$ represents each neighbour $j$ by its spatial offset $\mathbf{z}_{j}-\mathbf{z}_{i}$ . The dashed box marks the largest offset magnitude $\delta_{\max}$ in the graph. Right: The map in ( 1 ) rescales this box to the unit square. Node $i$ lands at the centre $\tfrac{1}{2}\mathbf{1}$ , node $j$ lands at $\mathbf{u}_{ij}$ in blue, and the largest offset reaches the boundary.

Panel of Figure 4 — Figure 4: From polynomials to safe Padé basis functions, Eqs. ( 3 )–( 5 ), with the same numerator $P$ and denominator $Q$ in all three panels. (a) Two polynomials with learnable coefficients; $Q$ has a root $t^{\ast}$ inside the domain. (b) The plain rational function $R=P/Q$ diverges at $t^{\ast}$ , and any root of $Q$ can move into the domain during training. (c) In the safe form the denominator $1+|Q|$ (gray) never drops below one (dashed), so $B$ stays bounded and well defined for every setting of the coefficients; where $Q$ vanishes, $B$ simply equals $P$ .

	DGMC ( Fey et al., 2020 )			NMT ( Pourhadi & Swoboda, 2026 )
Method	Params $\downarrow$	PascalVOC $\uparrow$	SPair-71k $\uparrow$	Params $\downarrow$	PascalVOC $\uparrow$	SPair-71k $\uparrow$
Spline baselines
SplineCNN ( $K{=}25$ )	9.50M	$73.00\pm 0.40$	$76.18\pm 0.64$	28.17M	$83.10\pm 0.36$	$82.10\pm 0.11$
SplineCNN ( $K{=}9$ )	3.74M	$72.78\pm 0.40$	$76.04\pm 0.43$	10.84M	$82.66\pm 0.30$	$81.54\pm 0.31$
SplineCNN ( $K{=}4$ )	1.93M	$71.48\pm 0.40$	$71.92\pm 0.38$	5.42M	$80.10\pm 0.46$	$77.79\pm 0.40$
MLP learned-basis control

Table 1: Semantic keypoint matching on PascalVOC and SPair-71k. Accuracy (%) follows the protocol of each pipeline and is the mean

\pm

std over 5 seeds, and best , second , and third mark the three highest means per column. Parameter counts exclude the VGG16 backbone and cover the whole DGMC model but only the NMT graph-convolution layers, which are larger because they use 648 channels instead of the 256 and 128 of DGMC. All other columns report the best test epoch.

Panel of Figure 5 — Figure 5: Tasks with pseudo-coordinate graphs used in our experiments. (a) Semantic keypoint matching on SPair-71k: keypoints of two images of the same category are graph nodes, and corresponding keypoints (same color, 8 of 20 shown) are matched across instances with different viewpoint and appearance. (b) Shape correspondence on FAUST: every vertex of a triangulated human scan is matched to its vertex on another person in a different pose; colors encode the ground-truth correspondence, and dashed arcs show example matches. (c) Event-based vision: object moving through a scene triggers a stream of events. The events of a short time window (gray plane) trace the contour of the car; each event is a graph node (orange) connected to its space-time neighbors, and the pseudo-coordinates are relative $(x,y,t)$ offsets.

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