Soft-Argmax for the Projective Plane via the Veronese Embedding

Soft-argmax is a widely used readout operator for differentiable coordinate regression, turning heatmaps into predictions by averaging candidate coordinates. While equivariant architectures account for symmetries in feature processing, the readout that turns evidence into a prediction, and the loss on that prediction, must respect the topology of the target domain as well. The parameters of undirected lines are a case in which such a symmetry breaks coordinate averaging. Each line has two sign-related representatives, so no coordinate domain admits an average that respects proximity between lines, and we prove that any readout averaging signed offsets can miss a target on which the evidence concentrates. We introduce Veronese soft-argmax, a parameter-free readout that averages sign-invariant quadratic features and recovers the line from their average. The same representation yields the training objective, whose value bounds the error of the recovered line. We prove that predictions approach the target as evidence concentrates and give sufficient conditions for a unique prediction, on real projective spaces of any dimension. Experiments on synthetic lines show that readout and objective each contribute, and our approach transfers to horizon estimation in real images. The construction applies wherever a heatmap ranges over candidates in a real projective space, and in particular to targets with antipodal symmetry.