# Soft-Argmax for the Projective Plane via the Veronese Embedding

arXiv: https://arxiv.org/abs/2609.00521 · HTML: https://mlsanity.com/paper/2609.00521/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 20 references that are in the map.
- First posted on arXiv 1 Sep 2026.

## Builds on (most cited first, 9 of 9)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [An Analysis of SVD for Deep Rotation Estimation](https://mlsanity.com/paper/2006.14616/) (2020)
- [Numerical Coordinate Regression with Convolutional Neural Networks](https://mlsanity.com/paper/1801.07372/) (2018)
- [Building Deep Networks on Grassmann Manifolds](https://mlsanity.com/paper/1611.05742/) (2016)
- [Human Pose Regression by Combining Indirect Part Detection and Contextual Information](https://mlsanity.com/paper/1710.02322/) (2017)
- [Deep Hough Transform for Semantic Line Detection](https://mlsanity.com/paper/2003.04676/) (2020)
- [CyCNN: A Rotation Invariant CNN using Polar Mapping and Cylindrical Convolution Layers](https://mlsanity.com/paper/2007.10588/) (2020)
- [Dictionary Learning and Sparse Coding on Grassmann Manifolds: An Extrinsic Solution](https://mlsanity.com/paper/1310.4891/) (2013)
- [Expanding the Family of Grassmannian Kernels: An Embedding Perspective](https://mlsanity.com/paper/1407.1123/) (2014)

## Abstract (arXiv)

From horizon detection to fibre structures in X-ray imaging, many vision tasks recover lines via peak detection in Hough space H=S1×ℝ, the domain of orientation-offset pairs (θ,ρ). Differentiable pipelines extract coordinates via soft-argmax, a probability-weighted average that is only meaningful in a globally linear space. However, (θ,ρ) and (θ+π,-ρ) describe the same undirected line, so H double-covers the space of undirected lines H/mathbbZ2: a M"obius strip, obtained by identifying each pair under mathbbZ2 action. Soft-argmax operates on the cover H, but since H/mathbbZ2 admits no linear structure, it tears geometrically adjacent lines apart. Thus we need a mathbbZ2-invariant embedding of lines into a linear space, on which soft-argmax is well-defined. We achieve this by parametrising lines via unit-norm homogeneous vectors ℓ=(1+ρ2)-1/2(cosθ,sinθ,-ρ)top∈ℝ3 and applying the Veronese map v2(ℓ)=ℓℓtop that satisfies v2(ℓ)=v2(-ℓ). This descends continuously to an embedding of the quotient H/mathbbZ2 into the linear space Sym2(ℝ3), where the antipodal ambiguity vanishes. Line extraction becomes a barycentre in Sym2(ℝ3), projected back via its leading eigenvector. We validate our Veronese soft-argmax in a Hough transform-based network across all resolvable lines, confirming uniform and seam-free recovery. We further derive that the L2-loss on isometrically weighted Veronese embeddings equals the squared chordal distance between lines in projective space, enabling a geometrically precise training objective.