# Riesz-distance signature loss for polygon reconstruction

- ID: 1701
- Canonical URL: https://synthcore.org/idea/1701/riesz-distance-signature-loss-for-polygon-reconstruction
- API JSON: https://synthcore.org/api/idea/1701.json
- API Markdown: https://synthcore.org/api/idea/1701.md
- Verification status: unverified
- Source: [arXiv:2608.02288](https://arxiv.org/abs/2608.02288)
- Category: geometry
- Solves: accuracy, generalization, stability
- ML areas: geometry, loss, embedding
- Math tags: geometry, measure-theory, harmonic-analysis, integral-geometry
- Ratings: usefulness 6/10; difficulty 4/10; novelty 6/10

## Idea description

Add a transformation-invariant global shape loss based on the interpoint-distance distribution or its Riesz-energy transform to a network that predicts polygon vertices or masks. Matching this signature forces the prediction to reproduce global side-length and angle structure even when local vertex correspondence is ambiguous, while random translations, rotations, and reflections require no alignment preprocessing. The uniqueness guarantee applies to generic polygonal domains, so this should be used as an auxiliary constraint rather than as a universal metric for arbitrary shapes.

## Mathematical statement

The paper defines the interpoint distance distribution as the measure-valued cumulative distribution \(\Psi_X(r)=\int_X\int_X\mathbf{1}\{\lVert x-y\rVert\le r\}\,dx\,dy\), where \(X\) is a compact planar body, \(x,y\in X\), \(\lVert x-y\rVert\) is Euclidean distance, and \(dx\,dy\) is area measure. It defines the Riesz energy or Brylinski beta function \(B_X(z)=\int_X\int_X\lVert x-y\rVert^z\,dx\,dy\), initially convergent for \(\operatorname{Re}z>-\dim X\), with meromorphic continuation for polygonal or smooth \(X\). The paper proves that a generic polygonal domain is determined up to Euclidean isometry by \(\Psi_X\), equivalently by \(B_X\); for convex domains the chord-length distribution is also sufficient. We use a finite set of real exponents \(z_k>-2\) as a stable sampled Riesz signature. For a predicted polygon \(\widehat X\) and target polygon \(X\), estimate \(B_X(z_k)\) by uniformly sampled interior points and minimize a normalized discrepancy. The generic-identification theorem does not imply exact identification for degenerate or nongeneric shapes, so the signature must supplement a task loss.

## Key formulas

- $$\Psi_X(r)=\int_X\int_X \mathbf{1}\{\lVert x-y\rVert\le r\}\,dx\,dy$$
- $$B_X(z)=\int_X\int_X \lVert x-y\rVert^{z}\,dx\,dy,\qquad \operatorname{Re}z>-\dim X$$
- $$\widehat B_X(z_k)=\frac{|X|^2}{N(N-1)}\sum_{i\ne j}\bigl(\lVert x_i-x_j\rVert+\varepsilon\bigr)^{z_k},\qquad x_i\sim\operatorname{Unif}(X)$$
- $$\mathcal L_{\mathrm{Riesz}}(X,\widehat X)=\sum_{k=1}^{K}w_k\left|\frac{\widehat B_X(z_k)-\widehat B_{\widehat X}(z_k)}{|\widehat B_X(z_k)|+\tau}\right|^2$$

## Implementation notes

Integrate this at the output of a polygon-reconstruction network: the network receives an image or point cloud and predicts an ordered vertex tensor \(V\in\mathbb R^{m\times2}\), or a differentiable rasterized polygon mask. During training, compute the ordinary supervised loss, such as vertex loss after matching, mask BCE/IoU loss, or Chamfer loss, and add \(\lambda\mathcal L_{\mathrm{Riesz}}\). First sample \(N\) points uniformly inside each target and predicted polygon; for a polygon, triangulate it and sample triangles proportional to area, using reparameterized random variables for differentiable prediction-side sampling. For exponents choose, for example, \(z_k\in\{-1.5,-1,-0.5,0,0.5,1,2\}\), with \(\varepsilon\approx10^{-4}\) in normalized image coordinates and \(\tau\approx10^{-6}\). Pseudocode: `x=sample_uniform(target); y=sample_uniform(predicted); Dx=pdist(x)+eps; Dy=pdist(y)+eps; Bx=area(target)^2*mean(Dx**z[k]); By=area(pred)^2*mean(Dy**z[k]); Lsig=sum(w[k]*abs((Bx-By)/(abs(Bx)+tau))**2); L=Ltask+lambda*Lsig`. Use symmetric pair subsampling if \(N^2\) is expensive, and stop gradients through target samples and target area. The Monte Carlo estimator is the engineering approximation; the paper supplies the invariant \(\Psi_X\)/\(B_X\) construction and generic identifiability, not a finite-sample error guarantee. Start with a small ResNet-to-8-vertex decoder on procedurally generated random polygons, including random rotations and translations, and compare vertex-only, Chamfer-only, and combined losses at fixed training FLOPs. Success is lower aligned Chamfer distance and fewer self-intersecting or badly distorted polygons under occlusion and random pose, without harming per-step throughput by more than roughly 20%.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.