Hyperbolic Mesh Optimization Loss

• The approach defines loss functions based on hyperbolic volume energy and the Lobachevsky function to promote equilateral triangle configurations in mesh optimization.
• It employs a two-stage optimization that splits convex angle-sum constraints from non-convex holonomy adjustments, ensuring both interior regularity and accurate boundary preservation.
• Extensions with pre-shape calculus and polyconvex hyperelastic losses further guarantee minimizer existence, prevent degeneracies, and enable robust deep mesh learning.

Hyperbolic mesh optimization loss refers to a family of objective functionals and penalty terms used in mesh improvement, inference, or learning tasks where the mesh structure, or its features, is optimized within or with respect to a hyperbolic (negatively curved) geometric context. Such loss functions are closely tied to the inherent properties of hyperbolic space, exploiting its curvature, volume growth, and boundary separation to promote desirable mesh characteristics—such as hierarchical structure preservation, element regularity, avoidance of degeneracy, and increased worst-case uniformity. Several distinct but related approaches have been proposed in the literature, ranging from convex variational principles grounded in hyperbolic simplex volumes to learnable loss functions for deep neural mesh parameterizations.

1. Variational Structures: Hyperbolic Volume-Based Energy Functionals

One seminal formulation is the energy-based approach introduced for 2D triangle meshes tessellating planar regions, where the primary objective is to maximize an energy functional directly related to the hyperbolic volume of ideal 3-simplices (Sun et al., 2013). For a single triangle with interior angles $\alpha, \beta, \gamma$ satisfying $\alpha + \beta + \gamma = \pi$, the triangle energy is given by

$$E(t) = \Lambda(\alpha) + \Lambda(\beta) + \Lambda(\gamma)$$

where $\Lambda(x)$ is the Lobachevsky function,

$\Lambda(x) = -\int_0^x \ln|2\sin t|\,dt$

This functional is strictly concave over angle assignments. For a mesh $T$ with triangle set $F$,

$\mathcal{E}(T) = \sum_{t\in F} E(t)$

Maximizing $\mathcal{E}$ over feasible angle structures (subject to linear angle-sum constraints and nonlinear holonomy constraints for embedding and boundary preservation) yields improved mesh regularity. The energy is maximized for equilateral triangles, tightly clustering both the minimum and maximum inner angles and aspect ratios of mesh elements relative to centroidal Voronoi tessellation (CVT) approaches.

The optimization problem is split into convex (angle sum) and non-convex (holonomy) stages, both tractable via interior-point algorithms. The first stage ensures interior regularity, while the second enforces boundary conditions via an auxiliary energy: $\mathcal{D}(A) = \sum_{i} (H(i, A) - H_i)^2$ where $H(i, A)$ denotes the holonomy constraint for vertex $i$. Experimental results confirm that the mesh angles and aspect ratios after optimization are confined to much tighter intervals than CVT-produced meshes, with superior $L_\infty$ (worst-case) performance.

2. Tangential and Normal Decomposition: Pre-Shape Calculus in Quality Tracking

The "pre-shape calculus" framework generalizes classical shape optimization by handling deformations in both normal and tangential mesh directions, thereby facilitating simultaneous mesh regularity and geometric fidelity (Luft et al., 2020, Luft et al., 2021). The pre-shape derivative decomposes as

$$\operatorname{PrShpDeriv}_J(\varphi)[V] = \langle g^N, V \rangle + \langle g^T, V \rangle$$

where $g^N$ governs shape changes and $g^T$ governs mesh parameterization or vertex density. This allows the design of losses or tracking objectives penalizing deviations from targeted cell size distributions (or other mesh qualities), which can be adapted to enforce hyperbolic metric requirements.

Numerical implementations employ gradient descent on the tangential component, preserving shape while optimizing mesh density, with convergence demonstrated in both 2D and 3D settings. The p-Laplacian metric, regularized via an $\epsilon$-parameter to ensure uniform ellipticity and numerical stability, is one such choice within pre-shape gradient systems.

3. Riemannian Metric and Degeneracy Avoidance: Barrier Properties

A loss function can be augmented by a penalty term $\varphi(Q)$ that diverges as mesh quality deteriorates—effectively repelling the optimizer from degenerate configurations (Herzog et al., 2021). The induced complete metric on the mesh configuration space is given by

$$g_{ab}^{\text{complete}} = \delta_a^b + \frac{\partial \varphi}{\partial (\operatorname{vec} Q)^a} \frac{\partial \varphi}{\partial (\operatorname{vec} Q)^b}$$

Mesh degeneracy (e.g., collapsed triangles) is thus pushed "infinitely far away" in the metric geometry, functioning as an intrinsic hyperbolic barrier. This approach stabilizes discrete shape optimization and enables large, safe mesh deformations. The theoretical guarantees extend to existence and uniqueness results for the penalized optimization problem, independent of the Riemannian metric chosen.

4. Hyperelastic and Polyconvex Losses for Guaranteed Mesh Quality

Mesh untangling and quality improvement tasks often exploit polyconvex hyperelastic energy functionals, which guarantee both existence of minimizers and barrier penalties against inverted elements (Garanzha et al., 2022). These functionals, written as

$E(x) = \int_\Omega f(J(x))\,d\xi$

where $J$ is the deformation Jacobian and $f$ a polyconvex density (e.g., mixing shape and volume distortion), support robust schemes for mesh regularization under nonlinear constraints. Optimization proceeds via a continuation method starting with inversion-correcting regularization and proceeding to quasi-isometric stiffening, driving all element distortions below a tunable threshold.

5. Loss Functions for Hyperbolic Manifold Learning and Mesh Feature Embedding

Hyperbolic mesh optimization loss has also appeared as a learnable loss in deep neural systems for mesh regression and representation learning. In 3D human mesh recovery leveraging temporal motion priors, mesh vertex coordinates are embedded in hyperbolic space and the mesh optimization loss is computed as the mean absolute difference (L1 norm) between hyperbolic projections of ground truth and predicted meshes (Zhang et al., 21 Oct 2025): $$L_{\text{hymesh}} = \frac{1}{V} \sum_{i=1}^{V} \left\| \widehat{M}_{\text{gt}}^{(i)} - \widehat{M}_{\text{pre}}^{(i)} \right\|_1$$ where $\widehat{M} = \exp_0(M)$ denotes the exponential map projecting mesh vertices into the Poincaré ball. This loss operates solely on the hyperbolic embedding, directly enforcing both spatial accuracy and hierarchical smoothness. It is weighted alongside other structural mesh losses to achieve a superior trade-off between pose realism and temporal consistency in mesh recovery tasks.

6. Penalty Functions and Surrogate Testing in Mesh Regression

Overfitting in mesh-structured regression tasks can be managed using a Laplace-operator-based diffusion loss computed on staggered mesh configurations (Bigarella, 9 Jul 2025). Although implemented in Euclidean settings, this approach is extendable to hyperbolic meshes via the Laplace-Beltrami operator. Differences between the true Laplacian (on training mesh nodes) and the predicted Laplacian (on staggered probe nodes) provide an entropy metric penalizing oscillatory, nonphysical features—a plausible implication for future hyperbolic mesh optimization loss designs targeting generalization rather than just data fitting.

7. Connections and Extensions: Hierarchical Metric Learning and Deep Recommendation

Recent advances in hyperbolic metric learning and deep recommendation models have proposed loss functions (triplet loss, proxy-based cluster loss) that leverage the geometric separation and exponential capacity of hyperbolic spaces (Keller-Ressel, 2020, Yusupov et al., 16 Aug 2025, Saeki et al., 7 Oct 2025). While not directly mesh-centric, these works suggest that representing mesh nodes, features, or hierarchical clusters in hyperbolic space and formulating losses using Busemann functions, Lorentzian distances, or combined Euclidean/Hyperbolic proxy distances can unlock increased representational power and stability, especially in large-scale multi-class or hierarchical settings.

Summary Table: Representative Hyperbolic Mesh Optimization Losses

Approach	Loss Function	Key Metric/Barrier
Hyperbolic volume energy (Sun et al., 2013)	$\mathcal{E}(T) = \sum_t E(t)$ with $E(t)$ via $\Lambda$	Enforces equilateral triangles
Complete metric penalty (Herzog et al., 2021)	$\varphi(Q)$ blowing up at degeneracy; $g_{ab}^{\text{complete}}$	Infinite distance barrier
Polyconvex hyperelastic functional (Garanzha et al., 2022)	$E(x) = \int f(J(x))\,d\xi$	Existence, injectivity
Hyperbolic mesh L1 loss (Zhang et al., 21 Oct 2025)	$$L_{\text{hymesh}} = (1/V)\sum ||\exp_0(M_{\text{gt}}^{(i})$$ – $\exp_0(M_{\text{pre}}^{(i)})||_1$	Direct mesh embedding optimization

Impact and Applications

Hyperbolic mesh optimization losses offer a principled and mathematically robust strategy for improving mesh quality, regularity, and generalizability across tasks involving geometric modeling, PDE-constrained optimization, deep learning of shape features, and hierarchical structure inference. By exploiting the intrinsic geometry of negatively curved spaces, these losses avoid common pitfalls such as mesh degeneracy, overfitting, and loss of hierarchical context. They also provide a natural framework for learning and inferring on data that is best described as hierarchical, tree-like, or residing in non-Euclidean manifolds.

This class of losses continues to provide fertile ground for advances in mesh-aware deep learning, geometric PDE solvers, and other applications requiring fine topological and geometric control over complex, high-dimensional mesh representations.