Geom-Regularizer

• Geom-Regularizer is a family of techniques that apply differential geometry, such as curvature and spectral alignment, to regularize neural and variational models.
• It leverages analytic geometric quantities and convex penalties to enforce smoothness, robustness, and interpretable inductive biases across various applications.
• These methods yield practical improvements in artifact suppression, 3D surface fidelity, and reconstruction quality in complex computer graphics and geometric learning tasks.

Geom-Regularizer is a broad term applied to a family of regularization strategies that utilize geometric structure, invariants, or analytic properties (e.g., curvature, star-body gauges, spectral alignment, and more) to constrain neural or variational models in computer graphics, geometric learning, inverse problems, and generative modeling. These methods span from direct differential-geometry-based penalties in neural fields to global structural regularization in geodesic distance computation, adversarially-learned critic-based functionals, and geometry-aware augmentations for 3D representations. Geom-Regularizer frameworks are characterized by their explicit incorporation of analytic geometric quantities, often providing interpretable control over regularity, robustness, and inductive biases beyond what data-driven priors afford.

1. Foundations: Geometric Structure in Regularization

Geom-Regularizer methodologies exploit specific geometric features to regularize optimization problems. These may involve (a) differential properties—such as curvature or gradient alignment—in neural representations of fields or surfaces (Ehret et al., 2022), (b) star-body gauges and dual mixed volumes for learning regularizers via variational adversarial objectives (Leong et al., 29 Aug 2024), (c) convex penalties on distances and their derivatives over (Riemannian) manifolds (Edelstein et al., 2023), or (d) spectral properties of Jacobians in unsupervised representation learning (Ramesh et al., 2018).

The class of geometric regularizers contrasts with purely data-driven or "black-box" regularization (e.g., dropout), in that they encode precise geometric behavior or invariants, such as smoothness, local minimality of curvature, or directional steering, within the learning objective or constrained optimization.

2. Differential Geometry-Based Regularization in Neural Fields

Regularization of neural radiance fields (NeRFs) and related volumetric models provides a canonical example. By constraining the learned volumetric function $F:\mathbb{R}^3 \times S^2 \to (c,\sigma)$ to be infinitely differentiable (using $C^\infty$ Softplus activations), it becomes feasible to evaluate differential operators such as gradient, divergence, and Hessian through automatic differentiation (Ehret et al., 2022).

Explicit curvature terms are constructed using closed-form expressions for mean curvature

$$H(x) = -\nabla\cdot\left( \frac{\nabla F(x)}{\|\nabla F(x)\|} \right)$$

and Gaussian curvature

$$K(x) = \frac{\nabla F(x)^\top\,\mathrm{adj}(H(F)(x))\,\nabla F(x)}{\|\nabla F(x)\|^4}$$

where $n(x)$ is the unit surface normal and $H(F)$ the Hessian. The regularizer aggregates (clipped) curvature over samples: $$L_{\rm curv}(\kappa) = \mathbb{E}_{x \in \mathcal S} [\min(|\gamma(x)|, \kappa)]$$ with $\gamma$ denoting mean or Gaussian curvature. The final loss used in VolSDF-based NeRFs is

$$L = L_{RGB} + \lambda_{SDF}\,\mathbb{E}_{x}[(\|\nabla F(x)\|-1)^2] + \lambda_{\rm curv}\,L_{\rm curv}(\kappa)$$

Curvature regularization penalizes high-frequency artifact surfaces (wiggles, floaters) and encourages geometrically plausible, smooth geometry, especially under sparse or noisy supervision (Ehret et al., 2022).

3. Convex and Variational Geometric Regularizers in Optimization

Beyond neural fields, geometric regularization operates directly on optimization problems over geometric domains such as surfaces or manifolds. A representative instance is the convex framework for regularized geodesic distances, where the objective is to minimize

$$E(u) = \int_M F(\nabla u(x), x)\,d\mathrm{Vol}(x) - \int_M u(x)\,d\mathrm{Vol}(x)$$

subject to $|\nabla u(x)| \leq 1$ outside a source set $E$ and $u(x) \leq 0$ on $E$ (Edelstein et al., 2023). The regularizer $F$ can be chosen as:

• Dirichlet (quadratic) smoothing: $F(\xi, x) = \frac\alpha2|\xi|^2$;
• Vector-field alignment: $$F(\xi, x) = \frac\alpha2 [|\xi|^2 + \beta \langle V(x), \xi\rangle^2]$$ for alignment to a line field $V(x)$;
• Hessian smoothing: $$F(\nabla u(x), x) = \frac\alpha2 |\nabla^2 u(x)|_F^2$$.

Efficient solution is achieved via ADMM, where the gradient, constraint projection, and quadratic solve admit scalable GPU/CPU implementations (Edelstein et al., 2023). These geometric regularizers yield globally smooth, robust geodesic-like distances, with guarantees of well-posedness, uniqueness, and convergence.

4. Geometric Regularization in Generative Models and Spectral Alignment

Geom-Regularizer also appears in generative modeling for representation learning. In "A Spectral Regularizer for Unsupervised Disentanglement," the objective is to align the leading right singular vectors of the generator's Jacobian $J_G(z)$ with canonical axes, encouraging local disentanglement (Ramesh et al., 2018). By approximating top-$k$ singular vectors with masked power method and minimizing

$$\lambda\,\mathbb{E}_{z \sim p_z}\left[ R_k(z) \right]$$

where $R_k(z)$ penalizes misalignment, one enforces geometrically meaningful latent traversals. This geometric spectral regularization is lightweight and improves the interpretability and independence of latent directions.

5. Star-Body Gauges and Dual Mixed Volume Regularization

A rigorous geometric-analytic perspective emerges in the theory of critic-based regularizer learning, interpreted as learning star-body gauges via dual Brunn-Minkowski theory (Leong et al., 29 Aug 2024). A star body $K$ with gauge $\|x\|_K$ and radial function $\rho_K(u)$ induces an adversarial regularizer: $$F(K) = \mathbb{E}_{D_r}\left[\|x\|_K\right] - \mathbb{E}_{D_n}\left[\|x\|_K\right]$$ This can be expressed as a dual mixed volume $d\,\widetilde{V}_{-1}(L_{r,n}, K)$ for certain data-dependent star bodies $L_{r,n}$, and exact extremality conditions for optimal $K$ are provably characterized. Neural architectures parameterizing such star-body gauges require positive homogeneity, continuity, and injectivity (Leong et al., 29 Aug 2024).

6. Geometric Regularizers in 3D Gaussian Representations

Geom-Regularizer strategies manage both primitive shape and global surface fidelity in 3D Gaussian Splatting. For example, ARGS introduces:

• Effective rank regularization on each 3D Gaussian, using the entropy of normalized singular values to penalize degenerate ("needle-like") or collapsed shapes, favoring "disk-like" primitives,
• Neural SDF co-training with Eikonal regularization and SDF-Gaussian consistency losses to globally align Gaussians to a smooth surface manifold; losses include:

$$L_{eff} = \lambda_{erank} \sum_k \Big[ \max(-\log(erank(G_k) - 1 + \epsilon), 0) + s_{k3} \Big]$$

$$L_{eik} = \lambda_{eik} \mathbb{E}_{x}[ (\|\nabla f(x)\|_2 - 1)^2 ]$$

$$L_{sdf-G} = \lambda_{sdf-G} \mathbb{E}_{x \sim Gauss}[ f(x)^2 ]$$

yielding improved surface consistency, mesh coverage, artifact suppression, and rendering metrics (Lee et al., 29 Aug 2025).

In equirectangular omnidirectional settings (ErpGS), geometric regularization penalizes discrepancies between normals computed by Gaussian splatting and normals derived from rendered depth maps, weighted by color gradient and distortion-aware area, improving both accuracy and smoothness in the presence of strong ERP distortions (Ito et al., 26 May 2025).

7. Empirical Outcomes and Implementation Considerations

Empirical studies across multiple works confirm that Geom-Regularizer frameworks consistently improve robustness, geometric plausibility, representation quality, and downstream task performance:

• Curvature regularization in differential neural fields yields up to 1 dB PSNR gain under sparse supervision, with little computational overhead for first-order terms (Ehret et al., 2022).
• Geodesic regularization using convex penalties exhibits mesh-independence, parameter calibration tractability, and competitive accuracy even under severe remeshing or noise (Edelstein et al., 2023).
• Star-body-based critic regularizers yield explicit optimizers and sample-complexity guarantees; specific neural network conditions ensuring gauge properties are identified (Leong et al., 29 Aug 2024).
• Spectral geometric regularization enables improved linear disentanglement in GAN latent representations at manageable computation cost (Ramesh et al., 2018).
• 3DGS-based geometric penalties, in ARGS and ErpGS, improve artifact suppression, completeness and fine geometric fidelity, with moderate additional compute (Lee et al., 29 Aug 2025, Ito et al., 26 May 2025).
• In camera pose regression, training-time geometric consistency losses (Pose/Descriptor reprojection, RANSAC-based pose alignment) close a large portion of the accuracy gap to slow, correspondence-based approaches, with zero inference penalty—a paradigm shift in geometric vision (Li et al., 27 Sep 2025).

Practical implementation leverages automatic differentiation for all differential geometric terms, scalable ADMM optimization for convex geometric regularization, and SVD or power-iteration methods for low-dimensional Jacobian spectral analysis.

8. Scope, Limitations, and Theoretical Advances

Geom-Regularizer techniques offer unified control over geometric prior strength, allow flexible adaptation via problem-specific design (e.g., regularizer choice, hyperparameters), and enjoy theoretical guarantees where convexity and analytic structure are preserved (Edelstein et al., 2023, Leong et al., 29 Aug 2024). However, challenges remain:

• Debugging or tuning geometric regularization may require nontrivial geometric insight, especially in the presence of noise, nonconvexity, or lack of ground-truth geometric supervision.
• Over-regularization can introduce envelope artifacts or smooth away important sharp features; hyperparameter selection (e.g., curvature clipping thresholds, entropy penalties) entails a familiar bias-variance trade-off (Ehret et al., 2022).
• Certain geometric losses, such as star-body gauges, can be nonconvex and may mandate specialized optimization techniques (proximal-point, weak convexity, etc.) (Leong et al., 29 Aug 2024).
• Initialization and scheduling (e.g., turning on geometric losses only after initial convergence) are critical for stability in 3DGS and related frameworks (Lee et al., 29 Aug 2025, Ito et al., 26 May 2025).

Advances include precise links between variational regularizer learning and convex geometric analysis, robust and scalable solvers for diverse geometric PDEs, and new neural architectures tailored for geometric inductive bias. Future work may extend these concepts to non-Euclidean data, higher-order differential constraints, or more expressive classes of geometric functionals.

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Relevant references:

• (Ehret et al., 2022): "Regularization of NeRFs using differential geometry"
• (Edelstein et al., 2023): "A Convex Optimization Framework for Regularized Geodesic Distances"
• (Leong et al., 29 Aug 2024): "The Star Geometry of Critic-Based Regularizer Learning"
• (Ramesh et al., 2018): "A Spectral Regularizer for Unsupervised Disentanglement"
• (Lee et al., 29 Aug 2025): "ARGS: Advanced Regularization on Aligning Gaussians over the Surface"
• (Ito et al., 26 May 2025): "ErpGS: Equirectangular Image Rendering enhanced with 3D Gaussian Regularization"
• (Li et al., 27 Sep 2025): "GeLoc3r: Enhancing Relative Camera Pose Regression with Geometric Consistency Regularization"