While conventional data are represented as discrete vectors, Implicit Neural Representations (INRs) utilize neural networks to represent data points as continuous functions. By incorporating a shared network that maps latent vectors to individual functions, one can model the distribution of functional data, which has proven effective in many applications, such as learning 3D shapes, surface reflectance, and operators. However, the infinite-dimensional nature of these representations makes them prone to overfitting, necessitating sufficient regularization. Naïve regularization methods -- those commonly used with discrete vector representations -- may enforce smoothness to increase robustness but result in a loss of data fidelity due to improper handling of function coordinates. To overcome these challenges, we start by interpreting the mapping from latent variables to INRs as a parametrization of a Riemannian manifold. We then recognize that preserving geometric quantities -- such as distances and angles -- between the latent space and the data manifold is crucial. As a result, we obtain a manifold with minimal intrinsic curvature, leading to robust representations while maintaining high-quality data fitting. Our experiments on various data modalities demonstrate that our method effectively discovers a well-structured latent space, leading to robust data representations even for challenging datasets, such as those that are small or noisy.

Abstract Reducing the triangle count in complex 3D models is a basic geometry preprocessing step in graphics pipelines such as efficient rendering and interactive editing. However, most existing mesh simplification methods exhibit a few issues. Firstly, they often lead to self‐intersections during decimation, a major issue for applications such as 3D printing and soft‐body simulation. Second, to perform simplification on a mesh in the wild, one would first need to perform re‐meshing, which often suffers from surface shifts and losses of sharp features. Finally, existing re‐meshing and simplification methods can take minutes when processing large‐scale meshes, limiting their applications in practice. To address the challenges, we introduce a novel GPU‐based mesh optimization approach containing three key components: (1) a parallel re‐meshing algorithm to turn meshes in the wild into watertight, manifold, and intersection‐free ones, and reduce the prevalence of poorly shaped triangles; (2) a robust parallel simplification algorithm with intersection‐free guarantees; (3) an optimization‐based safe projection algorithm to realign the simplified mesh with the input, eliminating the surface shift introduced by re‐meshing and recovering the original sharp features. The algorithm demonstrates remarkable efficiency, simplifying a 2‐million‐face mesh to 20k triangles in 3 seconds on RTX4090. We evaluated the approach on the Thingi10K dataset and showcased its exceptional performance in geometry preservation and speed. https://seonghunn.github.io/pamo/