不用神经网络，用多项式插值实现高效函数表示
efunc: An Efficient Function Representation without Neural Networks
- 用径向基函数插值多项式构建紧凑函数表示
- 计算时间与内存低于传统框架10%以下
- 适合需要轻量化表示的图形与工程应用
函数拟合/逼近在计算机图形学及其他工程应用中具有基础作用。尽管近期研究探索了神经网络方法，但这些方法通常依赖大量参数的架构，限制了实际应用。本文提出一种无需神经网络的高效函数逼近方法。首先构建连续函数建模的新框架，多数现有工作可在此框架下统一表述。随后引入基于径向基函数插值多项式的紧凑函数表示，避免了神经网络和复杂分层数据结构。同时开发了内存高效的CUDA优化算法，使计算时间与内存消耗降至传统自动微分框架的10%以下。通过在3D有符号距离函数（SDFs）上的大量实验验证，该表示方法在参数量显著更少的情况下，性能可媲美或超越当前最优技术（如八叉树/哈希网格方法）。
原文摘要 · Abstract (English)
Function fitting/approximation plays a fundamental role in computer graphics and other engineering applications. While recent advances have explored neural networks to address this task, these methods often rely on architectures with many parameters, limiting their practical applicability. In contrast, we pursue high-quality function approximation using parameter-efficient representations that eliminate the dependency on neural networks entirely. We first propose a novel framework for continuous function modeling. Most existing works can be formulated using this framework. We then introduce a compact function representation, which is based on polynomials interpolated using radial basis functions, bypassing both neural networks and complex/hierarchical data structures. We also develop memory-efficient CUDA-optimized algorithms that reduce computational time and memory consumption to less than 10% compared to conventional automatic differentiation frameworks. Finally, we validate our representation and optimization pipeline through extensive experiments on 3D signed distance functions (SDFs). The proposed representation achieves comparable or superior performance to state-of-the-art techniques (e.g., octree/hash-grid techniques) with significantly fewer parameters.
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