距离查询的可微近似
Differentiable Approximations for Distance Queries
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中文总结 AI 辅助
本文针对几何查询问题，提出了一种可微的欧氏距离乘法(1+ε)近似方案，基于单位划分框架与新型近似Voronoi图实现，达到了最优的时间空间复杂度。
中文摘要 AI 辅助
基于梯度的优化技术的广泛应用，推动了各类经典算法向适配学习流程的可微求解器改进。本文研究传统几何查询问题的增强方案，使其输出包含几何函数及其梯度。具体而言，我们研究了针对ℝᵈ中点集P的距离查询这一基础问题，该问题也是学习算法各类相似性度量的核心基础。本文的主要成果是得到了ℝᵈ\backslash P中所有点处均可微的、与P的欧氏距离的乘法(1+ε)近似，其梯度与海森矩阵范数具有渐近最优界，且对应数据结构的存储与查询时间达到近似最近邻搜索的最优水平。该近似通过单位划分框架实现为正则化距离，该框架会在合适定义的空间覆盖下，高效融合多个局部近似，形成平滑的全局近似。为以便于融合的方式获取局部距离近似，我们开发了一种基于简单点定位数据结构的新型近似Voronoi图，省略了升维变换与射线投射步骤，简化了相关实现。
英文摘要
The widespread use of gradient-based optimization has motivated the adaptation of various classical algorithms into differentiable solvers compatible with learning pipelines. In this paper, we investigate the enhancement of traditional geometric query problems such that the result consists of both the geometric function as well as its gradient. Specifically, we study the fundamental problem of distance queries against a set of points $P$ in $\mathbb{R}^d$, which also underlies various similarity measures for learning algorithms. The main result of this paper is a multiplicative $(1+\varepsilon)$-approximation of the Euclidean distance to $P$ which is differentiable at all points in $\mathbb{R}^d \setminus P$ with asymptotically optimal bounds on the norms of its gradient and Hessian, from a data structure with storage and query time matching state-of-the-art results for approximate nearest-neighbor searching. The approximation is realized as a regularized distance through a partition-of-unity framework, which efficiently blends multiple local approximations, over a suitably defined covering of space, into a smooth global approximation. In order to obtain the local distance approximations in a manner that facilitates blending, we develop a new approximate Voronoi diagram based on a simple point-location data structure, simplifying away both the lifting transformation and ray shooting.
