Persistence-based topological optimization deforms a point cloud $X \subset \mathbb{R}^d$ by minimizing objectives of the form $L(X) = \ell(\mathrm{Dgm}(X))$, where $\mathrm{Dgm}(X)$ is a persistence diagram. In practice, optimization is limited by two coupled issues: persistent homology is typically computed on subsamples, and the resulting topological gradients are highly sparse, with only a few anchor points receiving nonzero updates. Motivated by diffeomorphic interpolation, which extends sparse gradients to smooth ambient vector fields via Reproducing Kernel Hilbert Space (RKHS) interpolation, we propose a more scalable pipeline that improves both subsampling and gradient extension. We introduce subsampling via random slicing, a lightweight scheme that promotes iteration-wise geometric coverage and mitigates density bias. We further replace the costly kernel solve with a fast Nadaraya-Watson (NW) Gaussian convolution, producing a globally defined smooth update field at a fraction of the computational cost, while being more suited for topological optimization tasks. We provide theoretical guarantees for NW smoothing, including anchor approximation bounds and global Lipschitz estimates. Experiments in $2$D and $3$D show that combining random slicing with NW smoothing yields consistent speedups and improved objective values over other baselines on common persistence losses.

翻译：基于持久性的拓扑优化通过最小化形如 $L(X) = \ell(\mathrm{Dgm}(X))$ 的目标函数来变形点云 $X \subset \mathbb{R}^d$，其中 $\mathrm{Dgm}(X)$ 是持久性图。在实践中，优化受到两个耦合问题的限制：持久同调通常基于子样本计算，且由此产生的拓扑梯度高度稀疏，只有少数锚点接收非零更新。受微分同胚插值的启发（该方法通过再生核希尔伯特空间（RKHS）插值将稀疏梯度扩展为光滑的全局向量场），我们提出了一种更可扩展的流程，改进了子采样和梯度扩展两个环节。我们引入基于随机切片的子采样方法，这是一种轻量级方案，能促进迭代中的几何覆盖并减轻密度偏差。我们进一步用快速的Nadaraya-Watson（NW）高斯卷积替代了昂贵的核求解，以极低的计算成本生成全局定义的光滑更新场，同时更适合拓扑优化任务。我们为NW平滑提供了理论保证，包括锚点近似界和全局Lipschitz估计。在2D和3D上的实验表明，将随机切片与NW平滑相结合，在常见持久性损失函数上相比其他基线方法能实现一致的速度提升和更优的目标值。