用拓扑损失自动选择核密度估计的最优带宽
Unsupervised Learning of Density Estimates with Topological Optimization
- 基于拓扑特征设计损失函数，实现无监督带宽优化
- 在多维数据上优于传统方法，避免过平滑或欠平滑
- 适合需要自动密度估计的科研与工程场景
核密度估计是机器学习、贝叶斯推断、随机动力学和信号处理中广泛应用的关键技术。然而，无监督密度估计需调优关键超参数——核带宽。带宽选择至关重要，它通过过平滑或欠平滑影响拓扑特征的保留。拓扑数据分析可数学量化高维空间中的连通分量、环路、空洞等拓扑特性，即使无法可视化密度估计。本文提出一种基于拓扑损失函数的无监督学习方法，实现带宽的自动、无监督最优选择，并在不同维度下与经典方法对比，验证其有效性。
原文摘要 · Abstract (English)
Kernel density estimation is a key component of a wide variety of algorithms in machine learning, Bayesian inference, stochastic dynamics and signal processing. However, the unsupervised density estimation technique requires tuning a crucial hyperparameter: the kernel bandwidth. The choice of bandwidth is critical as it controls the bias-variance trade-off by over- or under-smoothing the topological features. Topological data analysis provides methods to mathematically quantify topological characteristics, such as connected components, loops, voids et cetera, even in high dimensions where visualization of density estimates is impossible. In this paper, we propose an unsupervised learning approach using a topology-based loss function for the automated and unsupervised selection of the optimal bandwidth and benchmark it against classical techniques -- demonstrating its potential across different dimensions.
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