通过无bin多维积分进行密度估计
Density Estimation via Binless Multidimensional Integration
AI总结：
本文提出BMTI方法，通过计算邻近数据点的log密度差并进行多维积分，实现非参数、鲁棒且数据高效的密度估计，适用于高维空间。
摘要：
我们介绍了无bin多维热力学积分（BMTI）方法，用于非参数、鲁棒且数据高效的密度估计。BMTI通过初始计算邻近数据点之间的log密度差，随后利用最大似然公式对这些差异进行积分，权重为相关不确定性。该方法是统计物理中热力学积分技术在多维设置中的扩展。该方法利用流形假设，在不定义显式坐标映射的情况下估计内在数据流形中的量。它不依赖任何binning或空间划分，而是基于自适应带宽选择过程构建邻域图。BMTI克服了传统非参数密度估计器的局限性，即使在高维嵌入空间中也能有效重建平滑轮廓。该方法在多种复杂合成高维数据集上进行了测试，显示出优于传统估计器的性能，并在化学物理文献中的现实数据集上进行了基准测试。
英文摘要：
We introduce the Binless Multidimensional Thermodynamic Integration (BMTI) method for nonparametric, robust, and data-efficient density estimation. BMTI estimates the logarithm of the density by initially computing log-density differences between neighbouring data points. Subsequently, such differences are integrated, weighted by their associated uncertainties, using a maximum-likelihood formulation. This procedure can be seen as an extension to a multidimensional setting of the thermodynamic integration, a technique developed in statistical physics. The method leverages the manifold hypothesis, estimating quantities within the intrinsic data manifold without defining an explicit coordinate map. It does not rely on any binning or space partitioning, but rather on the construction of a neighbourhood graph based on an adaptive bandwidth selection procedure. BMTI mitigates the limitations commonly associated with traditional nonparametric density estimators, effectively reconstructing smooth profiles even in high-dimensional embedding spaces. The method is tested on a variety of complex synthetic high-dimensional datasets, where it is shown to outperform traditional estimators, and is benchmarked on realistic datasets from the chemical physics literature.
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