论文标题
非欧几里得空间中分布差异的半非偏见估计
Semi-Nonparametric Estimation of Distribution Divergence in Non-Euclidean Spaces
论文作者
论文摘要
本文探讨了使用独立和相同分布的样品探讨估算或近似总变化距离以及拓扑样本空间内的概率度量的差异。我们的重点是实用场景，样品空间对欧几里得空间的子集是同型的，而特定的同构态尚不清楚。我们提出的方法依赖于积分概率指标，并在通用繁殖Hilbert Spaces（RKHSS）中具有证人功能。我们开发的估计器包括可学习的参数函数，将样品空间映射到欧几里得空间，并与欧几里得空间中定义的通用内核配对。这种方法有效地克服了直接在非欧几里得空间上构建通用内核的挑战。此外，我们设计的估计值表明了渐近的一致性，我们提供了详细的统计分析，从而阐明了它们的实际实施。
This paper explores methods for estimating or approximating the total variation distance and the chi-squared divergence of probability measures within topological sample spaces, using independent and identically distributed samples. Our focus is on the practical scenario where the sample space is homeomorphic to subsets of Euclidean space, with the specific homeomorphism remaining unknown. Our proposed methods rely on the integral probability metric with witness functions in universal reproducing kernel Hilbert spaces (RKHSs). The estimators we develop consist of learnable parametric functions mapping the sample space to Euclidean space, paired with universal kernels defined in Euclidean space. This approach effectively overcomes the challenge of constructing universal kernels directly on non-Euclidean spaces. Furthermore, the estimators we devise demonstrate asymptotic consistency, and we provide a detailed statistical analysis, shedding light on their practical implementation.
