学习随机点积图后续推断的子流形，第一部分：理论
Learning Submanifolds for Subsequent Inference on Random Dot Product Graphs, Part 1: Theory
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中文总结 AI 辅助
提出半监督决策框架，利用Isomap学习随机点积图潜在位置的未知低维支撑流形，并证明随辅助数据增加，其风险收敛于最优预言规则。
中文摘要 AI 辅助
我们提出一个针对随机点积图受限推断的框架，其中潜在位置位于一个未知的低维支撑流形上。对于一般决策问题，我们提出半监督决策规则，利用辅助数据学习支撑流形。具体而言，我们的规则使用Isomap流形学习程序构建观测图的低维欧几里得表示，在该空间中，一个等距不变函数将点的配置映射到动作。我们研究当从未知支撑流形采样的辅助数据量增加时，所提规则的行为。我们证明，随着辅助样本量的增加，半监督规则的风险收敛于一个预言规则的风险，该预言规则依赖于可从支撑流形提取的最大量的低维欧几里得结构。示例、应用和模拟研究留待后续文章。
英文摘要
We propose a framework for restricted inference on random dot product graphs whose latent positions lie on an unknown low-dimensional support manifold. For general decision problems, we propose semisupervised decision rules that use auxiliary data to learn the support manifold. Specifically, our rules use the Isomap manifold learning procedure to construct a low-dimensional Euclidean representation of the observed graph, in which space an isometrically invariant function maps configurations of points to actions. We study the behavior of the proposed rules as the quantity of auxiliary data sampled from the unknown support manifold increases. We show that, as the auxiliary sample size increases, the risk of the semisupervised rule converges to the risk of an oracle rule that relies on the maximal amount of low-dimensional Euclidean structure that can be extracted from the support manifold. Examples, applications, and simulation studies are deferred to a sequel.
发表机构
- Indiana University（印第安纳大学）
- Johns Hopkins University（约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。
