双曲流形回归
Hyperbolic Manifold Regression
摘要：
几何表示学习最近在几个机器学习环境中显示出很大的希望，从关系学习到语言处理和生成模型。在这项工作中，我们认为执行流形值回归到双曲空间作为一个中间组件的一些相关的机器学习应用程序的问题。特别是，通过将预测树的节点的问题公式化为双曲空间中的流形回归任务，我们对两个具有挑战性的任务提出了新的观点：1）通过标签嵌入的分层分类和2）双曲表示的分类扩展。为了解决回归问题，我们考虑了以前的方法，并提出了两种在计算上更有利的新方法：一种是由目标空间的测地线提供信息的参数深度学习模型，另一种是我们还证明了超额风险界限的非参数核方法。我们的实验表明，利用双曲几何的策略是有前途的。特别是，在分类扩展设置中，我们发现基于双曲线的估计显著优于在环境欧氏空间中进行回归的方法。
英文摘要：
Geometric representation learning has recently shown great promise in several machine learning settings, ranging from relational learning to language processing and generative models. In this work, we consider the problem of performing manifold-valued regression onto an hyperbolic space as an intermediate component for a number of relevant machine learning applications. In particular, by formulating the problem of predicting nodes of a tree as a manifold regression task in the hyperbolic space, we propose a novel perspective on two challenging tasks: 1) hierarchical classification via label embeddings and 2) taxonomy extension of hyperbolic representations. To address the regression problem we consider previous methods as well as proposing two novel approaches that are computationally more advantageous: a parametric deep learning model that is informed by the geodesics of the target space and a non-parametric kernel-method for which we also prove excess risk bounds. Our experiments show that the strategy of leveraging the hyperbolic geometry is promising. In particular, in the taxonomy expansion setting, we find that the hyperbolic-based estimators significantly outperform methods performing regression in the ambient Euclidean space.
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