Recent research indicates that the performance of machine learning models can be improved by aligning the geometry of the latent space with the underlying data structure. Rather than relying solely on Euclidean space, researchers have proposed using hyperbolic and spherical spaces with constant curvature, or combinations thereof, to better model the latent space and enhance model performance. However, little attention has been given to the problem of automatically identifying the optimal latent geometry for the downstream task. We mathematically define this novel formulation and coin it as neural latent geometry search (NLGS). More specifically, we introduce a principled method that searches for a latent geometry composed of a product of constant curvature model spaces with minimal query evaluations. To accomplish this, we propose a novel notion of distance between candidate latent geometries based on the Gromov-Hausdorff distance from metric geometry. In order to compute the Gromov-Hausdorff distance, we introduce a mapping function that enables the comparison of different manifolds by embedding them in a common high-dimensional ambient space. Finally, we design a graph search space based on the calculated distances between candidate manifolds and use Bayesian optimization to search for the optimal latent geometry in a query-efficient manner. This is a general method which can be applied to search for the optimal latent geometry for a variety of models and downstream tasks. Extensive experiments on synthetic and real-world datasets confirm the efficacy of our method in identifying the optimal latent geometry for multiple machine learning problems.

翻译：最新研究表明，通过使隐空间的几何结构与底层数据结构对齐，可以提升机器学习模型的性能。研究者不再仅依赖欧几里得空间，而是提出使用具有恒定曲率的双曲空间、球面空间或其组合来更好地建模隐空间并增强模型性能。然而，针对如何自动识别下游任务的最优隐式几何结构这一问题，目前关注甚少。我们从数学角度正式定义了这一新概念，并将其命名为"神经隐式几何搜索"（NLGS）。具体而言，我们提出了一种基于原则的方法，该方法以最少的查询评估量，搜索由恒定曲率模型空间乘积构成的隐式几何结构。为实现这一目标，我们基于度量几何中的格罗莫夫-豪斯多夫距离，提出了候选隐式几何结构间距离的新定义。为计算格罗莫夫-豪斯多夫距离，我们引入了一种映射函数，通过将不同流形嵌入共同的高维环境空间来实现流形间的比较。最后，我们根据候选流形间的计算距离设计图搜索空间，并采用贝叶斯优化以查询高效的方式搜索最优隐式几何结构。作为一种通用方法，本方案可应用于各类模型及下游任务的最优隐式几何结构搜索。在合成数据集和真实世界数据集上的大量实验证实了该方法在识别多种机器学习问题最优隐式几何结构方面的有效性。