In this work, we investigate the problem of learning distance functions within the query-based learning framework, where a learner is able to pose triplet queries of the form: ``Is $x_i$ closer to $x_j$ or $x_k$?'' We establish formal guarantees on the query complexity required to learn smooth, but otherwise general, distance functions under two notions of approximation: $\omega$-additive approximation and $(1 + \omega)$-multiplicative approximation. For the additive approximation, we propose a global method whose query complexity is quadratic in the size of a finite cover of the sample space. For the (stronger) multiplicative approximation, we introduce a method that combines global and local approaches, utilizing multiple Mahalanobis distance functions to capture local geometry. This method has a query complexity that scales quadratically with both the size of the cover and the ambient space dimension of the sample space.

翻译：本文研究了在基于查询的学习框架下学习距离函数的问题，其中学习者能够提出如下形式的三元组查询：“$x_i$ 是否比 $x_k$ 更接近 $x_j$？” 针对两种近似概念——$\omega$-加性近似与$(1 + \omega)$-乘性近似，我们建立了学习平滑（但其他方面具有一般性）距离函数所需查询复杂度的严格保证。对于加性近似，我们提出了一种全局方法，其查询复杂度在样本空间有限覆盖的规模上呈二次方增长。对于（更强的）乘性近似，我们引入了一种结合全局与局部策略的方法，该方法利用多个马氏距离函数来捕捉局部几何结构。此方法的查询复杂度随覆盖规模与样本空间环境维度的乘积呈二次方增长。