Many techniques in machine learning attempt explicitly or implicitly to infer a low-dimensional manifold structure of an underlying physical phenomenon from measurements without an explicit model of the phenomenon or the measurement apparatus. This paper presents a cautionary tale regarding the discrepancy between the geometry of measurements and the geometry of the underlying phenomenon in a benign setting. The deformation in the metric illustrated in this paper is mathematically straightforward and unavoidable in the general case, and it is only one of several similar effects. While this is not always problematic, we provide an example of an arguably standard and harmless data processing procedure where this effect leads to an incorrect answer to a seemingly simple question. Although we focus on manifold learning, these issues apply broadly to dimensionality reduction and unsupervised learning.

本文通过一个简单的例子，警告我们在使用机器学习等技术进行测量时需要注意度量代表的物理现象与测量工具的度量值并不相同；本文提供的样例表明，在某些情况下这一问题可能导致我们对问题做出不正确的回答，从而提示我们在使用机器学习等技术进行测量时需要更加谨慎地进行数据处理。

论柏拉图洞中的流形学习——兼论流形学习与物理现象