  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.
