On Manifold Learning in Plato's Cave: Remarks on Manifold Learning and Physical Phenomena

Area of Science:

• Machine Learning
• Data Science
• Applied Mathematics

Background:

• Many machine learning techniques infer low-dimensional manifold structures from data.
• These methods often lack explicit models of the physical phenomenon or measurement apparatus.
• Understanding the relationship between measurement geometry and phenomenon geometry is crucial.

Purpose of the Study:

• To present a cautionary example of metric deformation in manifold learning.
• To illustrate how data processing can introduce discrepancies between measurement and phenomenon geometry.
• To demonstrate potential pitfalls in dimensionality reduction and unsupervised learning.

Main Methods:

• Mathematical analysis of metric deformation in measurement spaces.
• Illustrative example using a standard data processing procedure.
• Focus on manifold learning techniques.

Main Results:

• A straightforward and generalizable metric deformation effect was identified.
• A benign data processing step was shown to yield incorrect results due to this effect.
• The discrepancy between measurement and underlying phenomenon geometry is non-trivial.

Conclusions:

• The geometry of measurements can significantly differ from the geometry of the underlying phenomenon.
• Standard data processing can inadvertently introduce errors in manifold learning.
• These issues are broadly relevant to dimensionality reduction and unsupervised learning.