On the Selection of Tuning Parameters for Patch-Stitching Embedding Methods

TL;DR

This paper introduces a stress-minimization approach for selecting tuning parameters in patch-stitching embedding methods, addressing the challenge of unsupervised parameter tuning in multidimensional scaling and manifold learning.

Contribution

It proposes a simple, stress-based criterion for tuning parameter selection and uncovers a new bias-variance tradeoff phenomenon in embedding methods.

Findings

Stress minimization effectively guides parameter choice.

Identification of a bias-variance tradeoff in patch-stitching embeddings.

Application to both network localization and manifold learning.

Abstract

While classical scaling, just like principal component analysis, is parameter-free, other methods for embedding multivariate data require the selection of one or several tuning parameters. This tuning can be difficult due to the unsupervised nature of the situation. We propose a simple, almost obvious, approach to supervise the choice of tuning parameter(s): minimize a notion of stress. We apply this approach to the selection of the patch size in a prototypical patch-stitching embedding method, both in the multidimensional scaling (aka network localization) setting and in the dimensionality reduction (aka manifold learning) setting. In our study, we uncover a new bias--variance tradeoff phenomenon.