Structure-based hyperparameter selection with Bayesian optimization in multidimensional scaling

We introduce the structure optimized proximity scaling (STOPS) framework for hyperparameter selection in parametrized multidimensional scaling and extensions (proximity scaling; PS). The selection process for hyperparameters is based on the idea that we want the configuration to show a certain structural quality (c-structuredness). A number of structures and how to measure them are discussed. We combine the structural quality by means of c-structuredness indices with the PS badness-of-fit measure in a multi-objective scalarization approach, yielding the Stoploss objective. Computationally we suggest a profile-type algorithm that first solves the PS problem and then uses Stoploss in an outer step to optimize over the hyperparameters. Bayesian optimization with treed Gaussian processes as a an apt and efficient strategy for carrying out the outer optimization is recommended. This way, hyperparameter tuning for many instances of PS is covered in a single conceptual framework. We illustrate the use of the STOPS framework with three data examples.

Automated summary

In this paper, the authors introduce the structure optimized proximity scaling (STOPS) framework for hyperparameter selection in parametrized multidimensional scaling and extensions (proximity scaling; PS). By combining c-structuredness indices with the PS badness-of-fit measure, a multi-objective scalarization approach called Stoploss objective is proposed. A profile-type algorithm is suggested for the computational implementation, and Bayesian optimization with treed Gaussian processes is recommended for outer hyperparameter optimization. The STOPS framework is demonstrated with three data examples.