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.