Visual Diagnostics for Constrained Optimisation with Application to Guided Tours

TL;DR

This paper introduces visual diagnostics for constrained optimization in projection pursuit, enhancing the guided tour method for high-dimensional data visualization and providing tools to diagnose and improve optimization procedures.

Contribution

It develops new visual diagnostic tools and a data object for analyzing constrained optimization in guided tours, aiding in understanding and fixing optimization issues.

Findings

Diagnostics revealed issues with existing optimization methods.

Implemented diagnostics improved optimization performance.

Tools are available in the R package ferrn.

Abstract

A guided tour helps to visualise high-dimensional data by showing low-dimensional projections along a projection pursuit optimisation path. Projection pursuit is a generalisation of principal component analysis, in the sense that different indexes are used to define the interestingness of the projected data. While much work has been done in developing new indexes in the literature, less has been done on understanding the optimisation. Index functions can be noisy, might have multiple local maxima as well as an optimal maximum, and are constrained to generate orthonormal projection frames, which complicates the optimization. In addition, projection pursuit is primarily used for exploratory data analysis, and finding the local maxima is also useful. The guided tour is especially useful for exploration, because it conducts geodesic interpolation connecting steps in the optimisation and shows how the projected data changes as a maxima is approached. This work provides new visual diagnostics for examining a choice of optimisation procedure, based on the provision of a new data object which collects information throughout the optimisation. It has helped to diagnose and fix several problems with projection pursuit guided tour. This work might be useful more broadly for diagnosing optimisers, and comparing their performance. The diagnostics are implemented in the R package, ferrn.