Interactive Composition Operators: An Alternative Approach for Selecting Linear Embedding Parameters

Keywords

Star Coordinates, Multivariate Projections, Composition Operators, Multidimensional Data

Abstract

Linear embeddings support interactive visual exploration by mapping high-dimensional (nD) data into a two-dimensional space. Despite their popularity, selecting meaningful projection parameters remains a key challenge due to the infi nite 2n-dimensional parameter space. Once an informative projection is found, users often seek similar ones that emphasize specifi c items differently while preserving global structure. For instance: Do clusters become outliers under slight changes? Can grouped items separate—or merge—through parameter adjustments? Which changes to the embedding parameters lead to such projections — and do they exist at all? Answering these questions effi ciently is critical for effective visual search. Yet, current methods—such as projection tours or manual parameter tuning—are time-consuming and risk overlooking important views, including those of specifi c interest. We propose Composition Operators, a mathematical foundation for a novel set-of-point manipulation concept for linear embeddings—such as Star Coordinates—as an alternative approach to selecting informative embedding parameters in a more controllable manner with respect to the desired outcome. Users specify item-based constraints on the projection result; the corresponding 2n parameters are then derived automatically, eliminating the need to exhaustively search the entire parameter space to get a similar outcome. Neither the embedding space nor the set of parameters is altered – only the mechanism for navigating and selecting parameters is redefi ned. We provide closed-form solutions for this and demonstrate our interactive prototype on nD datasets from the UCI repository.