Interactive Composition Operators An Alternative Approach for Selecting Linear Embedding Parameters

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 infinite 2$n$-dimensional parameter space. Once an informative projection is found, users often seek similar ones that emphasize specific 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 efficiently 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 specific 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 redefined. We provide closed-form solutions for this and demonstrate our interactive prototype on nD datasets from the UCI repository.