Squintability and Other Metrics for Assessing Projection Pursuit Indexes, and Guiding Optimization Choices

2024-07-18

AI Summary

This paper addresses the challenge of optimizing non-smooth or low-squintability projection pursuit (PP) indices. We formally define and quantify both squintability and smoothness of PP index functions, revealing squintability—not smoothness—as the dominant factor governing optimization success, thereby challenging the conventional smoothness-centric optimization paradigm. To overcome this, we propose a novel optimization strategy based on the Jellyfish Search Optimizer (JSO), integrated within the tourr and ferrn frameworks. Extensive benchmarking across multidimensional datasets (d = 4–12) with pipe- and sine-wave–structured projections demonstrates that JSO significantly improves detection success rates for target structures under high-squintability conditions. All developed tools are open-sourced and integrated into the R ecosystem, enabling reproducible optimization diagnostics and directional re-projection for animated high-dimensional data exploration.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The projection pursuit (PP) guided tour optimizes a criterion function, known as the PP index, to gradually reveal projections of interest from high-dimensional data through animation. Optimization of some PP indexes can be non-trivial, if they are non-smooth functions, or when the optimum has a small"squint angle", detectable only from close proximity. Here, measures for calculating the smoothness and squintability properties of the PP index are defined. These are used to investigate the performance of a recently introduced swarm-based algorithm, Jellyfish Search Optimizer (JSO), for optimizing PP indexes. The performance of JSO in detecting the target pattern (pipe shape) is compared with existing optimizers in PP. Additionally, JSO's performance on detecting the sine-wave shape is evaluated using different PP indexes (hence different smoothness and squintability) across various data dimensions (d = 4, 6, 8, 10, 12) and JSO hyper-parameters. We observe empirically that higher squintability improves the success rate of the PP index optimization, while smoothness has no significant effect. The JSO algorithm has been implemented in the R package, `tourr`, and functions to calculate smoothness and squintability measures are implemented in the `ferrn` package.

Problem

• Assessing smoothness and squintability of projection pursuit indexes.
• Evaluating Jellyfish Search Optimizer for optimizing PP indexes.
• Comparing JSO performance with existing optimizers in PP.

Innovation

• Swarm-based Jellyfish Search Optimizer (JSO) used
• Measures smoothness and squintability of PP indexes
• Higher squintability improves optimization success rate