An analysis of classical multidimensional scaling with applications to clustering

Area of Science:

• Statistics
• Data Science
• Machine Learning

Background:

• Classical multidimensional scaling (CMS) is a prevalent dimension reduction method.
• Theoretical analysis of CMS statistical performance is limited.
• Understanding CMS embedding quality is crucial for downstream tasks.

Purpose of the Study:

• To develop a theoretical framework for assessing the quality of samples embedded by CMS.
• To establish conditions for successful clustering of noisy data using CMS.
• To provide a foundation for advanced statistical analyses following CMS.

Main Methods:

• Developing a theoretical framework to analyze CMS embedding quality.
• Deriving signal-to-noise ratio scaling conditions for accurate clustering.
• Utilizing simulation studies to validate theoretical findings.

Main Results:

• Established theoretical conditions for CMS to recover cluster labels in noisy data.
• Demonstrated that derived scaling conditions are sharp and precise.
• Validated the methodology through simulations and real-world data.

Conclusions:

• CMS, when combined with distance-based clustering, can effectively recover cluster labels under specific signal-to-noise ratios.
• The theoretical framework provides a robust basis for analyzing CMS performance.
• The methodology shows promise for applications in gene expression, single-cell sequencing, and natural language processing.