Local coordinates alignment with global preservation for dimensionality reduction.

Area of Science:

• Data Science
• Machine Learning
• Computer Vision

Background:

• Nonlinear dimensionality reduction (NLDR) is crucial for analyzing high-dimensional data.
• Alignment-based NLDR methods map data to low-dimensional spaces while preserving local isometry.
• Existing methods struggle to fully maintain relationships between data patches during alignment.

Purpose of the Study:

• To propose a novel NLDR method, local coordinates alignment with global preservation (LCAGP).
• To enhance the preservation of global dataset structure in NLDR.
• To address limitations of existing alignment-based NLDR techniques.

Main Methods:

• Introduction of topology-preserving landmarks (TPLs) for global structure preservation and patch construction.
• Derivation of low-dimensional TPL coordinates using an existing global structure-preserving method.
• Alignment of local patch coordinates in tangent spaces to a global space using landmarks as references (landmarks-based alignment).

Main Results:

• The proposed landmarks-based alignment offers a closed-form solution without constraints.
• It overcomes deficiencies in global metrics and undesired rescaling common in methods with unit covariance constraints.
• Experiments on synthetic and real-world data confirm the algorithm's effectiveness.

Conclusions:

• LCAGP effectively preserves both local and global data structures in nonlinear dimensionality reduction.
• The method provides a robust and constraint-free alignment approach.
• This advancement offers improved performance for high-dimensional data analysis.