Sufficient dimension reduction for classification using principal optimal transport direction

Area of Science:

• Statistics
• Machine Learning
• Data Science

Background:

• Sufficient dimension reduction (SDR) is a key supervised dimension reduction technique.
• Existing SDR methods often perform poorly with categorical responses, particularly binary ones.
• There is a need for robust SDR methods applicable to diverse data types.

Purpose of the Study:

• To propose a novel method for estimating the sufficient dimension reduction subspace (SDR subspace) for categorical response data.
• To address the limitations of current SDR methods when dealing with binary or categorical outcomes.
• To establish a connection between sufficient dimension reduction, support vector machines, and optimal transport.

Main Methods:

• Developed a new estimation method for the SDR subspace using optimal transport.
• Introduced the Principal Optimal Transport Direction (POTD) method.
• Estimated the SDR subspace basis using principal directions of optimal transport coupling between data categories.

Main Results:

• POTD effectively estimates the SDR subspace for categorical response data.
• The study reveals theoretical links between SDR, support vector machines, and optimal transport.
• Asymptotic analysis confirms POTD's exclusive estimation of the SDR subspace under error-free class labels.
• Empirical evaluations demonstrate POTD's superior performance compared to state-of-the-art linear dimension reduction methods.

Conclusions:

• Principal Optimal Transport Direction (POTD) offers a powerful new approach for sufficient dimension reduction with categorical data.
• The POTD method provides a robust and effective alternative to existing techniques, especially for binary response variables.
• This research bridges concepts in optimal transport and dimension reduction, opening avenues for further statistical and machine learning advancements.