A convex model for nonnegative matrix factorization and dimensionality reduction on physical space.

Area of Science:

• Data science
• Machine learning
• Convex optimization

Background:

• Matrix factorization is crucial for data analysis.
• Learning meaningful dictionaries from data is challenging.
• Existing methods often lack interpretability or require constraints.

Purpose of the Study:

• To propose a collaborative convex framework for nonnegative matrix factorization (NMF).
• To develop a method for learning a physically meaningful dictionary from data.
• To apply the framework to hyperspectral endmember identification and blind source separation.

Main Methods:

• A collaborative convex framework for factoring data matrix X into AS.
• Using l(1, ∞) regularization for dictionary selection from data X.
• Employing alternating minimization initialized by the convex model solution to relax constraints.

Main Results:

• The framework guarantees a physically meaningful dictionary and dimensionality reduction.
• l(1, ∞) regularization provides an exact convex relaxation of l(0) for distinct noise-free data.
• The method successfully identifies hyperspectral endmembers and abundances and separates NMR data.

Conclusions:

• The proposed convex framework offers an effective approach for sparse matrix factorization and dictionary learning.
• The method provides interpretable results and is applicable to diverse scientific domains.
• This work advances data analysis techniques in hyperspectral imaging and biomedical signal processing.