Generalized Matrix Local Low Rank Representation by Random Projection and Submatrix Propagation

Area of Science:

• Data Science
• Computational Mathematics
• Machine Learning

Background:

• Matrix low rank approximation reduces data redundancy.
• Local methods are superior to global methods (e.g., SVD) for uncovering interpretable structures.
• Existing local methods fail to detect patterns with diverse mean structures.

Purpose of the Study:

• Introduce a novel computational framework, Random Probing based submatrix Propagation (RPSP).
• Address the limitations of current methods in detecting general local low rank patterns.
• Provide an effective solution for the general matrix local low rank representation problem.

Main Methods:

• RPSP detects local low rank patterns by propagating from small low rank submatrices.
• The initial submatrices are identified using a random projection approach.
• Theoretical underpinnings are based on random projection theories.

Main Results:

• RPSP outperforms state-of-the-art methods on synthetic datasets.
• The method robustly identifies low rank matrices with similar means to the background.
• RPSP effectively handles heteroscedastic noise and multiple co-existing patterns.

Conclusions:

• RPSP offers a robust and effective solution for general local low rank representation.
• The method demonstrates significant improvements over existing techniques.
• RPSP successfully identifies interpretable local low rank matrices in real-world applications.