Physics-Informed Matrix Factorization Operator.

Area of Science:

• Machine Learning
• Physics-Informed Machine Learning
• Data Science

Background:

• Matrix factorization is a core machine learning technique, but is sensitive to data quality and noise.
• Existing methods rely on mathematical decomposition, lacking physical interpretability and robustness.
• Noise in data can significantly degrade the performance and reliability of matrix factorization models.

Purpose of the Study:

• To introduce a novel Physics-informed Matrix Factorization (PiMF) operator.
• To enhance matrix factorization by incorporating physical laws, specifically the law of conservation of energy.
• To improve the robustness and interpretability of matrix factorization, particularly for noisy data.

Main Methods:

• Developed the PiMF operator using the heat conduction equation to formulate an energy objective function.
• Ensured the PiMF operator retains the decomposition meaning of mathematical models while satisfying physical interpretability.
• Demonstrated the consistency between the energy objective function and the mathematical model for feasibility verification.

Main Results:

• The PiMF operator effectively suppresses noise by adhering to physical principles.
• Solutions derived from PiMF incorporate both mathematical and physical knowledge, enhancing generalization for complex and noisy data.
• Experimental results for classification and clustering tasks show significant advantages of PiMF, especially on noisy datasets.

Conclusions:

• Physics-informed Matrix Factorization (PiMF) offers a robust and interpretable alternative to traditional methods.
• The energy decline perspective validates the physical interpretability of the PiMF operator.
• PiMF enhances the practicability of matrix factorization, proving its value for real-world applications with noisy data.