L₁-Norm Low-Rank Matrix Decomposition by Neural Networks and Mollifiers.

Area of Science:

• Numerical Analysis
• Machine Learning
• Computer Vision

Background:

• Low-rank approximation with missing entries is crucial in data analysis but challenging due to the non-smooth L1-norm cost function.
• Existing optimization methods for this problem are often complex or inefficient, hindering practical applications.

Purpose of the Study:

• To develop an efficient and accurate optimization method for low-rank matrix approximation with missing entries.
• To address the non-smoothness of the L1-norm cost function and improve computational performance.

Main Methods:

• Smoothing the non-smooth L1-norm cost function using a mollifier with tunable parameters.
• Employing a recurrent neural network to optimize the smoothed cost function.
• Implementing the mollifying process via a filtering procedure for enhanced speed.

Main Results:

• The proposed method demonstrates competitive performance against state-of-the-art techniques on synthetic datasets.
• Analysis reveals one mollifier parameter is critical for efficiency and accuracy, while the other is less influential.
• The algorithm exhibits mild memory requirements, suitable for large-scale matrix decomposition.

Conclusions:

• The novel smoothed cost function and recurrent neural network approach effectively optimizes low-rank matrix approximation with missing data.
• The method is computationally efficient and memory-friendly, making it viable for real-world applications like structure from motion.