Robust Generalized Low Rank Approximations of Matrices.

Area of Science:

• Data Science
• Matrix Analysis
• Machine Learning

Background:

• Low rank structure is key for dimensionality reduction, noise removal, and data completion.
• Generalized Low Rank Approximations of Matrices (GLRAM) is efficient but sensitive to noise and outliers.
• A robust version of GLRAM has been lacking.

Purpose of the Study:

• Introduce Robust GLRAM (RGLRAM) to address GLRAM's sensitivity to noise and outliers.
• Develop an efficient and robust method for low rank approximation.
• Extend the robust approach to tensor data.

Main Methods:

• Formulated RGLRAM as an l1-norm optimization problem.
• Employed Augmented Lagrange Multipliers (ALM) to solve the optimization problem.
• Derived an iterative scheme and analyzed its weak convergence.

Main Results:

• RGLRAM successfully recovers low rank and sparse components from synthetic data.
• Demonstrated superior denoising and compression performance on corrupted facial images.
• Investigated RGLRAM's sensitivity to initialization, generalization ability, and runtime.

Conclusions:

• RGLRAM provides a robust alternative to GLRAM, effectively handling noise and outliers.
• The method shows promise for applications in data denoising, compression, and tensor analysis.
• Further research can explore RGLRAM's generalization and optimization.