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Efficient computation of robust weighted low-rank matrix approximations using the L1 norm
Anders Eriksson1, Anton van den Hengel
1Department of Computer Science, The University of Adelaide, North Terrace, SA 5005, Australia. anders.eriksson@adelaide.edu.au
Summary
This study introduces a novel low-rank matrix approximation method, a generalization of the Wiberg algorithm, designed to handle outliers and missing data effectively. The new approach minimizes the L1 norm for robust matrix factorization, outperforming traditional methods in practical scenarios.
Area of Science:
- Computer Vision
- Numerical Analysis
- Optimization
Background:
- Low-rank matrix approximation is crucial for many computer vision algorithms.
- Singular Value Decomposition (SVD) is a common method but fails with data outliers or missing values.
- Real-world data frequently contains such imperfections, limiting SVD's applicability.
Purpose of the Study:
- To develop a robust low-rank matrix approximation method adaptable to imperfect data.
- To generalize the Wiberg algorithm for enhanced matrix factorization.
- To minimize the L1 norm in rank-constrained factorization, accommodating missing data.
Main Methods:
- The proposed method generalizes the Wiberg algorithm.
- It formulates rank-constrained factorization minimizing the L1 norm.
- Exploits the differentiability of linear programs for efficient computation.
Main Results:
- The algorithm efficiently handles matrices with outliers and missing elements.
- Experimental results on synthetic and real data demonstrate effectiveness.
- The method is implementable using existing optimization software.
Conclusions:
- The presented algorithm offers a robust alternative to SVD for low-rank approximation.
- It effectively addresses the common challenges of outliers and missing data.
- The approach provides a computationally efficient solution for practical applications.
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