General, Nested, and Constrained Wiberg Minimization.

Area of Science:

• Optimization
• Computer Vision
• Machine Learning

Background:

• The Wiberg matrix factorization efficiently decomposes matrices by optimizing only one factor.
• Extensions to L1 minimization have been developed, but applicability to more general functions is limited.

Purpose of the Study:

• To generalize the Wiberg algorithm for minimizing L1 norms of general nonlinear functions f(U, V).
• To introduce nested Wiberg algorithms for complex optimization problems like L1 projective bundle adjustment.
• To extend Wiberg minimization to handle nonlinear constraints, demonstrated with Constrained Wiberg Minimization for Multiple Instance Learning (CWM-MIL).

Main Methods:

• Generalizing the Wiberg approach to minimize ||Y - f(U, V)||1 for functions nonlinear in two variable sets.
• Developing nested Wiberg algorithms for problems with three sets of variables, such as L1 projective bundle adjustment.
• Implementing Constrained Wiberg Minimization for Multiple Instance Learning (CWM-MIL) to handle nonlinear constraints.

Main Results:

• Demonstrated a practical Wiberg algorithm for L1 bundle adjustment.
• Showcased a nested Wiberg algorithm for L1 projective bundle adjustment, solving for camera matrices, points, and projective depths.
• Validated CWM-MIL for efficiently removing variables from constrained optimization problems.

Conclusions:

• The generalized Wiberg algorithm effectively handles L1 minimization for a broader class of nonlinear functions.
• Nested and constrained Wiberg methods offer efficient solutions for complex computer vision and machine learning tasks.
• Experimental comparisons against successive linear programming highlight the advantages of the Wiberg approach.