Eigendecomposition-Free Training of Deep Networks for Linear Least-Square Problems.

Area of Science:

• Computer Vision
• Deep Learning
• Linear Algebra

Background:

• Classical computer vision problems often involve solving linear least-squares problems via eigendecomposition.
• Integrating geometric priors into deep learning can improve performance.
• Directly differentiating eigendecomposition in neural networks leads to numerical instability.

Purpose of the Study:

• To develop an eigendecomposition-free deep learning approach for computer vision.
• To enable explicit encoding of geometric principles within neural networks.
• To overcome the practical limitations of differentiating eigendecomposition.

Main Methods:

• Introduced a novel deep network training method that avoids explicit eigendecomposition.
• The loss function depends on the eigenvector associated with a zero eigenvalue of a predicted matrix.
• Validated the approach on outlier rejection and denoising tasks.

Main Results:

• The proposed method shows significantly improved robustness compared to explicit differentiation.
• Demonstrated state-of-the-art results on practical computer vision problems.
• Empirically observed better convergence properties.

Conclusions:

• The eigendecomposition-free approach offers a more stable and effective way to incorporate geometric constraints in deep learning.
• This method advances the state-of-the-art in computer vision tasks requiring geometric reasoning.