Generalization Analysis of CNNs for Classification on Spheres.

Area of Science:

• Machine Learning
• Computer Vision
• Artificial Intelligence

Background:

• Deep learning, particularly Convolutional Neural Networks (CNNs), excels at classification tasks.
• Theoretical understanding of CNN generalization ability remains limited.
• Binary classification on spheres presents challenges due to non-smooth target functions.

Purpose of the Study:

• To develop a theoretical generalization analysis for deep CNN algorithms in binary classification.
• To investigate the approximation capabilities of CNNs for non-smooth functions in L_p spaces.
• To establish generalization bounds and learning rates for excess misclassification error.

Main Methods:

• Function approximation in L_p spaces (1 ≤ p ≤ ∞) for non-smooth target functions.
• Utilizing efficient cubature formulas on spheres.
• Applying tools from spherical analysis and approximation theory.
• Deriving generalization bounds and learning rates for CNNs.

Main Results:

• Provided rates of L_p approximation for functions within Sobolev spaces.
• Established generalization bounds for deep CNN classification algorithms.
• Determined learning rates for the excess misclassification error of CNNs.

Conclusions:

• The study offers a novel theoretical framework for understanding deep CNN generalization.
• The findings contribute to the theoretical foundation of deep learning for classification tasks.
• The developed methods and bounds are applicable to binary classification problems on spheres.