Some comparisons of complexity in dictionary-based and linear computational models.

Area of Science:

• Computational mathematics
• Machine learning theory

Background:

• Traditional linear regression uses fixed function sets, limiting flexibility.
• Neural networks offer adaptable function approximation by adjusting function parameters.

Purpose of the Study:

• Compare model complexity requirements for linear vs. variable-basis approximators.
• Analyze neural networks' performance against linear models in function approximation.
• Investigate theoretical advantages of neural networks in high-dimensional settings.

Main Methods:

• Nonlinear approximation techniques.
• Integral representations tailored for computational units.
• Comparison of worst-case error bounds for variable-basis and linear approximators.

Main Results:

• Neural networks exhibit lower model complexity than linear models.
• Variable-basis models, including neural networks, can outperform linear approximators.
• Theoretical analysis supports neural network superiority in specific approximation cases.

Conclusions:

• Neural networks provide a more flexible and often more accurate function approximation method.
• Despite optimization difficulties, neural networks' reduced complexity is advantageous, particularly in high dimensions.
• Variable-basis models represent a powerful alternative to traditional linear approximation methods.