On Generalizations of the Nonwindowed Scattering Transform.

Area of Science:

• Mathematics
• Signal Processing
• Harmonic Analysis

Background:

• Wavelet scattering transforms are powerful tools for analyzing signals.
• Existing methods have limitations in handling certain transformations like rotations and diffeomorphisms.
• Generalizing these transforms is crucial for broader applications in machine learning and data analysis.

Purpose of the Study:

• To generalize finite depth wavelet scattering transforms.
• To provide rigorous mathematical formulations and proofs for these generalized transforms.
• To extend the framework to handle rotational symmetries and actions of diffeomorphisms.

Main Methods:

• Formulating wavelet scattering transforms as norms of cascaded continuous wavelet transforms and nonlinearities.
• Developing and proving norms for the resulting operators.
• Demonstrating Lipschitz continuity with respect to diffeomorphisms in specific settings.
• Constructing rotation-invariant and rotation-equivariant operators.

Main Results:

• Generalization of finite depth wavelet scattering transforms.
• Well-definedness and Lipschitz continuity of the generalized operators under specific conditions.
• Development of operators invariant and equivariant to rotations.
• Provides a theoretical foundation for applying scattering transforms in more complex scenarios.

Conclusions:

• The generalized wavelet scattering transforms offer enhanced capabilities for signal analysis.
• The theoretical framework supports the development of more robust and versatile signal processing tools.
• This work paves the way for improved feature extraction in the presence of geometric transformations.