Group Invariant Scattering (1101.2286v3)

Abstract: This paper constructs translation invariant operators on L2(R^d), which are Lipschitz continuous to the action of diffeomorphisms. A scattering propagator is a path ordered product of non-linear and non-commuting operators, each of which computes the modulus of a wavelet transform. A local integration defines a windowed scattering transform, which is proved to be Lipschitz continuous to the action of diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform which is translation invariant. Scattering coefficients also provide representations of stationary processes. Expected values depend upon high order moments and can discriminate processes having the same power spectrum. Scattering operators are extended on L2 (G), where G is a compact Lie group, and are invariant under the action of G. Combining a scattering on L2(R^d) and on Ld (SO(d)) defines a translation and rotation invariant scattering on L2(R^d).

• The paper presents a scattering propagator that constructs translation invariant operators using cascaded wavelet modulus and local integration.
• It demonstrates Lipschitz continuity with respect to small diffeomorphisms, ensuring stable and robust performance in signal analysis.
• It extends the framework to compact Lie groups, achieving combined translation and rotation invariance for enhanced signal classification.

Group Invariant Scattering

The paper "Group Invariant Scattering" by Stéphane Mallat introduces a robust framework for constructing translation invariant operators on $L^2(\mathbb{R}^d)$, which are Lipschitz continuous with respect to the action of diffeomorphisms. This framework is predicated on the construction of a scattering propagator, a path ordered product of non-linear and non-commuting operators that compute the modulus of a wavelet transform. Through local integration, a windowed scattering transform that is translation invariant is defined. This transform extends to build invariants under the action of compact Lie groups and rotations, developing a dual scattering mechanism on $L^2(SO(d))$ to achieve rotation and translation invariance in $L^2(\mathbb{R}^d)$.

Summary of Key Contributions

1. Translation and Lipschitz Continuity:
   • The paper first addresses constructing translation invariant operators that maintain Lipschitz continuity to the action of small diffeomorphisms. It does so by splitting frequencies into dyadic packets via a wavelet transform which, although not inherently translation invariant, upon applying a scattering procedure that combines convolutions and modulus operations along multiple paths, achieves a translation invariant operator.
2. Scattering Propagator:
   • A scattering propagator is defined as an ordered path product of the modulus of wavelet transforms. The resulting windowed scattering transform is shown to preserve norms and is Lipschitz continuous to $C^2$ diffeomorphisms.
3. High Order Moments and Stationary Processes:
   • It is demonstrated that scattering coefficients provide robust representations of stationary processes. These coefficients depend on high order moments and can discriminate between processes with identical power spectra.
4. Extension to Lie Groups:
   • The framework is extended to $L^2(G)$ where $G$ is a compact Lie group. Scattering transforms invariant to actions of $G$ are built, enabling a combined invariant scattering transform on $L^2(\mathbb{R}^d)$ and $L^2(SO(d))$ to achieve both translation and rotation invariance.
5. Expected Scattering and Random Deformations:
   • The paper further extends the notion of scattering to the context of stochastic processes. It defines the expected scattering transform and explores its continuity under random deformations. This section underscores how the proposed scattering mechanism adds resilience to random deformations, highlighting its application potential in signal and image analysis where stochastic deformations are prevalent.

Numerical Results and Computational Considerations

The numerical experiments provided demonstrate the energy propagation among dyadic frequency packets, illustrating how scattering operators transform the structured information within signals. The decay of path energy for various structured and noise signals illustrates the transform's ability to maintain high frequency resolutions through interference patterns captured by cascading modulus computations over paths.

Implications for Theoretical and Practical Developments

The theoretical implications of this work are profound. The insights into constructing invariants for a broad class of transformations open potential advancements for signal processing, particularly in fields requiring robustness to geometric transformations. The use of wavelet transforms cascaded through scattering mechanisms provides a rigorous way to analyze high-dimensional signals while maintaining critical invariants.

From a practical standpoint, the applications discussed, including audio and image classification, underscore the utility of invariant scattering transforms. The tools and software provided, accessible via the referenced repository, facilitate further exploration and application by the research community.

The conjectures on convergence and path classifications provide avenues for future research. Specifically, validating the conjecture on the scattering transform's strong convergence and its implications on the theoretical bounds of discriminative capabilities remains an engaging challenge.

Conclusion

Stéphane Mallat’s "Group Invariant Scattering" proposes a structured approach to constructing invariant transforms under translation, rotation, and compact group actions while preserving Lipschitz continuity to small diffeomorphism deformations. This work bridges gaps between theoretical signal processing and practical applications, offering a robust toolset for complex signal classification and analysis, paving the way for advancements in the understanding and processing of high-dimensional, invariant signal representations.