Harmonic Networks

We conclude Module 2 by analyzing the seminal Harmonic Networks (H-Nets) paper by Worrall et al. (2017). This work was one of the first to successfully train deep steerable CNNs using the complex circular harmonic basis.

1. H-Nets as Steerable CNNs

Although the paper predates the general theory of "Steerable CNNs" (Weiler et al.), H-Nets are exactly a specific instance of this framework:

• Group: $SO(2)$ (2D discrete and continuous rotations).
• Basis: Complex Circular Harmonics $\Psi_m(\theta) = e^{im\theta}$.
• Features: Complex-valued feature maps. A complex number $z = r e^{i\phi}$ naturally encodes a magnitude $r$ (feature strength) and an orientation $\phi$.

2. The Equivariance Condition

The paper derives a constraint for rotation equivariance, which they call the Equivariance Condition. It states that if an input feature map has rotation frequency $m$ and the filter has rotation frequency $k$, the output feature map must have frequency $m+k$.

This is exactly the Clebsch-Gordan condition for the group $SO(2)$, where the tensor product of irreducible representations corresponds to addition of frequencies: $e^{im\theta} \cdot e^{ik\theta} = e^{i(m+k)\theta}$.

3. Significance and Results

Harmonic Networks demonstrated that it is possible to achieve state-of-the-art results on rotation-sensitive tasks (like Rotated MNIST) without data augmentation. By hard-coding the symmetry into the architecture, the network becomes extremely data-efficient and robust, generalizing to orientations it has never seen during training.

This concludes Module 2. In Module 3, we will generalize all these concepts to 3D and the group $SO(3)$.