Feature Fields & Induced Representations

This lecture introduces the formal definition of Feature Fields and Induced Representations, which are the primary objects manipulated by Steerable CNNs.

1. Definition of a Feature Field

A feature field is a map $f: \mathbb{R}^d \to V$, assigning a feature vector $v \in V$ to every point in space $\mathbf{x} \in \mathbb{R}^d$.

This "Type" dictates how the local feature vectors transform when the global image is transformed.

2. Induced Representations

When we apply a geometric transformation $g$ (e.g., a rotation) to a feature field, two things happen simultaneously. This combined action is called the Induced Representation $\pi(g)$:

$$ [\pi(g)f](x) = \rho(g) \cdot f(g^{-1}x) $$

1. Domain Transformation ($g^{-1}x$): The field is moved to a new location. Pixel $x$ grabs the value from $g^{-1}x$.
2. Fiber Transformation ($\rho(g)$): The vector at that pixel is itself transformed (e.g., rotated) by the representation $\rho(g)$.

3. Intuition: Scalar vs. Vector Fields

Let's visualize this with two common examples:

• Scalar Field (Type-0): Think of a grayscale image or a temperature map. If you rotate the map, the values move to new locations, but the values themselves (temperature) don't change. Here, $\rho(g) = 1$ (trivial representation).
• Vector Field (Type-1): Think of a wind velocity map. If you rotate the map by 90 degrees, the vectors move to new locations, AND the vectors themselves must rotate by 90 degrees to remain physically consistent. Here, $\rho(g) = \mathbf{R}$ (standard rotation matrix).

4. Steerable Feature Fields

In Steerable CNNs, our feature maps are Steerable Feature Fields. The "fibers" (pixel vectors) are vectors of Fourier coefficients. The representation $\rho(g)$ is the block-diagonal matrix of Irreps we derived in the previous lectures.

This is mathematically equivalent to the "Regular" feature maps (functions on the group) but expressed in the Fourier basis.