steerable cnns
**Steerable CNNs** are **convolutional neural networks where filters are defined as linear combinations of a steerable basis — typically Gaussian derivatives, circular harmonics, or spherical harmonics — enabling the output feature maps to be analytically rotated to any orientation without pixel resampling or interpolation artifacts** — providing exact continuous rotation equivariance by construction rather than the approximate discrete rotation equivariance achieved through data augmentation or filter rotation on pixel grids.
**What Are Steerable CNNs?**
- **Definition**: A steerable CNN (Cohen & Welling, 2016; Weiler & Cesa, 2019) constrains convolutional filters to be linear combinations of a steerable basis — a set of basis functions with the mathematical property that rotating the function is equivalent to applying a known linear transformation to the expansion coefficients. This means the output of a steerable filter at any rotation can be computed by transforming the coefficients rather than re-evaluating the filter at rotated pixel positions.
- **Steerable Basis**: In 2D, circular harmonics ($e^{im heta}$) form a steerable basis — rotating a circular harmonic of order $m$ by angle $alpha$ simply multiplies the coefficient by $e^{imalpha}$. In 3D, spherical harmonics ($Y_l^m$) play the same role. Restricting filters to these bases guarantees that rotation acts linearly and predictably on feature maps.
- **Feature Field Types**: Steerable CNNs classify features by their rotation order — scalar fields (type-0, invariant under rotation), vector fields (type-1, rotate as 2D/3D vectors), and higher-order tensor fields (type-2+, rotate as tensors). Each layer maps between specified feature types through constrained kernel spaces that preserve equivariance.
**Why Steerable CNNs Matter**
- **Exact Continuous Equivariance**: Standard approaches to rotation equivariance — data augmentation (training with rotated copies) or discrete group convolution (testing 4 or 8 orientations) — are approximate. Steerable CNNs provide exact equivariance for all continuous rotations, including the infinite orientations between discrete samples, through mathematical construction rather than empirical approximation.
- **Orientation Detection**: Steerable CNNs produce oriented feature maps that encode not just "what is here" but "what is here and which direction is it pointing." This is critical for medical imaging (detecting oriented structures like blood vessels, muscle fibers), satellite imagery (road direction, river flow), and texture analysis (fabric weave, crystal grain orientation).
- **Parameter Efficiency**: By constraining filters to the steerable basis, the effective parameter count is reduced — the network does not waste capacity learning the same pattern at multiple orientations independently. A steerable filter with $K$ basis functions replaces $R imes K$ parameters in a rotation-augmented standard CNN, where $R$ is the number of rotation samples.
- **Mathematical Foundation**: Steerable CNNs are grounded in representation theory of symmetry groups, providing a principled mathematical framework for understanding what "equivariance" means and how to achieve it. This theoretical foundation enables systematic construction of equivariant architectures for arbitrary symmetry groups.
**Steerable CNN Features**
| Feature Type | Rotation Behavior | Physical Analog |
|-------------|-------------------|-----------------|
| **Type-0 (Scalar)** | Invariant — no change under rotation | Temperature, pressure, energy |
| **Type-1 (Vector)** | Rotates as a 2D/3D vector | Velocity, force, gradient |
| **Type-2 (Matrix)** | Rotates as a rank-2 tensor | Stress, strain, diffusion tensor |
| **Type-$l$ (General)** | Transforms via Wigner D-matrices of order $l$ | Multipole moments, angular distributions |
**Steerable CNNs** are **mathematically rotating filters** — analyzing orientation at every spatial position with infinite angular precision by exploiting the algebraic structure of rotation groups, providing the exact continuous equivariance that approximate methods can only estimate.