Steerable Networks

Why Does This Exist?

Discrete group CNNs click into four orientations and improvise between them. For applications where every angle occurs — rotated text, aerial tracking, molecular imaging — those gaps are where the errors live. Finer sampling (eight, sixteen orientations) closes the gaps by brute force and multiplies memory each time.

Steerable networks solve rotation continuously. Filters are built from basis functions with known rotation behavior (circular harmonics), so steering a filter to any angle is an exact change of coefficients, not an interpolation. The discrete sibling is group equivariant CNNs; the shared principle is geometric deep learning.

The one idea you need first. A steerable basis is a small filter set closed under rotation: rotating any basis filter yields a linear mix of the set, never something outside it. The classic order-one basis on the circle is $\cos\theta$ and $\sin\theta$. A filter $w(\theta) = a\cos\theta + b\sin\theta$ steered to direction $\phi$ is exactly $\cos(\theta - \phi)$, obtained by setting $a = \cos\phi$ and $b = \sin\phi$ — no resampling, no approximation.

Think of It Like This

How It Actually Works

Each layer stores features as geometric types: scalars (unchanged by rotation), vectors (rotate as arrows), and higher-order tensors. Convolution kernels are expanded in a basis matched to the input and output types, with learnable coefficients per basis element. Because the basis transforms exactly, the whole layer commutes with continuous rotation.

Nonlinearities must respect types: a pointwise ReLU applied to vector components independently breaks equivariance, so steerable networks use norm-based gating (scale the vector by a function of its length, which preserves direction). Pooling over orientation at the end yields invariance when the task needs it.

Worked example

Steer the order-one filter to $\phi = 30$ degrees. Coefficients: $a = \cos 30^\circ \approx 0.866$, $b = \sin 30^\circ = 0.5$. The steered filter is $w(\theta) = 0.866\cos\theta + 0.5\sin\theta$. Evaluate where it peaks by testing $\theta = 30^\circ$:

$w(30^\circ) = 0.866 \times 0.866 + 0.5 \times 0.5 = 0.75 + 0.25 = 1.0$

Test $\theta = 120^\circ$ (ninety degrees away): $\cos 120^\circ = -0.5$, $\sin 120^\circ \approx 0.866$, so $w = 0.866 \times (-0.5) + 0.5 \times 0.866 = -0.433 + 0.433 = 0$. Peak exactly at the steering angle, zero exactly orthogonal — the continuous guarantee in two lines of arithmetic.

Code

The point of this snippet is the steering law: two coefficients reproduce the filter at any angle, and the numbers confirm the peak and the null.

1import math2
3
4def steered_response(theta_deg: float, phi_deg: float) -> float:5    a = math.cos(math.radians(phi_deg))  # basis weights for the target angle6    b = math.sin(math.radians(phi_deg))7    t = math.radians(theta_deg)8    return a * math.cos(t) + b * math.sin(t)9
10
11print(round(steered_response(30, 30), 6))  # -> 1.012print(round(steered_response(120, 30), 6))  # -> 0.0

Watch Out For

The Quick Version

• Steerable networks expand filters in rotation-closed bases, so any angle is an exact coefficient remix.
• The worked fixture steers to $30$ degrees with weights $(0.866, 0.5)$, peaking at $1.0$ on target and $0$ orthogonal.
• Unlike discrete group CNNs, there are no gaps between sampled angles.
• Sharp patterns need higher harmonics, each costing channels, so match basis order to the data.
• Nonlinearities must preserve feature type; plain per-component ReLU voids the guarantee.

What to Read Next

• Group Equivariant CNNs
• Spherical CNNs
• Geometric and Equivariant Deep Learning