Example: Histopathology

We demonstrate the practical utility of Group CNNs with an application to Histopathology: the detection of mitotic cells in tumor tissue sections.

1. The Problem: Mitosis Detection

Pathologists diagnose cancer grades by counting "mitotic figures"—cells undergoing division. These cells have a distinct appearance but can be arbitrarily rotated. A "healthy" cell or a "mitotic" cell remains effectively the same biological entity regardless of its orientation under the microscope.

2. Architecture: Invariant via Equivariant

Although the final task is invariant, we solve it by being equivariant throughout the network.

• Lifting Layer: The input patch is lifted to a group feature map (positions + orientations).
• Group Convolutions: We process these maps, detecting patterns of patterns at specific relative poses. Point-wise non-linearities (like ReLU) are applied, which preserve equivariance.
• Projection (Pooling): Only at the very end, we perform a Global Max Pooling over the rotation axis. This collapses the group feature map to a spatial map (or a single vector) that is invariant to the input's rotation.

3. Results: Data Efficiency & Stability

We compared Regular Group CNNs (G-CNN) against standard CNNs trained with data augmentation.

Sample Efficiency

G-CNNs are significantly more sample-efficient. A G-CNN trained on only 25% of the data achieves comparable or better performance than a standard CNN trained on 100% of the data with augmentation. This is because the G-CNN shares weights across rotations—it doesn't need to "learn" that a rotated edge is still an edge; it knows this by design.

Geometric Stability

If we rotate the input image and look at the predicted class probability:

• Standard CNN: The prediction fluctuates wildly (e.g., Healthy -> Mitotic -> Healthy) as the image rotates.
• Groups CNN: The prediction remains stable and constant across rotations (up to discretization artifacts).

4. Other Domains

The same principles apply to other domains:

• Medical Imaging: Vessel segmentation, lung nodule detection (scale equivariance).
• Audio Analysis: Pitch shifts in audio (e.g., a word spoken with high vs. low pitch) can be modeled as scaling of the waveform. Scale-equivariant CNNs can recognize sounds regardless of pitch.