Sampling spread is reduced with larger numbers of shapes #2489
Description
Running the same parameters with a small number of shapes (e.g. 6) and a large number of shapes (e.g. 139) results in a very different amount of particle spreading during initialization:
6 shapes:
139 shapes:
Some theories on why this is the case:
- This might be caused by the mean energy correspondence formulation becoming effectively stiffer as the number of shapes increases, because the mean shape becomes increasingly rigid and unresponsive to individual particle movements.
When a particle on shape s moves by delta, the mean shifts by only delta/N. The effective correspondence gradient that shape s's particle sees is:
x_s - mean ≈ ((N-1)/N) * (deviation of shape s from other shapes)
| Shapes (N) | Mean shift per particle move | Effective gradient retention |
|---|---|---|
| 6 | 1/6 = 16.7% | 5/6 = 83.3% |
| 139 | 1/139 = 0.7% | 138/139 = 99.3% |
With 6 shapes, the mean is elastic -- it follows particles as they spread, reducing the pull-back force. With 139 shapes, the mean is rigid -- it barely moves, and each particle faces a strong pull back toward a nearly fixed target.
Surface diversity amplifies the effect
The rigid mean becomes even more problematic with diverse shapes:
- With few similar shapes, particles on all shapes tend to spread in similar directions (surfaces are alike). The mean moves with them, and the correspondence gradient stays small.
- With many diverse shapes, particles on each shape want to spread in shape-specific directions dictated by different surface geometries. The mean averages these conflicting movements and barely moves in any particular direction. Every shape's particles face a strong pull-back toward this nearly-fixed mean.