I wonder if this could be used anywhere as an alternative to Lloyd's algorithm. I have a hobby project which uses Lloyd's to "relax" an icosohedral equal area mesh on the surface of a sphere [1]. I'm going to try to swap it out with an adaptation of this and see if it's much better or if I can afford more iterations with this method.
Edit: tried it, and it’s definitely not better for this use case. e-means wins by needing fewer passes over a huge dataset, but relaxing a mesh on a sphere has no dataset: each Voronoi cell’s centroid can be computed directly from its geometry. Mini-batches just swap that for a noisy sampled estimate of the same thing. It has the same objective and the same optimum, so at best it matches Lloyd. In practice it was much slower and left the grid noisy, and with small sample budgets it even tangled the mesh.
Since it's basically a linear interpolation between Lloyd's algorithm and not moving any points it will be at least as good but gains might be limited without careful tuning.
Edit: in fact you might end up going faster by moving the other way, set alpha = -0.1 and see what happens!
This is a great example of why p=np is sometimes the wrong question. For this np hard problem approximate solutions are good enough, and if you get them 45x faster that's even better!
The best class of algorithms in this category are those where you know for certain that one answer is impossible and the other is highly probable. Like Bloom Filters, and if you squint, consistent hashing. I wish we had more algorithms of this sort and books about them. If I were a professor I’d be tempted to propose teaching a class.
When I was just starting out, client-server was still a bit of a specialty, mostly reserved for gaming and large institutions. Most of my education was in single task computing. If the university hadn’t offered a class in distributed computing I wouldn’t have gotten much in that direction except in independent study.
In the time since then, we seem to have focused much less on seconds per task and more on tasks per second, which results in more concern about statistical spreads, and queuing mechanisms to manage the variability between a hit and a miss. It would be better if this sort of computation were more organized into a discipline.
My personal favourite is compressive sensing, sparse signals beat the Nyquist sampling limit, by what seems like an unbounded amount 1000x, 1000000x, etc, because all real world stuff we care about lives on a low dimensional manifold, so unlike the noise.
These kinds of algorithms tend to have equivalent descriptions as expectation maximisation or simulated annealing. I wonder if it's possible to understand their algorithm in that sense.
Though I suppose it may be as simple as expectation maximization with a friction term, since Sj = sj - a (sj - Sj).
Importantly they note that this update rule keeps the centroid within the convex hull, which is pretty important.
Edit: tried it, and it’s definitely not better for this use case. e-means wins by needing fewer passes over a huge dataset, but relaxing a mesh on a sphere has no dataset: each Voronoi cell’s centroid can be computed directly from its geometry. Mini-batches just swap that for a noisy sampled estimate of the same thing. It has the same objective and the same optimum, so at best it matches Lloyd. In practice it was much slower and left the grid noisy, and with small sample budgets it even tangled the mesh.
[1] https://webgcm.app/explain/grid#nudging-the-cells-into-shape
Edit: in fact you might end up going faster by moving the other way, set alpha = -0.1 and see what happens!
When I was just starting out, client-server was still a bit of a specialty, mostly reserved for gaming and large institutions. Most of my education was in single task computing. If the university hadn’t offered a class in distributed computing I wouldn’t have gotten much in that direction except in independent study.
In the time since then, we seem to have focused much less on seconds per task and more on tasks per second, which results in more concern about statistical spreads, and queuing mechanisms to manage the variability between a hit and a miss. It would be better if this sort of computation were more organized into a discipline.
That's impressive!
Though I suppose it may be as simple as expectation maximization with a friction term, since Sj = sj - a (sj - Sj).
Importantly they note that this update rule keeps the centroid within the convex hull, which is pretty important.
Hope it gets added to scipy - it’s much more available in the envs many use than facebooks own faiss library.