becomes an empty set.
The result of this algorithm is a cluster groups defined by level curves ????ₐ, for a = 1, … , M, where M is the number of cluster sets.
The Algorithm in ActionIn all of these examples, we treat the data as elements in (x,p) space, e,g.
a pixels of a two dimensional image is a point (x,p).
The first example is using Hamiltonian-based clustering to flocks of birds, similar to the example presented in (Casagrande, Sassano, & Astolfi, 2012).
Birds are grouped into three different groups using the algorithm described above.
The second example is clustering spirals.
There are three spirals and we demonstrate how the clustering algorithm can identify and group data points in their spiral groups accordingly.
This is a demonstration of how using level sets of a Hamiltonian function can help group points into three distinct spirals.
The third example is an extension of the second one and it consists of clustering a spiral on a sphere.
This uses a modification to the Hamiltonian and Hamilton’s equation, making use of the Hamiltonian’s symmetry.
The result of such algorithm can be seen in the following figure:This example demonstrates how Hamiltonian-based clustering can be used for applications when the surface is curved.
This does not rely on transforming the points to a suitable space, the algorithm works directly on the surface of the sphere.
ConclusionUsing Hamiltonian dynamics is different than other methods, and it relies heavily on calculus and symplectic geometry, which I intentionally avoided, and with complex mathematics comes some various advantages.
This method is still nascent and there are so many research avenues need to be explored to develop algorithms for data mining.
One unexplored area is the role of conserved quantities and how they can be utilised potentially for object identification.
I am planning on working on these things in the foreseeable future, so watch this space!Thanks for reading!ReferencesD.
Sassano and A.
Astolfi, “Hamiltonian-Based Clustering: Algorithms for Static and Dynamic Clustering in Data Mining and Image Processing,” in IEEE Control Systems Magazine, vol.