How unsupervised learning can help us detecting phase transitions and emergent phenomena. 2D Ising Model and PCA.
This article was inspired in the paper Discovering Phase Transitions with Unsupervised Learning by Lei Wang [1].
One of the most intriguing phenomena within the natural sciences is phase transition or critical phenomena, which can usually be seen in the transition between states of matter such as solid, liquid and gas. However, it is a phenomenon that transcends the usual and connects the brain, a sheet of ferromagnetic material and population dynamics, among others [2,3]. However, not always those emergent patterns of the phase transitions are easily detectable. Therefore, unsupervised learning techniques can be employed to help with this task. In this article, I'll use as a toy model, for the phase transition recognition with Principal Component Analysis (PCA), one of the simplest models for critical phenomena analysis, a.k.a the 2D Ising model. It was introduced in 1920 by Wilhelm Lenz to describe ferromagnetic materials in a simplified way.
[1] Lei Wang. (2016). Discovering Phase Transitions with Unsupervised Learning. Physical Review B, 94(19),195105. https://doi.org/10.1103/PhysRevB.94.195105
[2] Krkošek, M., & Drake, J. M. (2014). On signals of phase transitions in salmon population dynamics. Proceedings. Biological sciences, 281(1784), 20133221. https://doi.org/10.1098/rspb.2013.3221
[3] Steyn-Ross, M. L., Steyn-Ross, D. A., & Sleigh, J. W. (2004). Modelling general anaesthesia as a first-order phase transition in the cortex. Progress in biophysics and molecular biology, 85(2–3), 369–385. https://doi.org/10.1016/j.pbiomolbio.2004.02.001