Rigid geometry solves “curse of dimensionality” effects in clustering methods: An application to omics data

The quality of samples preserved long term at ultralow temperatures has not been adequately studied. To improve our understanding, we need a strategy to analyze protein degradation and metabolism at subfreezing temperatures. To do this, we obtained liquid chromatography-mass spectrometry (LC/MS) data of calculated protein signal intensities in HEK-293 cells. Our first attempt at directly clustering the values failed, most likely due to the so-called "curse of dimensionality". The clusters were not reproducible, and the outputs differed with different methods. By utilizing rigid geometry with a prime ideal I-adic (p-adic) metric, however, we rearranged the sample clusters into a meaningful and reproducible order, and the results were the same with each of the different clustering methods tested. Furthermore, we have also succeeded in application of this method to expression array data in similar situations. Thus, we eliminated the "curse of dimensionality" from the data set, at least in clustering methods. It is possible that our approach determines a characteristic value of systems that follow a Boltzmann distribution.