Algebraic Approach To Data Processing: Techniques And Applications

In many areas of human knowledge, symmetries and invariances play an important role. In fundamental physics, starting with Relativity Theory, new physical theories have been formulated in terms of invariances and of the corresponding transformation groups – i.e., in terms what a mathematician would call an algebraic approach. In engineering, devices like wind tunnels, which are based on scale-invariance, enable us to test smaller-scale models of the actual designs. In biological sciences, symmetries and invariances are extremely important in analyzing the shape and functioning of living beings, from mammals to viruses. Invariance and symmetry – in the form of fairness – are an extremely important topic in social sciences. Because of ubiquity of invariances, it is reasonable to take them into account when processing data corresponding to different domains. In this thesis, we show, on examples from various application domains –physics, engineering, medicine, economics, social sciences, education, even mathematics itself – that the algebraic approach is indeed very helpful in data processing, both in providing theoretical justifications for heuristic techniques and in coming up with new more efficient data processing methods. We also show that algebraic approach is helpful not only in specific applications, but also in analyzing and developing computational methods leading to these applications – methods ranging from deep learning to fuzzy and probabilistic techniques to other promising techniques such as DNA computing.