I wrote my master's thesis at the Insitute of Theoretical Physics of the University of Heidelberg, at the Institute for Physics of the University of Freiburg and was supported by (as well as located at) the Institute for Frontier Areas of Psychology and Mental Health in Freiburg.
In the thesis I develop the possibility to treat Neural Networks (NN) in an algebraic way, akin to physical theories, based on which I explore the range of "non-classical" behavior which may arise in the dynamics of the NN. Also, a mathematical method which may amend conventional tools in analyzing NN-data is constructed.
Title: The mathematical structure of measurements, observables and states on Neural Networks
Abstract: This thesis studies the mathematical consequences of a notion of "measurement" on NNs, which consists of both presentation of an input and observation of the output. Once a definition of NNs is chosen, which is general enough to cover a multitude of different NN-models, observables are constructed which represent measurements abstractly. They are found to behave (in light of the fact of NNs being classical systems) unexpected, which is a consequence of this choice of measurement. The properties of the observables are studied. E.g., it is evaluated whether they form a C*-algebra, which is expected from both classical and non-classical physical systems. In a second part, the relation between the notion of state, which is generated by this concept of measurement, and several consecutive measurements is investigated. Mathematical concepts are proposed which relate the two. This might ultimately amend mathematical tools in experimental situations.
Be invited to have a look into the thesis: Download.