Gaussian fuzzy theoretic analysis for variational learning of nested compositions

M. Kumar, S. Singh, B. Freudenthaler. Gaussian fuzzy theoretic analysis for variational learning of nested compositions. International Journal of Approximate Reasoning, volume 131, pages 1-29, DOI, 4, 2021.

  • Mohit Kumar
  • Sukhvir Singh
  • Bernhard Freudenthaler
JournalInternational Journal of Approximate Reasoning

This paper introduces a variational analysis approach to the learning of a deep model formed via a nested composition of mappings. The fuzzy sets, being characterized by Gaussian type of membership functions, are used to represent unknown functions associated to the layers of the model. The learning of the deep model would require a quantification of the uncertainties on the signals across the layers of the deep model. We derive analytically the mathematical expressions for membership functions using variational optimization to quantify the uncertainties on variables. The most significant feature of the learning approach is that all of the unobserved variables and parameters, associated to the deep model, are averaged out where the averages are computed taking into account the uncertainties (on variables and parameters). The uncertainties are quantified by means of fuzzy sets with membership functions optimally learned from the observed data. A rigorous mathematical treatment of the learning problem results in the development of a competitive classification algorithm. The study is a theoretical contribution to the field of fuzzy machine learning, nevertheless, offering practical machine learning algorithms.