Decomposing neural networks as mappings of correlation functions

Physical Review Research, vol. 4

Abstract

Understanding the functional principles of information processing in deep neural networks continues to be a challenge, in particular for networks with trained and thus nonrandom weights. To address this issue, we study the mapping between probability distributions implemented by a deep feed-forward network. We characterize this mapping as an iterated transformation of distributions, where the nonlinearity in each layer transfers information between different orders of correlation functions. This allows us to identify essential statistics in the data, as well as different information representations that can be used by neural networks. Applied to an XOR task and to MNIST, we show that correlations up to second order predominantly capture the information processing in the internal layers, while the input layer also extracts higher-order correlations from the data. This analysis provides a quantitative and explainable perspective on classification.

Authors 6

1. Kirsten Fischer Aachen

   RWTH Aachen University · Forschungszentrum Jülich · Jülich Aachen Research Alliance

   Affiliation as printed

   Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, 52425 Jülich, Germany

   RWTH Aachen University, 52062 Aachen, Germany

2. Alexandre René Aachen Department of Physics Faculty 1

   RWTH Aachen University · Forschungszentrum Jülich · Jülich Aachen Research Alliance · University of Ottawa

   Affiliation as printed

   Department of Physics, Faculty 1, RWTH Aachen University, 52074 Aachen, Germany

   Department of Physics, University of Ottawa, K1N 6N5 Ottawa, Canada

   Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, 52425 Jülich, Germany

3. Christian Keup Aachen

   RWTH Aachen University · Forschungszentrum Jülich · Jülich Aachen Research Alliance

   Affiliation as printed

   Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, 52425 Jülich, Germany

   RWTH Aachen University, 52062 Aachen, Germany

4. Moritz Layer Aachen

   RWTH Aachen University · Forschungszentrum Jülich · Jülich Aachen Research Alliance

   Affiliation as printed

   Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, 52425 Jülich, Germany

   RWTH Aachen University, 52062 Aachen, Germany

5. David Dahmen

   Forschungszentrum Jülich · Jülich Aachen Research Alliance

   Affiliation as printed

   Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, 52425 Jülich, Germany

6. Moritz Helias Aachen Department of Physics Faculty 1

   RWTH Aachen University · Forschungszentrum Jülich · Jülich Aachen Research Alliance

   Affiliation as printed

   Department of Physics, Faculty 1, RWTH Aachen University, 52074 Aachen, Germany

   Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, 52425 Jülich, Germany