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@INPROCEEDINGS{Fischer:909968,
      author       = {Fischer, Kirsten and Rene, Alexandre and Keup, Christian
                      and Layer, Moritz and Dahmen, David and Helias, Moritz},
      title        = {{S}tatistical decomposition of neural networks:
                      {I}nformation transfer between correlation functions},
      reportid     = {FZJ-2022-03558},
      year         = {2022},
      abstract     = {Uncovering principles of information processing in neural
                      systems continues to be an active field of research. For the
                      visual system it is well known that it processes signals in
                      a hierarchical manner [1,2]. Commonly used models in machine
                      learning that perform hierarchical computations are
                      feed-forward networks. We here study deep feed-forward
                      networks with the aim of deducing general functional aspects
                      of such systems. These networks implement a mapping between
                      probability distributions, where the probability
                      distribution is iteratively transformed from layer to layer.
                      We develop a formalism for expressing signal transformations
                      in each layer as transfers of information between different
                      orders of correlation functions (see Fig. (a)). We show that
                      the processing within internal network layers is captured by
                      correlations up to second order. In addition, we demonstrate
                      how the input layer also extracts higher order correlations
                      from the data. Thus, by presenting different correlation
                      orders in the input, we identify key statistics in the data
                      (see Fig. (b)-(d)). As a next step, we consider recurrent
                      time-continuous networks, reminiscent of biological neuronal
                      networks (NeuralODEs, [3]). We derive a Fokker-Planck
                      equation describing the evolution of the probability
                      distribution. This formulation allows us to study
                      time-dependent information flow between different
                      interaction terms. In summary, this work provides insights
                      into functional principles of information processing in
                      neural networks.References:[1] Hubel, D. H., $\&$ Wiesel, T.
                      N. (1962). Receptive fields, binocular interaction and
                      functional architecture in the cat's visual cortex. The
                      Journal of physiology, 160(1), 106.[2] Zhuang, C., Yan, S.,
                      Nayebi, A., Schrimpf, M., Frank, M. C., DiCarlo, J. J., $\&$
                      Yamins, D. L. (2021). Unsupervised neural network models of
                      the ventral visual stream. Proceedings of the National
                      Academy of Sciences, 118(3), e2014196118.[3] Chen, R. T.,
                      Rubanova, Y., Bettencourt, J., $\&$ Duvenaud, D. K. (2018).
                      Neural ordinary differential equations. Advances in neural
                      information processing systems, 31.},
      month         = {Sep},
      date          = {2022-09-14},
      organization  = {Bernstein conference, Berlin
                       (Germany), 14 Sep 2022 - 16 Sep 2022},
      subtyp        = {After Call},
      cin          = {INM-6 / IAS-6 / INM-10},
      cid          = {I:(DE-Juel1)INM-6-20090406 / I:(DE-Juel1)IAS-6-20130828 /
                      I:(DE-Juel1)INM-10-20170113},
      pnm          = {5232 - Computational Principles (POF4-523) / 5234 -
                      Emerging NC Architectures (POF4-523) / RenormalizedFlows -
                      Transparent Deep Learning with Renormalized Flows
                      (BMBF-01IS19077A) / MSNN - Theory of multi-scale neuronal
                      networks (HGF-SMHB-2014-2018) / ACA - Advanced Computing
                      Architectures (SO-092) / neuroIC002 - Recurrence and
                      stochasticity for neuro-inspired computation
                      (EXS-SF-neuroIC002) / GRK 2416:  MultiSenses-MultiScales:
                      Novel approaches to decipher neural processing in
                      multisensory integration (368482240)},
      pid          = {G:(DE-HGF)POF4-5232 / G:(DE-HGF)POF4-5234 /
                      G:(DE-Juel-1)BMBF-01IS19077A /
                      G:(DE-Juel1)HGF-SMHB-2014-2018 / G:(DE-HGF)SO-092 /
                      G:(DE-82)EXS-SF-neuroIC002 / G:(GEPRIS)368482240},
      typ          = {PUB:(DE-HGF)24},
      url          = {https://juser.fz-juelich.de/record/909968},
}