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@INPROCEEDINGS{Bouss:1007805,
      author       = {Bouss, Peter and Nestler, Sandra and Rene, Alexandre and
                      Helias, Moritz},
      title        = {{D}imensionality reduction with normalizing flows},
      school       = {RWTH Aachen},
      reportid     = {FZJ-2023-02199},
      year         = {2022},
      note         = {Copyright: © (2022) Bouss P, Nestler S, René A, Helias M},
      abstract     = {Despite the large number of active neurons in the cortex,
                      for various brain regions, the activity of neural
                      populations is expected to live on a low-dimensional
                      manifold [1]. Among the most common tools to estimate the
                      mapping to this manifold, along with its dimension, are many
                      variants of principal component analysis [2]. Despite their
                      apparent success, these procedures have the disadvantage
                      that they assume only linear correlations and that their
                      performance, when used as a generative model, is poor.To be
                      able to fully learn the statistics of neural activity and to
                      generate artificial samples, we make use of normalizing
                      flows (NFs) [3, 4, 5]. These neural networks learn a
                      dimension-preserving estimator of the data probability
                      distribution. They are outstanding in comparison to
                      generative adversarial networks (GANs) and variational
                      autoencoders (VAEs) for their simplicity ‒ only one
                      invertible network is learned ‒ and for their exact
                      estimation of the likelihood due to tractable Jacobians at
                      each building block.We aim to modify NFs such that they can
                      discriminate relevant (in manifold) from noise (out of
                      manifold) dimensions. To this end, we penalize the
                      participation of each single latent variable in the
                      reconstruction of the data through the inverse mapping
                      (following a different reasoning than [6]). We can thus not
                      only give an estimate of the dimensionality of the activity
                      sub-space but also describe the underlying manifold without
                      the need to discard any information.We prove the validity of
                      our modification on controlled data sets of different
                      complexity. We emphasize, in particular, differences between
                      affine and additive coupling layers in normalizing flows
                      [7], and show that the former lead to pathologies when the
                      data topology is non-trivial, or when the data set is
                      composed of classes with different volumes. We further
                      illustrate the power of our modified NFs by reconstructing
                      data using only a few dimensions.We finally apply this
                      technique to identify manifolds in EEG recordings from a
                      dataset showing high gamma activity (described in [8]),
                      obtained from 128 electrodes during four different movement
                      tasks.AcknowledgementsThis project is funded by the Deutsche
                      Forschungsgemeinschaft (DFG, German Research Foundation) -
                      368482240/GRK2416; and by the German Federal Ministry for
                      Education and Research (BMBF Grant 01IS19077A to
                      Jülich).References [1] Gao, P., Trautmann, E., Yu, B.,
                      Santhanam, G., Ryu, S., Shenoy, K., $\&$ Ganguli, S. (2017).
                      A theory of multineuronal dimensionality, dynamics and
                      measurement. BioRxiv, 214262., 10.1101/214262 [2] Gallego,
                      J. A., Perich, M. G., Miller, L. E., $\&$ Solla, S. A.
                      (2017). Neural manifolds for the control of movement.
                      Neuron, 94(5), 978-984., 10.1016/j.neuron.2017.05.025 [3]
                      Dinh, L., Krueger, D., $\&$ Bengio, Y. (2014). Nice:
                      Non-linear independent components estimation. arXiv preprint
                      arXiv:1410.8516., 10.48550/arXiv.1410.8516 [4] Dinh, L.,
                      Sohl-Dickstein, J., $\&$ Bengio, S. (2016). Density
                      estimation using real nvp. arXiv preprint arXiv:1605.08803.,
                      10.48550/arXiv.1605.08803 [5] Kingma, D. P., $\&$ Dhariwal,
                      P. (2018). Glow: Generative flow with invertible 1x1
                      convolutions. Advances in neural information processing
                      systems, 31. [6] Cunningham, E., Cobb, A., $\&$ Jha, S.
                      (2022). Principal manifold flows. arXiv preprint
                      arXiv:2202.07037., 10.48550/arXiv.2202.07037 [7] Behrmann,
                      J., Vicol, P., Wang, K. C., Grosse, R., $\&$ Jacobsen, J. H.
                      (2021). Understanding and mitigating exploding inverses in
                      invertible neural networks. In International Conference on
                      Artificial Intelligence and Statistics (pp. 1792-1800).
                      PMLR. [8] Schirrmeister, R. T., Springenberg, J. T.,
                      Fiederer, L. D. J., Glasstetter, M., Eggensperger, K.,
                      Tangermann, M., ... $\&$ Ball, T. (2017). Deep learning with
                      convolutional neural networks for EEG decoding and
                      visualization. Human brain mapping, 38(11), 5391-5420.,
                      10.1002/hbm.23730},
      month         = {Sep},
      date          = {2022-09-13},
      organization  = {Bernstein Conference, Berlin
                       (Germany), 13 Sep 2022 - 16 Sep 2022},
      subtyp        = {After Call},
      keywords     = {Computational Neuroscience (Other) / Data analysis, machine
                      learning, neuroinformatics (Other)},
      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          = {5231 - Neuroscientific Foundations (POF4-523) / 5232 -
                      Computational Principles (POF4-523) / GRK 2416 - GRK 2416:
                      MultiSenses-MultiScales: Neue Ansätze zur Aufklärung
                      neuronaler multisensorischer Integration (368482240) /
                      RenormalizedFlows - Transparent Deep Learning with
                      Renormalized Flows (BMBF-01IS19077A)},
      pid          = {G:(DE-HGF)POF4-5231 / G:(DE-HGF)POF4-5232 /
                      G:(GEPRIS)368482240 / G:(DE-Juel-1)BMBF-01IS19077A},
      typ          = {PUB:(DE-HGF)24},
      doi          = {10.12751/NNCN.BC2022.104},
      url          = {https://juser.fz-juelich.de/record/1007805},
}