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Area of Science:

  • Computational neuroscience
  • Machine learning theory
  • Dynamical systems theory

Background:

  • Deep neural networks (DNNs) exhibit complex behaviors.
  • Understanding their internal decision-making processes is crucial.
  • Input perturbations can significantly alter DNN outputs.

Purpose of the Study:

  • To analyze the effect of small input perturbations on DNN outputs.
  • To draw an analogy between DNNs and dynamical systems.
  • To visualize the geometric structures learned by DNNs.

Main Methods:

  • Calculating finite-time Lyapunov exponents for DNNs.
  • Exploring the geometrical interpretation of these exponents in input space.
  • Identifying structures related to classification boundaries.

Main Results:

  • Maximal Lyapunov exponents form geometric structures in input space.
  • These structures resemble coherent structures found in dynamical systems.
  • Ridges of high positive exponents delineate regions associated with different classes.

Conclusions:

  • DNNs construct specific geometries in input space during learning.
  • Lyapunov exponents provide a tool to visualize and understand these learned geometries.
  • This approach sheds light on the fundamental mechanisms of DNN learning.