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Information Geometry and Neural Netowrks

This work delves into the intersection of information geometry and neural networks, emphasizing the role of orthogonal decomposition in analyzing firing rates and higher-order correlations in neural activity. It addresses algebraic singularities and the dynamics of learning in multiplayer perceptrons, offering a comprehensive framework for understanding neural coding and firing mechanisms. Utilizing tools from statistics and combinatorics, the paper presents critical insights into the Riemannian structure of probability distributions and dual affine connections, contributing to advancements in AI and neuroscience.

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Information Geometry and Neural Netowrks

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  1. Information Geometryand Neural Netowrks Shun-ichi Amari RIKEN Brain Science Institute Orthogonal decomposition of rates and (higher-order) correlations Synchronous firing and higher correlations Algebraic singularities caused by multiple stimuli Dynamics of learning in multiplayer perceptrons

  2. Information Geometry Systems Theory Information Theory Statistics Neural Networks Combinatorics Physics Information Sciences Math. AI Riemannian Manifold Dual Affine Connections Manifold of Probability Distributions

  3. Information Geometry ? Riemannian metric Dual affine connections

  4. Manifold of Probability Distributions

  5. Two Structures Riemannian metric and affine connection Fisher information

  6. Riemannian Structure

  7. AffineConnection covariant derivative straight line

  8. Independent Distributions

  9. Neural Firing ----firing rate ----covariance higher-order correlations orthogonal decomposition

  10. Riemannian metric dual affine connections Pythagoras theorem Dual geodesics Information Geometry of Higher-Order Correlations ----orthogonal decomposition

  11. Correlations of Neural Firing firing rates correlations orthogonal coordinates

  12. ００１１０００１０１１０１０１００１００１１０１０００１０１１０１００１０１０firing rates:correlation—covariance?

  13. Independent Distributions

  14. Pythagoras Theorem p correlations D[p:r] = D[p:q]+D[q:r] q r p,q: same marginals r,q: same correlations independent estimation correlation testing invariant under firing rates

  15. 01100101……. 110001011001……. 101000111100……. 1001 No pairwise correlations, Triplewise correlation

  16. Pythagoras Decomposition of KL Divergence only pairwise independent

  17. Higher-Order Correlations

  18. Synfiring andHigher-Order Correlations Amari, Nakahara, Wu, Sakai

  19. Population and Synfire Neurons

  20. Population and Synfire s

  21. Synfiring

  22. Bifurcation : independent---single delta peak pairwise correlated higher-order correlation！ r

  23. Shun-ichi Amari RIKEN Brain Science Institute amari@brain.riken.go.jp Collaborators: Si Wu Hiro Nakahara Field Theory of Population Coding

  24. Population Coding and Neural Field

  25. f (z-x) x Population Encoding r(z) z

  26. b Noise

  27. Probability Model

  28. Fisher information Cramer-Rao

  29. Fourier Analysis

  30. Fisher Information

  31. Dynamics of Neural Fields Shaping Detecting Decoding

  32. How the Brain Solves Singularity in Population Coding S. Amari and H. Nakahara RIKEN Brain Science Institute

  33. Neural Activity

  34. Parameter Space

  35. synfiring resolves singularity

  36. synfiring mechanism common multiplicative noise

  37. S.Amari and H.Nagaoka,Methods of Information GeometryAMS &Oxford Univ Press, 2000

  38. Mathematical Neurons

  39. ｙ Multilayer Perceptrons

  40. Multilayer Perceptron neuromanifold space of functions

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