Geometric structures for statistical models

• Fisher-Rao manifolds (Riemannian)
• Hilbert/Birkhoff geometry (Finsler)
• Bregman manifolds (dual Hessian)
• dual α-statistical structures
• semi-Riemannian structures

• Fisher-Rao manifolds: 2403.10089, 2302.08175
  
  • Hyperbolic geometry:
    • Hyperbolic Voronoi diagrams: 0903.3287 1210.8234 2402.04163
    • Fisher-Rao multivariate Gaussian manifold: 2403.10089
    • Fisher-Rao Cauchy manifold: 2006.07020
    • Cayley-Klein geometry: Classification with mixtures of curved Mahalanobis metrics
  • Riemannian smallest enclosing ball: 1101.4718
• Finsler geometry: medians and means https://arxiv.org/abs/1011.6076
• Hilbert/Birkhoff geometry:
  • Simplex domain (categorical distributions): 2203.11434
  • Elliptope domain (correlation matrices): 1704.00454
  • Symmetric positive-definite cone (SPDs, MVNs): 2307.10644
  • Siegel domain, complex domain including SPD: 2004.08160
• Bregman manifolds:
  • Dual geometry of convex conjugate functions: 1910.03935
  • Monte Carlo information geometry (when potential functions are not in closed form): 1803.07225
  • Applications to mixture families (Jensen-Shannon centroid): 1912.00610
  • Applications to exponential families (Chernoff information): 2207.03745 2207.03745
• Dual statistical structures:
  • PDF "The many faces of information geometry." Not. Am. Math. Soc 69.1 (2022): 36-45.
  • PDF "An elementary introduction to information geometry." Entropy 22.10 (2020): 1100.
• Semi-Riemannian structures (stochastic NNs, singularities): 1905.11027
• Aitchison geometry (compositional data, CODA): 2203.11434

Statistical models

• Hyperboloid and Poincaré Distributions (2205.13984)
• Product Jacobi-Theta Boltzmann machines (2303.05910)
• Lattice Gaussian distributions (2109.14920)