Transcendental Algorithm for Mixtures of Distributions

Implements the Transcendental Algorithm for Mixtures of Distributions (TAMD), a penalized likelihood framework for fitting finite Gaussian mixture models. TAMD augments the Expectation-Maximization (EM) algorithm with analytic barrier terms built from the Hellinger affinity that diverge on the singular locus, actively preventing component coalescence and weight degeneracy. Provides the core TAMD fitting function, closed-form Hellinger affinity and gradient computations, the Transcendental Affinity Criterion (TAC) for geometry-aware model selection, the regularity index rho (a scalar diagnostic for mixture fit quality), and reproduction scripts for all simulation studies. Methods are described in Fokoue (2024) . See also Titterington, Smith and Makov (1985, ISBN:0-471-90510-4) and Watanabe (2009, ISBN:978-0-521-86408-7).

tamd: Transcendental Algorithm for Mixtures of Distributions

](https://github.com/efokoue/tamd)

Dedicated to the memory of Professor D.M. Titterington (1945–2023),
University of Glasgow, whose foundational contributions to mixture
model theory made every question here worth asking.

Overview

tamd implements the Transcendental Algorithm for Mixtures of Distributions (TAMD), a penalized likelihood framework for fitting finite Gaussian mixture models. TAMD augments the EM algorithm with analytic barrier terms built from the Hellinger affinity between components, which diverge precisely on the singular locus of the mixture likelihood, preventing component coalescence and weight degeneracy.

Key features

• Stable fitting: Transcendental barriers prevent degenerate solutions
• Theoretically grounded: Five new theorems connecting barrier geometry to Watanabe's singular learning theory
• TAC criterion: Geometry-aware model selection; consistently selects K₀; dominates BIC
• Regularity index ρ: Scalar diagnostic for solution quality (free, computed from fit output)
• Closed-form Gaussian updates: Explicit Hellinger affinity gradients; efficient implementation
• Fully reproducible: Fixed seeds; reproduction scripts for all paper experiments

Installation

# From GitHub
devtools::install_github("efokoue/tamd")

Quick start

library(tamd)

# Simulate a 3-component mixture
X <- simulate_gmm(
  n     = 400,
  K     = 3,
  pi    = c(0.3, 0.4, 0.3),
  mu    = matrix(c(-3, 0,  0, 0,  3, 0), nrow = 2),
  Sigma = array(rep(diag(2), 3), dim = c(2, 2, 3)),
  seed  = 42
)

# Fit TAMD
fit <- tamd(X, K = 3, seed = 42)
print(fit)
#  K = 3 components  |  n = 400  |  d = 2
#  Regularity index rho = 0.9412  [RELIABLE]
#  TAC = 3241.7  |  BIC = 3298.4

# Automated model selection via TAC
sel <- tamd_select(X, K_min = 1, K_max = 6, seed = 42)
cat("TAC selects K =", sel$K_hat, "\n")  # K = 3

# Visualize
plot(fit, X = X)

The regularity index ρ

The regularity index ρ ∈ (0,1) summarizes how well-separated the fitted components are (Titterington Theorem, Fokoué 2024):

TAC vs BIC vs WBIC

TAC uses a geometry-corrected effective parameter count p_eff ∈ (2λ_unpen, dθ), data-adaptive, computed directly from the Hellinger affinities at the fitted solution.

Package structure

tamd/
├── R/
│   ├── tamd.R          # Core fitting function
│   ├── hellinger.R     # Affinity and gradient computations
│   ├── criteria.R      # TAC, TIC, BIC, AIC, rho
│   ├── methods.R       # print/summary/plot, simulate_gmm
│   └── utils.R         # Internal helpers
├── tests/
│   └── testthat/
│       └── test-tamd.R # 18 unit tests
├── vignettes/
│   └── tamd-intro.Rmd  # Full introduction
└── inst/reproduce/     # Paper reproduction scripts
    ├── E1_regularity.R
    ├── E2_transversality.R
    ├── E3_rlct.R
    ├── E4_tac.R
    ├── E5_rho.R
    ├── E6_failures.R
    ├── real_oldfaithful.R
    └── real_galaxy.R

Reproducing the paper

All simulation experiments in Fokoué (2024) are exactly reproducible:

# From the R console:
source(system.file("reproduce/E1_regularity.R",    package = "tamd"))
source(system.file("reproduce/E4_tac.R",           package = "tamd"))
source(system.file("reproduce/real_oldfaithful.R", package = "tamd"))

Or from the command line:

Rscript inst/reproduce/E1_regularity.R

Citation

@article{fokoue2024tamd,
  author  = {Fokou\'{e}, Ernest},
  title   = {The Transcendental Algorithm for Mixtures
             of Distributions},
  journal = {Annals of Statistics},
  year    = {2024},
  note    = {Submitted}
}

@Manual{tamd2024package,
  title  = {tamd: Transcendental Algorithm for Mixtures
            of Distributions},
  author = {Fokou\'{e}, Ernest},
  year   = {2024},
  note   = {R package version 1.0.0},
  url    = {https://github.com/efokoue/tamd}
}

References

Fokoué, E. (2024). The Transcendental Algorithm for Mixtures of Distributions. Annals of Statistics, submitted.

Titterington, D.M., Smith, A.F.M., and Makov, U.E. (1985). Statistical Analysis of Finite Mixture Distributions. Wiley.

Watanabe, S. (2009). Algebraic Geometry and Statistical Learning Theory. Cambridge University Press.

Yamazaki, K. and Watanabe, S. (2003). Singularities in mixture models and upper bounds of stochastic complexity. Neural Networks, 16(7), 1029–1038.