Phase-transition component discovery by deterministic annealing

Tracks the number of distinct mixture centroids as a function of temperature under mass-constrained deterministic annealing (Rose, Gurewitz and Fox 1990), a physics-derived alternative to information-criterion model selection. The system starts at a high temperature where all k_max centroids collapse to the data centroid (a single effective component) and is cooled along a geometric schedule; at each critical temperature a centroid bifurcates, so the number of distinct centroids grows in steps. The temperatures at which it grows are the phase transitions, and the count occupying the widest temperature range is the discovered component number.

Usage

gmm_anneal_path(
  x,
  k_max = 8L,
  sigma = NULL,
  t_high = NULL,
  t_low = NULL,
  n_steps = 80L,
  n_inner = 30L,
  w = NULL,
  perturb = 0.02,
  merge_tol = 0.1,
  ridge_eps = 1e-06,
  seed = 1L
)

Arguments

x: A numeric n by p matrix of samples, or a gmm_target carrying a samples matrix. For regime (iii) targets, pass an importance-resampled draw.
k_max: Maximum number of centroids tracked (the discovered count is at most k_max).
sigma: Reference scale: the shared covariance is sigma^2 * I. When NULL (the default) sigma is 1, so the first critical temperature is the largest eigenvalue of the data covariance.
t_high, t_low: Top and bottom of the temperature schedule. When NULL they default to 3 * t_critical_analytic and 0.05 * t_critical_analytic, bracketing the bifurcation cascade.
n_steps: Number of temperatures on the geometric schedule.
n_inner: Fixed-point iterations run at each temperature.
w: Optional length-n vector of non-negative observation weights (e.g. importance weights). Defaults to uniform.
perturb: Symmetry-breaking perturbation, as a fraction of the data scale, applied to the centroids at each temperature.
merge_tol: Two centroids count as distinct when their distance exceeds merge_tol times the data scale.
ridge_eps: Ridge added to the reference covariance for stability.
seed: Optional integer seed for the perturbations (the result is deterministic given a seed).

Value

A list with elements path (a data frame of temperature, n_effective and free_energy), critical_temperatures (the temperatures at which the count increased), first_critical_temperature (the first such, or NA if none was detected), t_critical_analytic (\(\lambda_{\max}(\Sigma^{-1} C)\)), k_selected (the widest-plateau component count), lambda_max and sigma.

Details

The first bifurcation has a closed-form critical temperature \(T_c = \lambda_{\max}(\Sigma^{-1} C)\), where \(C\) is the (weighted) data covariance and \(\Sigma = \sigma^2 I\) the shared reference covariance. This value is returned as t_critical_analytic and serves as an independent analytic check on the empirically detected first transition. Subsequent transitions have no comparably simple closed form, and the count is a diagnostic rather than a guarantee.

Annealing fixes the component covariance to the reference \(\Sigma\) so the temperature is the only scale; this is the clean isotropic regime in which the critical temperature is exact. For robust fitting under free covariances, use anneal = TRUE on fit_em_samples() or fit_kld_em() instead.

References

Rose, K., Gurewitz, E. and Fox, G. C. (1990) Statistical mechanics and phase transitions in clustering. Physical Review Letters 65(8), 945–948. doi:10.1103/PhysRevLett.65.945

Examples

set.seed(1)
x <- rbind(
  matrix(stats::rnorm(120, mean = -4), ncol = 2),
  matrix(stats::rnorm(120, mean =  4), ncol = 2)
)
path <- gmm_anneal_path(x, k_max = 4L, n_steps = 40L)
path$k_selected
#> [1] 2
path$first_critical_temperature
#> [1] 26.82229