Problem Statement

GMM provides probabilistic clustering with soft assignments, allowing each point to belong to multiple clusters with different probabilities. Currently missing from aprender.

Advantages:

• Soft clustering (probability distributions)
• Handles elliptical clusters (vs K-Means spherical)
• Probabilistic framework (can compute likelihood)
• Foundation for more advanced models

Proposed Solution

Implement GMM using Expectation-Maximization (EM) algorithm with EXTREME TDD.

Algorithm

Model: Mixture of k Gaussian distributions

• Each cluster: mean μ_k, covariance Σ_k, weight π_k

EM Algorithm:

1. E-step: Compute responsibilities (soft assignments)
   • γ_ik = P(cluster k | point i)
2. M-step: Update parameters
   • μ_k, Σ_k, π_k from weighted samples
3. Repeat until convergence

Implementation

Trait: UnsupervisedEstimator

API Design:

pub struct GaussianMixture {
    n_components: usize,
    covariance_type: CovarianceType,  // Full, Tied, Diag, Spherical
    means: Option<Matrix<f32>>,
    covariances: Option<Vec<Matrix<f32>>>,
    weights: Option<Vector<f32>>,
}

pub enum CovarianceType {
    Full,       // Each component has own full covariance matrix
    Tied,       // All components share same covariance
    Diag,       // Diagonal covariance (faster)
    Spherical,  // Isotropic covariance (like K-Means)
}

impl UnsupervisedEstimator for GaussianMixture {
    type Labels = Vec<usize>;  // Hard assignments
    fn fit(&mut self, x: &Matrix<f32>) -> Result<(), &'static str>;
    fn predict(&self, x: &Matrix<f32>) -> Vec<usize>;
}

// Additional methods
impl GaussianMixture {
    pub fn predict_proba(&self, x: &Matrix<f32>) -> Matrix<f32>;  // Soft assignments
    pub fn score(&self, x: &Matrix<f32>) -> f32;  // Log-likelihood
}

Success Criteria

• ✅ GaussianMixture with EM algorithm
• ✅ 4 covariance types
• ✅ predict() for hard assignments
• ✅ predict_proba() for soft assignments
• ✅ 18+ tests (including convergence tests)
• ✅ Zero clippy warnings
• ✅ Example: examples/gmm_clustering.rs

Estimated Effort

Timeline: 4-5 days
Complexity: High (matrix operations, numerical stability)