GMM Soft Assignment

A Gaussian mixture model (GMM) models the data distribution as a weighted sum of $K$ Gaussian components, $p(x) = \sum_{k=1}^{K} \pi_k \mathcal{N}(x \mid \mu_k, \Sigma_k)$, with mixing coefficients satisfying $\sum_k \pi_k = 1, \ \pi_k \ge 0$.Parameters are estimated with EM: the E-step computes responsibilities (posteriors, i.e. soft assignments) $\gamma_{ik} = \frac{\pi_k \mathcal{N}(x_i \mid \mu_k, \Sigma_k)}{\sum_{j=1}^{K} \pi_j \mathcal{N}(x_i \mid \mu_j, \Sigma_j)}$; the M-step re-estimates $N_k = \sum_i \gamma_{ik}$, $\pi_k = \tfrac{N_k}{N}$, $\mu_k = \tfrac{1}{N_k} \sum_i \gamma_{ik} x_i$, and $\Sigma_k = \tfrac{1}{N_k} \sum_i \gamma_{ik} (x_i - \mu_k)(x_i - \mu_k)^T$.As $\Sigma_k = \sigma^2 I$ with $\sigma \to 0$, responsibilities collapse to hard 0/1 assignments and the EM update reduces to cluster means — K-Means is exactly the zero-variance EM special case of GMM.

soft clustering and density estimation, anomaly detection (low-likelihood samples), speaker recognition, background modeling, and as an upgrade when K-Means underperforms; interviews ask for the responsibility formula and the K-Means-as-GMM derivation.

K-Means only produces hard clusters and assumes spherical shapes; GMM outputs a posterior probability for every sample (confidence and uncertainty), and per-component covariances fit elliptical clusters of varying scale and orientation. The cost: far more parameters — $d + \tfrac{d(d+1)}{2}$ covariance parameters per Gaussian — a non-convex likelihood that needs K-Means++ initialization and constrained covariances (diagonal/shared/regularized) to avoid singular degeneracy.

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