Stanford CS229 Machine Learning | Spring 2026 | Lecture 10: GMM (EM), PCA
This lecture provides a deep dive into two foundational unsupervised machine learning techniques: Gaussian Mixture Models (GMM) solved via the Expectation-Maximization (EM) algorithm, and Principal Component Analysis (PCA). The discussion emphasizes the mathematical underpinnings of these methods. For GMM, the EM algorithm is shown to solve for latent variables by constructing a tractable lower bound using Jensen's inequality. For PCA, the technique identifies directions of maximum variance by performing Eigen Decomposition on the data's covariance matrix, highlighting the critical need for proper data centering and scaling.
Key takeaways
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GMM via EM Algorithm
23:20
The Expectation-Maximization (EM) algorithm is used to estimate parameters ($ heta$) for GMM. It operates by constructing a lower bound $L( heta|T)$ of the log-likelihood, which allows optimization through alternating steps: E-step (calculating soft assignments/probabilities $Q_i(Z)$) and M-step (re-estimating parameters $oldsymbol{ heta}$).
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PCA Core Principle
1:03:20
PCA is a dimensionality reduction technique that finds orthogonal unit vectors (principal components, $U_k$) corresponding to the directions of maximal variance in the data. The process relies on finding the Eigen Decomposition ($oldsymbol{A} = oldsymbol{U} oldsymbol{ ext{diag}}(oldsymbol{ ext{eigenvalues}}) oldsymbol{U}^T$) of the covariance matrix.
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Mathematical Rigor (Jensen's Inequality)
27:10
The EM algorithm leverages Jensen's inequality to transform an intractable expectation into a manageable lower bound, enabling iterative optimization. The selection of the conditional probability $Q(Z)$ is key to achieving this tight lower bound.