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Learning Objectives
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Now · 1. Gaussian Mixture Models
Learning Objectives
- Understand GMMs for acoustic modeling
- Apply EM algorithm for GMM training
- Use GMM-HMM hybrid systems
- Gaussian distributions
- Week 4: HMMs
1. Gaussian Mixture Models
GMMs model complex probability distributions as weighted sum of Gaussians.
Components:
- K: Number of mixtures (typically 16-64 per state)
- pik: Component weights (sum to 1)
- muk: Mean vectors
- Sigmak: Covariance matrices (often diagonal for efficiency) Why GMMs? Single Gaussian can't capture complex, multi-modal distribution of speech features. GMMs can approximate any distribution with enough components.
2. EM Algorithm for GMM Training
E-step: Compute responsibility of each component for each data point.
M-step: Update parameters using weighted statistics.
- pik = sum(gamma_nk) / N
- muk = sum(gamma_nk * xn) / sum(gamma_nk)
- Sigmak = sum(gamma_nk * (xn-muk)(xn-muk)^T) / sum(gamma_nk)
3. GMM-HMM Architecture
Each HMM state uses a GMM for observation probability. Combined:
- HMM: Models temporal sequence (state transitions)
- GMM: Models acoustic variability (observation likelihood)
- Together: Powerful model for speech recognition
Q1: Why use GMMs instead of a single Gaussian per state?Speech features are multi-modal (different speakers, contexts produce different clusters of features). A single Gaussian can't capture this. GMMs can model complex, multi-modal distributions. Q2: What does the EM algorithm do for GMMs?Iteratively estimates GMM parameters from data. E-step: assign soft membership of each point to each component. M-step: update parameters (weights, means, covariances) based on membership. Converges to local optimum. Q3: Why use diagonal covariance matrices in GMMs?Full covariance has O(D^2) parameters per component (D = feature dimension ~39). Diagonal has O(D). Much fewer parameters, less data needed, faster computation. Features are decorrelated by DCT (MFCC extraction) making diagonal reasonable. Q4: What is the difference between GMM and GMM-HMM?GMM alone: models static distribution of features (bag of frames). GMM-HMM: models sequence, with each state using GMM. HMM adds temporal information - which feature distributions occur in which order. Q5: How many Gaussian components are typically used per state?Typically 16-64 components per state. More components = better modeling but more parameters. Selection depends on: amount of training data, computational resources, complexity of acoustic variability. Join Discord PreviousHidden Markov Models for SpeechNextAutomatic Speech Recognition (ASR)