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Questions and Answers
What is the underlying principle of the standard algorithm for K-means clustering?
What is the underlying principle of the standard algorithm for K-means clustering?
Expectation-Maximization (EM)
In Gaussian Mixture Modeling (GMM), what is updated in the Expectation step?
In Gaussian Mixture Modeling (GMM), what is updated in the Expectation step?
Cluster labels based on Gaussian distribution density
How is the log-likelihood optimized in GMM?
How is the log-likelihood optimized in GMM?
By fitting multiple Gaussian distributions
What is the purpose of constraining the covariance matrix in Gaussian Mixture Modeling?
What is the purpose of constraining the covariance matrix in Gaussian Mixture Modeling?
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What does the Maximization Step in EM clustering involve?
What does the Maximization Step in EM clustering involve?
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How is the log likelihood improved in the generic EM clustering algorithm?
How is the log likelihood improved in the generic EM clustering algorithm?
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What are some numerical issues in Gaussian Mixture Modeling?
What are some numerical issues in Gaussian Mixture Modeling?
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How can we avoid singularities in Gaussian Mixture Modeling?
How can we avoid singularities in Gaussian Mixture Modeling?
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What is the benefit of using the covariance of the entire data set as a prior in Gaussian Mixture Modeling?
What is the benefit of using the covariance of the entire data set as a prior in Gaussian Mixture Modeling?
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How can we address getting stuck in local optima and saddle points in Gaussian Mixture Modeling?
How can we address getting stuck in local optima and saddle points in Gaussian Mixture Modeling?
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What is a key difference in the stopping criteria between Gaussian Mixture Modeling and K-means clustering?
What is a key difference in the stopping criteria between Gaussian Mixture Modeling and K-means clustering?
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Study Notes
Expectation-Maximization (EM) in Clustering
- EM is the underlying principle of the standard algorithm for k-means
- The algorithm consists of four steps:
- Choose initial model parameters
- Expect latent variables (e.g., cluster assignment) from the data
- Update to maximize the likelihood of observing the data
- Repeat steps 2-3 until a stopping condition holds
Gaussian Mixture Modeling (GMM)
- GMM is an extension of k-means that allows for different cluster shapes and densities
- The algorithm consists of four steps:
- Choose initial centers, unit covariance, and uniform weight
- Expect cluster labels based on Gaussian distribution density
- Update Gaussian models with the mean, covariance matrix, and weight
- Repeat steps 2-3 until a stopping condition holds
Objective of GMM
- Optimize the log-likelihood of observing the data
- Fit multiple Gaussian distributions to the data
Multivariate Gaussian Density
- The probability density function of a multivariate Gaussian is a function of the mean, covariance matrix, and weight
- The covariance matrix controls the shape of the cluster
- A symmetric and positive semi-definite covariance matrix is required
- The covariance matrix can be:
- Rotated ellipsoid (A)
- Diagonal (“variance matrix”) ellipsoid (B)
- Scaled unit matrix, spherical (C)
- Different for each cluster or the same for all clusters
E-M Optimization
- The Expectation Step uses Bayes' rule and the law of total probability
- The Maximization Step uses weighted mean and weighted covariance to recompute cluster model
EM Clustering Algorithm
- The generic EM clustering algorithm is:
- E-step: compute the probability of each data point belonging to each cluster
- M-step: update the cluster model parameters
- The log likelihood is improved by both the E-step and the M-step
Numerical Issues in GMM
- Numerical problems can occur in GMM, such as:
- Unstable computation of the inverse of the covariance matrix
- Singular covariance matrix
- Numerical issues in computing the covariances
- Getting stuck in local optima and saddle points
- Improvements can be implemented to avoid these issues, such as:
- Adding a small constant to the diagonal of the covariance matrix to avoid singularities
- Using the covariance of the entire data set as prior
- Using numerically stable approaches for computing the covariance matrix
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Description
Test your knowledge of Expectation-Maximization (EM), the underlying principle of the standard algorithm for -means. Questions cover topics like choosing initial model parameters, latent variables, updating for maximizing likelihood, and more related to Gaussian Mixture Modeling (GMM).
