Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit...
Transcript of Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit...
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Gaussian Mixture Models and EM algorithm
Radek Danecek
![Page 2: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/2.jpg)
Gaussian Mixture Model
• Unsupervised method
• Fit multimodal Gaussian distributions
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Formal Definition
• The model is described as:
• The parameters of the model are:
• The training data is unlabeled – unsupervised setting
• Why not fit with MLE?
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Optimization problem
• Model:
• Apply MLE:• Maximize:
• Difficult, non convex optimization with constraints
• Use EM algorithm instead
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EM Algorithm for GMMs
• Idea: • Objective function:
• Split optimization of the objective into to parts
• Algorithm:• Initialize model parameters (randomly):
• Iterate until convergence: • E-step
• Assign cluster probabilities (“soft labels”) to each sample
• M-step
• Solve the MLE using the soft labels
![Page 6: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/6.jpg)
Initialization
• Initialize model parameters (randomly)
• Uniform for cluster probabilities
• Centers• Random
• K-means heuristics
• Covariances: • Spherical, according to empirical variance
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E-step
• For each data point and each cluster , compute the probabilitythat belongs to(given current model parameters)
“soft labels”
N
K
Probabilities of point n belonging to clusters 1…K (sum up to 1)
Sum determines the “weight” of cluster k
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E-step
• For each data point and each cluster , compute the probabilitythat belongs to(given current model parameters)
“soft labels”
![Page 9: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/9.jpg)
M-step
• Now we have “soft labels” for the data -> fall back to supervised MLE
• Optimize the log likelihood:• Instead of the original (difficult objective):
We optimize the following:
• Differentiate w.r.t.
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M-step
• Update model parameters:
• Update prior for each cluster: N
K
Probabilities of point n belonging to clusters 1…K (sum up to 1)
Sum determines the “weight” of cluster k
Normalized column-wise sum are priors for clusters 1…K
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M-step
• Update model parameters:
• Update mean and covariance of each cluster
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M-step
• Update model parameters:
• Update mean and covariance of each cluster
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EM Algorithm for GMMs
• Idea: • Objective function:
• Split optimization of the objective into to parts
• Algorithm:• Initialize model parameters (randomly):
• Iterate until convergence: • E-step
• Assign cluster probabilities (“soft labels”) to each sample
• M-step
• Find optimal parameters given the soft labels
![Page 14: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/14.jpg)
Overlapping clusters
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Overlapping clusters
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Unequal cluster size
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Imbalanced cluster size
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Sensitivity to Initialization
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Sensitivity to Initialization
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Sensitivity to Initialization
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Sensitivity to Initialization
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Sensitivity to Initialization
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Sensitivity to initialization
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Degenerate covariance
• The determinant of the covariance matrix tends to 0
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Practical Example – Color segmentation
• Input: an image
• Can be thought of as a dataset of 3D (color) samples
• Run 3D GMM clustering over
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Practical Example – Color segmentation
Input Image
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Practical Example – Color segmentation
3 clusters
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Practical Example – Color segmentation
4 clusters
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Practical Example – Color segmentation
5 clusters
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Practical Example – Color segmentation
7 clusters
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Practical Example – Color segmentation
8 clusters
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Practical Example – Color segmentation
9 clusters
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Practical Example – Color segmentation
10 clusters
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Practical Example – Color segmentation
15 clusters
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Practical Example – Color segmentation
20 clusters
![Page 36: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/36.jpg)
Practical Example – Color segmentation
Input Image
![Page 37: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/37.jpg)
EM Algorithm for GMMs
• Idea: • Objective function:
• Split optimization of the objective into to parts
• Algorithm:• Initialize model parameters (randomly):
• Iterate until convergence: • E-step
• Assign cluster probabilities (“soft labels”) to each sample
• M-step
• Find optimal parameters given the soft labels
![Page 38: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/38.jpg)
Generalized EM
• Idea: • Objective function:
• Split optimization of the objective into to parts
• Algorithm:• Initialize model parameters (randomly):
• Iterate until convergence: • E-step
• Assign cluster probabilities (“soft labels”) to each sample
• M-step
• Find optimal parameters given the soft labels
![Page 39: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/39.jpg)
Generalized EM
• Idea: • Objective function:
• Split optimization of the objective into to parts
• Algorithm:• Initialize model parameters (randomly):
• Iterate until convergence: • E-step
• Assign cluster probabilities (“soft labels”) to each sample
• M-step
• Find optimal parameters given the soft labels
![Page 40: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/40.jpg)
Generalized EM
• Idea: • Objective function:
• Split optimization of the objective into to parts
• Algorithm:• Initialize model parameters (randomly):
• Iterate until convergence: • E-step
• Assign cluster probabilities (“soft labels”) to each sample
• M-step
• Find optimal parameters given the soft labels
![Page 41: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/41.jpg)
Generalized EM
• Idea: • Objective function:
• Split optimization of the objective into to parts
• Algorithm:• Initialize model parameters (randomly):
• Iterate until convergence: • E-step
• Assign cluster probabilities (“soft labels”) to each sample
• M-step
• Find optimal parameters given the soft labels
![Page 42: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/42.jpg)
Generalized M-step
• What is the objective function?
• GMM:
• General:
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Exercise
• Consider a mixture of K multivariate Bernoulli distributions with parameters , where
• Multivariate Bernoulli distribution:
• Question 1: Write down the equation for the E-step update
hint GMM: Answer:
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Exercise
• Consider a mixture of K multivariate Bernoulli distributions with parameters , where
• Multivariate Bernoulli distribution:
• Question 1: Write down the equation for the E-step update
hint GMM: Answer:
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Exercise
• Consider a mixture of K multivariate Bernoulli distributions with parameters , where
• Multivariate Bernoulli distribution:
• Question 2: Write down the EM objective:
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Exercise
• Consider a mixture of K multivariate Bernoulli distributions with parameters , where
• Multivariate Bernoulli distribution:
• Question 2: Write down the EM objective:
![Page 47: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/47.jpg)
Exercise
• Multivariate Bernoulli distribution:
• EM objective:
![Page 48: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/48.jpg)
Exercise
• Multivariate Bernoulli distribution:
• EM objective:
![Page 49: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/49.jpg)
Exercise
• Multivariate Bernoulli distribution:
• EM objective:
![Page 50: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/50.jpg)
Exercise
• Multivariate Bernoulli distribution:
• EM objective:
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Exercise
• Question 3: Write down the M-step update
• Differentiate wrt.:
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Exercise
• Question 3: Write down the M-step update
• Differentiate wrt.:
![Page 53: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/53.jpg)
Exercise
• Question 3: Write down the M-step update
• Differentiate wrt.:
![Page 54: Gaussian Mixture Models and EM algorithm · Gaussian Mixture Model •Unsupervised method •Fit multimodal Gaussian distributions . Formal Definition •The model is described as:](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8b129bd926567bba30dd90/html5/thumbnails/54.jpg)
Summary
• EM algorithm is useful for fitting GMMs (or other mixtures) in an unsupervised setting
• Can be used for: • Clustering
• Classification
• Distribution estimation
• Outlier detection
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Other unsupervised clustering techniques
Source: https://scikit-learn.org/stable/modules/clustering.html
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Alternative for density estimation
• Kernel density estimation
Source: https://scikit-learn.org/stable/auto_examples/neighbors/plot_species_kde.html
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References
• Lecture slides/videos
• https://www.python-course.eu/expectation_maximization_and_gaussian_mixture_models.php
• https://scikit-learn.org/stable/modules/clustering.html