K-means and GMM

Network Intelligence and Analysis Lab
Clustering methods via EM algorithm
2014.07.10
SanghyukChun

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Machine Learning
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Training data
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Learning model
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Unsupervised Learning
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Training data without label
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Input data: 퐷퐷={푥푥1,푥푥2,…,푥푥푁푁}
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Most of unsupervised learning problems are trying to find hidden structure in unlabeled data
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Examples: Clustering, Dimensionality Reduction (PCA, LDA), …
Machine Learning and Unsupervised Learning
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Clustering
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Grouping objects in a such way that objects in the same group are more similar to each other than other groups
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Input: a set of objects (or data) without group information
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Output: cluster index for each object
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Usage: Customer Segmentation, Image Segmentation…
Unsupervised Learning and Clustering
Input
Output
Clustering
Algorithm
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K-means Clustering
Introduction
Optimization
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Intuition: data in same cluster has shorter distance than data which are in other clusters
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Goal: minimize distance between data in same cluster
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Objective function:
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퐽퐽=෍ 푛푛=1 푁푁 ෍ 푘푘=1 퐾퐾 푟푟푛푛푛퐱퐱퐧퐧−훍훍퐤퐤 2
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Where N is number of data points, K is number of clusters
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푟푟푛푛푛∈{0,1}is indicator variables where k describing which of the K clusters the data point 퐱퐱퐧퐧is assigned to
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훍훍퐤퐤is a prototype associated with the k-thcluster
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Eventually 훍훍퐤퐤is same as the center (mean) of cluster
K-means Clustering
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Objective function:
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푎푎푎푎푎푎푎푎푎푛푛{푟푟푛푛푛푛,훍훍퐤퐤}෍ 푛푛=1 푁푁 ෍ 푘푘=1 퐾퐾 푟푟푛푛푛퐱퐱퐧퐧−훍훍퐤퐤 2
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This function can be solved through an iterative procedure
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Step 1: minimize J with respect to the 푟푟푛푛푛, keeping 훍훍퐤퐤is fixed
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Step 2: minimize J with respect to the 훍훍퐤퐤, keeping 푟푟푛푛푛is fixed
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Repeat Step 1,2 until converge
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Does it always converge?
K-means Clustering –Optimization
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Biconvex optimization is a generalization of convex optimization where the objective function and the constraint set can be biconvex
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푓푓푥푥,푦푦is biconvex if fixing x, 푓푓푥푥y=푓푓푥푥,푦푦is convex over Y and fixing y, 푓푓푦푦푥푥=푓푓푥푥,푦푦is convex over X
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One way to solve biconvex optimization problem is that iteratively solve the corresponding convex problems
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It does not guarantee the global optimal point
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But it always converge to some local optimum
Optional –Biconvex optimization
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Step 1: minimize J with respect to the 푟푟푛푛푛, keeping 훍훍퐤퐤is fixed
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푟푟푛푛푛=ቊ1푖푘푘=푎푎푎푎푎푎푎푎푎푛푛푗푗퐱퐱퐧퐧−훍훍퐤퐤 ퟐퟐ 0표표표표표표표표표표표표표표표
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Step 2: minimize J with respect to the 훍훍퐤퐤, keeping 푟푟푛푛푛is fixed
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Derivative with respect to 훍훍퐤퐤to zero giving
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2Σ푛푛푟푟푛푛푛퐱퐱퐧퐧−훍훍퐤퐤=0
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훍훍퐤퐤=Σ푛푛푟푟푛푛푛푛퐱퐱퐧퐧 Σ푛푛푟푟푛푛푛푛
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훍훍퐤퐤is equal to the mean of all the data assigned to cluster k
K-means Clustering –Optimization
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Advantage of K-means clustering
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Easy to implement (kmeansin Matlab, kclusterin Python)
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In practice, it works well
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Disadvantage of K-means clustering
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It can converge to local optimum
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Computing Euclidian distance of every point is expensive
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Solution: Batch K-means
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Euclidian distance is non-robust to outlier
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Solution: K-medoidsalgorithms (use different metric)
K-means Clustering –Conclusion
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Mixture of Gaussians
Mixture Model
EM Algorithm
EM for Gaussian Mixtures
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Assumption: There are k components: 푐푐푖푖푖푖=1 푘푘
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Component 푐푐푖푖has an associated mean vector 휇휇푖푖
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Each component generates data from a Gaussian with mean 휇휇푖푖 and covariance matrix Σ푖푖
Mixture of Gaussians
휇휇1
휇휇2
휇휇3
휇휇4
휇휇5
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Represent model as linear combination of Gaussians
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Probability density function of GMM
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푝푝푥푥=෍ 푘푘=1 퐾퐾 휋휋푘푘푁푁푥푥휇휇푘푘,Σ푘푘
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푁푁푥푥휇휇푘푘,Σ푘푘=12휋휋푑푑/2Σ1/2exp{−12푥푥−휇휇⊤Σ−1푥푥−휇휇}
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Which is called a mixture of Gaussian or Gaussian Mixture Model
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Each Gaussian density is called component of the mixtures and has its own mean 휇휇푘푘and covariance Σ푘푘
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The parameters are called mixing coefficients (Σ푘푘휋휋푘푘=1)
Gaussian Mixture Model
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푝푝푥푥=Σ푘푘=1 퐾퐾휋휋푘푘푁푁푥푥휇휇푘푘,Σ푘푘, where Σ푘푘휋휋푘푘=1
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Input:
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The training set: 푥푥푖푖푖푖=1 푁푁
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Number of clusters: k
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Goal: model this data using mixture of Gaussians
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Mixing coefficients 휋휋1,휋휋2,…,휋휋푘푘
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Means and covariance: 휇휇1,휇휇2,…,휇휇푘푘;Σ1,Σ2,…,Σ푘푘
Clustering using Mixture Model
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푝푝푥푥퐺퐺=푝푝푥푥휋휋1,휇휇1,…=Σ푖푖푝푝푥푥푐푐푖푖푝푝(푐푐푖푖)=Σ푖푖휋휋푖푖푁푁(푥푥|휇휇푖푖,Σ푖푖)
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푝푝푥푥1,푥푥2,…,푥푥푁푁퐺퐺=Π푖푖푝푝(푥푥푖푖|퐺퐺)
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The log likelihood function is given by
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ln푝푝퐗퐗훑훑,훍훍,횺횺=෍ 푛푛=1 푁푁 ln෍ 푘푘=1 퐾퐾 휋휋푘푘푁푁퐱퐱퐧퐧훍훍퐤퐤,횺횺퐤퐤
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Goal: Find parameter which maximize log-likelihood
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Problem: Hard to compute maximum likelihood
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Solution: use EM algorithm
Maximum Likelihood of GMM
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EM algorithm is an iterative procedure for finding the MLE
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An expectation (E) step creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters
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A maximization (M) step computes parameters maximizing the expected log-likelihood found on the E step
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These parameter-estimates are then used to determine the distribution of the latent variables in the next E step.
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EM always converges to one of local optimums
EM (Expectation Maximization) Algorithm
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E-Step: minimize J with respect to the 푟푟푛푛푛, keeping 훍훍퐤퐤is fixed
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푟푟푛푛푛=ቊ1푖푘푘=푎푎푎푎푎푎푎푎푎푛푛푗푗퐱퐱퐧퐧−훍훍퐤퐤 ퟐퟐ 0표표표표표표표표표표표표표표표
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M-Step: minimize J with respect to the 훍훍퐤퐤, keeping 푟푟푛푛푛is fixed
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훍훍퐤퐤=Σ푛푛푟푟푛푛푛푛퐱퐱퐧퐧 Σ푛푛푟푟푛푛푛푛
K-means revisit: EM and K-means
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Let 푧푧푘푘is Bernoulli random variable with probability 휋휋푘푘
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푝푝푧푧푘푘=1=휋휋푘푘where Σ푧푧푘푘=1and Σ휋휋푘푘=1
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Because z use a 1-of-K representation, this distribution in the form
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푝푝푧푧=Π푘푘=1 퐾퐾휋휋푘푘 푧푧푘
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Similarly, the conditional distribution of x given a particular value for z is a Gaussian
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푝푝푥푥푧푧=Π푘푘=1 퐾퐾푁푁푥푥휇휇푘푘,Σ푘푘 푧푧푘
Latent variable for GMM
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The joint distribution is given by 푝푝푥푥,푧푧=푝푝푧푧푝푝(푥푥|푧푧)
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푝푝푥푥=Σ푧푧푝푝푧푧푝푝(푥푥|푧푧)=Σ푘푘휋휋푘푘푁푁(푥푥|휇휇푘푘,Σ푘푘)
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Thus the marginal distribution of x is a Gaussian mixture of the above form
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Now, we are able to work with joint distribution instead of marginal distribution
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Graphical representation of a GMMfor a set of N i.i.d. data points {푥푥푛푛} with corresponding latent variable{푧푧푛푛},where n=1,…,N
Latent variable for GMM
퐳퐳퐧퐧
푿푿풏풏
훑훑
흁흁
횺횺
N
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Conditional probability of z given x
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From Bayes’ theorem,
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훾훾푧푧푘푘≡푝푝푧푧푘푘=1퐱퐱=푝푝푧푧푘=1푝푝퐱퐱푧푧푘푘=1Σ푗푗=1 퐾퐾푝푝푧푧푗푗=1푝푝퐱퐱푧푧푗푗=1= 휋휋푘푘푁푁퐱퐱훍훍퐤퐤,횺횺퐤퐤 Σ푗푗=1 퐾퐾휋휋푗푗푁푁(퐱퐱|훍훍퐣퐣,횺횺퐣퐣)
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훾훾푧푧푘푘can also be viewed as the responsibility that component k takes for ‘explaining’ the observation x
EM for Gaussian Mixtures (E-step)
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Likelihood function for GMM
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ln푝푝퐗퐗훑훑,훍훍,횺횺=෍ 푛푛=1 푁푁 ln෍ 푘푘=1 퐾퐾 휋휋푘푘푁푁퐱퐱퐧퐧훍훍퐤퐤,횺횺퐤퐤
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Setting the derivatives of log likelihood with respect to the means 휇휇푘푘of the Gaussian components to zero, we obtain
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휇휇푘푘= 1N푘푘 ෍ 푛푛=1 푁푁 훾훾푧푧푛푛푛퐱퐱퐧퐧 where, 푁푁푘푘=Σ푛푛=1 푁푁훾훾(푧푧푛푛푛)
EM for Gaussian Mixtures (M-step)
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Setting the derivatives of likelihood with respect to the Σ푘푘to zero, we obtain
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횺횺풌풌= 1 푁푁푘푘 ෍ 푛푛=1 푁푁 훾훾푧푧푛푛푛퐱퐱퐧퐧−휇휇푘푘퐱퐱퐧퐧−휇휇푘푘 ⊤
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Maximize likelihood with respect to the mixing coefficient 휋휋by using a Lagrange multiplier, we obtain
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ln푝푝퐗퐗훑훑,훍훍,횺횺+휆휆(Σ푘푘=1 퐾퐾휋휋푘푘−1)
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휋휋푘푘=푁푁푘푁푁
EM for Gaussian Mixtures (M-step)
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휇휇푘푘,Σ푘푘,휋휋푘푘do not constitute a closed-form solution for the parameters of the mixture model because the responsibility 훾훾푧푧푛푛푛depend on those parameters in a complex way
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훾훾(푧푧푛푛푛)=휋휋푘푁푁퐱퐱훍훍퐤퐤,횺횺퐤퐤 Σ푗푗=1 퐾퐾휋휋푗푗푁푁(퐱퐱|훍훍퐣퐣,횺횺퐣퐣)
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In EM algorithm for GMM, 훾훾(푧푧푛푛푛)and parameters are iteratively optimized
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In E step, responsibilities or the posterior probabilities are evaluated by current values for the parameters
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In M step, re-estimate the means, covariances, and mixing coefficients using previous results
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Initialize the means 휇휇푘푘, covariancesΣ푘푘and mixing coefficient 휋휋푘푘, and evaluate the initial value of the log likelihood
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E step: Evaluate the responsibilities using the current parameter
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훾훾(푧푧푛푛푛)= 휋휋푘푘푁푁퐱퐱훍훍퐤퐤,횺횺퐤퐤 Σ푗푗=1 퐾퐾휋휋푗푗푁푁(퐱퐱|훍훍퐣퐣,횺횺퐣퐣)
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M step: Re-estimate parameters using the current responsibilities
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휇휇푘푘 푛푛푛푛푛푛=1N푘Σ푛푛=1 푁푁훾훾푧푧푛푛푛퐱퐱퐧퐧
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횺횺풌풌 풏풏풏풏풏풏=1 푁푁푘Σ푛푛=1 푁푁훾훾푧푧푛푛푛퐱퐱퐧퐧−휇휇푘푘퐱퐱퐧퐧−휇휇푘푘 ⊤
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휋휋푘푘 푛푛푛푛푛푛=푁푁푘푁푁
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푁푁푘푘=Σ푛푛=1 푁푁훾훾(푧푧푛푛푛)
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Repeat E step and M step until converge
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We can derive the K-means algorithm as a particular limit of EM for Gaussian Mixture Model
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Consider a Gaussian mixture model with covariance matrices are given by 휀휀퐼퐼, where 휀휀is a variance parameter and I is identity
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If we consider the limit휀휀→0, log likelihood of GMM becomes
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퐸퐸푧푧ln푝푝푋푋,푍푍휇휇,Σ,휋휋→−12=Σ푛푛Σ푘푘푟푟푛푛푛퐱퐱퐧퐧−훍훍퐤퐤 2+퐶퐶
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Thus, we see that in this limit, maximizing the expected complete- data log likelihood is equivalent to K-means algorithm
Relationship between K-means algorithm and GMM
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