### Table of Contents

This tutorial shows how to estiamte Gaussian
mixture model using the VlFeat implementation of
the *Expectation Maximization* (EM) algorithm.

A GMM is a collection of $K$ Gaussian distribution. Each
distribution is called a *mode* of the GMM and represents a
cluster of data points. In computer vision applications, GMM are
often used to model *dictionaries of visual words*. One
important application is the computation
of Fisher vectors encodings.

# Learning a GMM with expectation maximization

Consider a dataset containing 1000 randomly sampled 2D points:

```
numPoints = 1000 ;
dimension = 2 ;
data = rand(dimension,N) ;
```

The goal is to fit a GMM to this data. This can be obtained by
running the `vl_gmm`

function, implementing
the EM algorithm.

```
numClusters = 30 ;
[means, covariances, priors] = vl_gmm(data, numClusters) ;
```

Here `means`

, `covariances`

and `priors`

are respectively the means $\mu_k$, diagonal
covariance matrices $\Sigma_k$, and prior probabilities $\pi_k$ of
the `numClusters`

Gaussian modes.

These modes can be visualized on the 2D plane by plotting ellipses
corresponding to the equation:
\[
\{ \bx: (\bx-\mu_k)^\top \Sigma_k^{-1} (\bx-\mu_k) = 1 \}
\]
for each of the modes. To this end, we can use
the `vl_plotframe`

:

```
figure ;
hold on ;
plot(data(1,:),data(2,:),'r.') ;
for i=1:numClusters
vl_plotframe([means(:,i)' sigmas(1,i) 0 sigmas(2,i)]);
end
```

This results in the figure:

## Diagonal covariance restriction

Note that the ellipses in the previous example are axis
alligned. This is a restriction of the `vl_gmm`

implementation that imposes covariance matrices to be diagonal.

This is suitable for most computer vision applications, where estimating a full covariance matrix would be prohebitive due to the relative high dimensionality of the data. For example, when clustering SIFT features, the data has dimension 128, and each full covariance matrix would contain more than 8k parameters.

For this reason, it is sometimes desirable to globally decorrelated the data before learning a GMM mode. This can be obtained by pre-multiplying the data by the inverse of a square root of its covariance.

# Initializing a GMM model before running EM

The EM algorithm is a local optimization method, and hence
particularly sensitive to the initialization of the model. The
simplest way to initiate the GMM is to pick `numClusters`

data points at random as mode means, initialize the individual
covariances as the covariance of the data, and assign equa prior
probabilities to the modes. This is the default initialization
method used by `vl_gmm`

.

Alternatively, a user can specifiy manually the initial paramters
of the GMM model by using the `custom`

initalization
method. To do so, set
the `'Initialization'`

option to `'Custom'`

and
also the options `'InitMeans'`

, `'InitCovariances'`

and
`'IniPriors'`

to the desired values.

A common approach to obtain an initial value for these parameters is to run KMeans first, as demonstrated in the following code snippet:

```
numClusters = 30;
numData = 1000;
dimension = 2;
data = rand(dimension,numData);
% Run KMeans to pre-cluster the data
[initMeans, assignments] = vl_kmeans(data, numClusters, ...
'Algorithm','Lloyd', ...
'MaxNumIterations',5);
initCovariances = zeros(dimension,numClusters);
initPriors = zeros(1,numClusters);
% Find the initial means, covariances and priors
for i=1:numClusters
data_k = data(:,assignments==i);
initPriors(i) = size(data_k,2) / numClusters;
if size(data_k,1) == 0 || size(data_k,2) == 0
initCovariances(:,i) = diag(cov(data'));
else
initCovariances(:,i) = diag(cov(data_k'));
end
end
% Run EM starting from the given parameters
[means,covariances,priors,ll,posteriors] = vl_gmm(data, numClusters, ...
'initialization','custom', ...
'InitMeans',initMeans, ...
'InitCovariances',initCovariances, ...
'InitPriors',initPriors);
```

The demo scripts `vl_demo_gmm_2d`

and `vl_demo_gmm_3d`

also produce cute colorized figures
such as these: