K-means Clustering

The K-means algorithm is the first unsupervised method we encounter in the course. The previous supervised methods worked with a set of observations and labels $\mathcal{T} = \{(\mathbf{x}_1, y_1), \ldots, (\mathbf{x}_n, y_n)\}$. The unsupervised methods use unlabeled observations $\mathcal{T} = \{\mathbf{x}_1, \ldots, \mathbf{x}_n\}$. The goal is to infer properties of a probability distribution $p_X(\mathbf{x})$.

In low-dimensional problems ($|X| \leq 3$), we can estimate the probability density directly using variety of techniques (e.g. histograms, Parzen window). When we get prior information about the type of a particular probability distribution, we can directly estimate the distribution parameters.

Due to curse of dimensionality these methods fail in higher dimensions. When the dimensionality increases, the volume grows exponentially and the data becomes exponentially sparser. In such cases we use techniques that do not find the full underlying distribution $p_X(\mathbf{x})$, but instead describe the data in a more relaxed way.

The main groups of unsupervised methods are:

• dimensionality reduction methods
• clustering methods

The dimensionality reduction methods map data onto a lower-dimensional manifold while minimising the lost information. The clustering methods partition the observations into groups with small difference between the group samples.

The unsupervised methods have many applications including smart feature selection, data visualization, efficient data encoding, adaptive classification and image segmentation. Last but not least example is Google's PageRank algorithm.

The K-means algorithm is a popular clustering approach. The goal is to find a clustering $\{\mathcal{T}_k\}^K_{k=1}$ of the observations that minimizes the within-cluster sum of squares (WCSS): $$\{\mathbf{c}_1, \ldots, \mathbf{c}_K; \mathcal{T}_1, \ldots, \mathcal{T}_K\} = \underset{\text{all }\mathbf{c}^{'}_k, \mathcal{T}^{'}_k} {\operatorname{arg\,min}} \sum_{k=1}^{K} \sum_{\mathbf{x} \in \mathcal{T}^{'}_k} \| \mathbf{x} - \mathbf{c}^{'}_k \|^2_2,$$ where $K$ is a user specified number of clusters, $\mathcal{T}_1,\ldots,\mathcal{T}_K$ are sets of observations belonging to clusters $1,\ldots,K$ and each cluster is represented by its prototype $\mathbf{c}_k$.

The solution that K-means reaches often depends on the initialisation of the means and there is no guarantee that it reaches the global optimum. As you can see in the two examples below, the solution is not unique and it can happen that the algorithm reaches a local minimum (as in the lower example), where reassigning any point to some other cluster would increase WCSS, but where a better solution does exist.

The chance of getting trapped in a local minimum can be reduced by performing more trials of the K-means clustering and choosing the result that minimizes WCSS.

K-Means++ is the K-means with clever initialization of cluster centers. The motivation is to make initializations which make K-means more likely to end up in better local minima (minima with lower WCSS).

Your task is to implement the basic K-means algorithm as described in the box above. The function prototype is [c, means, x_dists] = k_means(x, k, max_iter, show), where x are observations, k is the number of clusters, c are indices of the found clusters, means are means of the clusters (see the function header in the template for more details).

Important (for automatic evaluation): Use Matlab's function randsample to generate indices for means initialization.

rng(0); % In order to obtain the same results, use this initialization of the random number generator
        % (don't put this into functions submitted to the upload system).
x = [16  12  50  96  34  59  22  75  26  51];
[cluster_idxs, means, sq_dists] = k_means(x, 3, Inf)
 
cluster_idxs =
     3  3  2  1  3  2  3  1  3  2
 
means =
   85.5000  53.3333  22.0000
 
sq_dists =
    36.0000  100.0000   11.1111  110.2500  144.0000   32.1111   0  110.2500   16.0000    5.4444

Limited number of iterations:

rng(0);
load image_data.mat
x = compute_measurements(images);
[~, means, ~] = k_means(x, 3, 4)
 
means =
 
   1.0e+03 *
 
    0.3608   -4.5668   -0.1417
   -2.3709    2.3758    0.6181

Test the algorithm on the measurements extracted from the unlabelled data:

• image_data.mat, which consists of images of three letters - H, L, T (here K = 3)
• the MNIST dataset mnist_image_data.mat of hand written numbers, K = 10

Inspect the results.

Implement the function [c, means, x_dists] = k_means_multiple_trials(x, k, n_trials, max_iter, show) which tries to avoid being trapped in local minimum by performing K-means several times and selecting the solution minimizing the WCSS.

Implement clever K-means initialization: K-means++. The function centers = k_meanspp(x, k) returns initial cluster prototypes for use in the already implemented K-means algorithm. You will first need to prepare idx = random_sample(weights) function, that picks randomly a sample based on the sample weights. See the following recipe and figure to illustrate the sampling process:

1. normalize the sample weights
2. generate a random number in the interval (0, 1)
3. select corresponding sample

Your implementation should return the same results:

>>> rng(0);
>>> distances = [10 2 4 8 3];
>>> idx = random_sample(distances)
>>> idx = random_sample(distances)
 
idx = 4
 
idx = 5

K-means++ favors initialization with cluster prototypes far from each other:

>>> rng(0);
>>> x = 1:100;
>>> means = k_meanspp(x, 3)	
 
means =    82    46     3

Inspect the K-means++ initialization for the data generated by gen_kmeanspp_data.m and compare it to a uniform random initialization of vanilla K-means. Save the figures of initializations to random_initialization.png and kmeanspp_initialization.png.

Apply K-means algorithm to the color quantization problem. The aim is to reduce the number of distinct colors in an image such that the resulting image is visually close to the source image. The algorithm is as follows:

1. Convert the image from RGB color space to L*a*b* color space. Differences between colors measured as euclidean distances in the three-dimensional L*a*b* space correspond better to human perception of color differences.
2. Randomly select 1000 pixels from the input image.
3. Perform K-means clustering of the selected pixels in the three-dimensional L*a*b* space (using k_means function with max_iter=Inf, without k_meanspp initialization).
4. Assign all image pixels to the found means.
5. Convert the resulting image from the L*a*b* space back to the RGB space.

You are supposed to implement a function im_q = quantize_colors(im, k) which performs the described color quantization. Run this function on geeks.png image and save the resulting quantized version as geeks_quantized.png. The right image below is a sample result (for k=8):

Important (for automatic evaluation): To generate indices of the 1000 random pixels, use this code: inds = randsample(size(im,1) * size(im,2), 1000);