Clustering & Dimensionality Reduction 273A Intro Machine Learning
What is Unsupervised Learning? In supervised learning we were given attributes & targets (e.g. class labels). In unsupervised learning we are only given attributes. Our task is to discover structure in the data. Example I: the data may be structured in clusters: Example II: the data may live on a lower dimensional manifold:
Why Discover Structure? Data compression: If you have a good model you can encode the data more cheaply. Example PCA: To encode the data I have to encode the x and y position of each data-case. However, I could also encode the offset and angle of the line plus the deviations from the line. Small numbers can be encoded more cheaply than large numbers with the same precision. This idea is the basis for model selection: The complexity of your model (e.g. the number of parameters) should be such that you can encode the data-set with the fewest number of bits (up to a certain precision). Clustering: represent every data-case by a cluster representative plus deviations. ML is often trying to find semantically meaningful representations (abstractions). These are good as a basis for making new predictions.
Clustering: K-means We iterate two operations: 1. Update the assignment of data-cases to clusters 2. Update the location of the cluster. Denote Denote Denote the assignment of data-case i to cluster c. the position of cluster c in a d-dimensional space. the location of data-case i Then iterate until convergence: 1. For each data-case, compute distances to each cluster and the closest one: 2. For each cluster location, compute the mean location of all data-cases assigned to it: Nr. of data-cases in cluster c Set of data-cases assigned to cluster c
K-means Cost function: Each step in k-means decreases this cost function. Often initialization is very important since there are very many local minima in C. Relatively good initialization: place cluster locations on K randomly chosen data-cases. How to choose K? Add complexity term: Or X-validation Or Bayesian methods and minimize also over K
Vector Quantization K-means divides the space up in a Voronoi tesselation. Every point on a tile is summarized by the code-book vector +. This clearly allows for data compression!
Mixtures of Gaussians K-means assigns each data-case to exactly 1 cluster. But what if clusters are overlapping? Maybe we are uncertain as to which cluster it really belongs. The mixtures of Gaussians algorithm assigns data-cases to cluster with a certain probability.
MoG Clustering Covariance determines the shape of these contours Idea: fit these Gaussian densities to the data, one per cluster.
EM Algorithm: E-step r is the probability that data-case i belongs to cluster c. is the a priori probability of being assigned to cluster c. Note that if the Gaussian has high probability on data-case i (i.e. the bell-shape is on top of the data-case) then it claims high responsibility for this data-case. The denominator is just to normalize all responsibilities to 1:
EM Algorithm: M-Step total responsibility claimed by cluster c expected fraction of data-cases assigned to this cluster weighted sample mean where every data-case is weighted according to the probability that it belongs to that cluster. weighted sample covariance
EM-MoG EM comes from expectation maximization. We won t go through the derivation. If we are forced to decide, we should assign a data-case to the cluster which claims highest responsibility. For a new data-case, we should compute responsibilities as in the E-step and pick the cluster with the largest responsibility. E and M steps should be iterated until convergence (which is guaranteed). Every step increases the following objective function (which is the total log-probability of the data under the model we are learning):