Semi-supervised Clustering
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1 Semi-supervised lustering BY: $\ S - MAI AMLT /2017 (S - MAI) Semi-supervised lustering AMLT / / 26
2 Outline 1 Semisupervised lustering 2 Semisupervised lustering/labeled Examples 3 Semisupervised lustering/onstraints 4 Semisupervised lustering/distance Learning (S - MAI) Semi-supervised lustering AMLT / / 26
3 Semisupervised lustering 1 Semisupervised lustering 2 Semisupervised lustering/labeled Examples 3 Semisupervised lustering/onstraints 4 Semisupervised lustering/distance Learning (S - MAI) Semi-supervised lustering AMLT / / 26
4 Semisupervised lustering Semi-supervised lustering Sometimes we have available some information about the dataset we are analyzing unsupervisedly ould be interesting to incorporate this information to the clustering process in order to: Bias the search of the algorithm toward the solutions more consistent with our knowledge Improve the quality of the result reducing the algorithm natural bias (predictivity/stability) (S - MAI) Semi-supervised lustering AMLT / / 26
5 Semisupervised lustering Semi-supervised lustering The information that we have available can be of different kinds: Sets of labeled instances onstrains among certain instances: Instances that have to be in the same group/instances that can not belong to the same group General information about the properties that the instances of a group have to hold (S - MAI) Semi-supervised lustering AMLT / / 26
6 Semisupervised lustering How to use supervised information Usually the use will depend on the model we can obtain 1 Begin with a prior model that changes how the search is performed 2 Bias the search, pruning the models that are not consistent with the supervised knowledge 3 Modify the similarity among instances to match the constraints imposed by the prior knowledge (S - MAI) Semi-supervised lustering AMLT / / 26
7 Semisupervised lustering/labeled Examples 1 Semisupervised lustering 2 Semisupervised lustering/labeled Examples 3 Semisupervised lustering/onstraints 4 Semisupervised lustering/distance Learning (S - MAI) Semi-supervised lustering AMLT / / 26
8 Semisupervised lustering/labeled Examples Semi supervised clustering using labeled examples Assuming that we have some labeled examples, these can be used to obtain an initial model We only have to know what examples belong to clusters, the actual clusters are not needed We can begin from this model the clustering process This means that we decide the starting point of the search using the supervised information This initial model changes the search and the final model (bias) This differs from semi-supervised learning from a supervised perspective, were the labels of some of the examples are known (S - MAI) Semi-supervised lustering AMLT / / 26
9 Semisupervised lustering/labeled Examples Semi supervised clustering using labeled examples Basu, Banerjee, Mooney, Semi supervised clustering by seeding, IML 2002 Algorithm based on K-means (spherical clusters based on prototypes) The usual initialization of K-means is by selecting randomly the initial prototype (there are other alternatives) Two proposals: Use the labeled examples to build the initial prototypes (seeding) Use the labeled examples to build the initial prototypes and constrain the model so the labeled examples are always in the initial clusters (seed and constraint) The initial prototypes give an initial probability distribution for the clustering (S - MAI) Semi-supervised lustering AMLT / / 26
10 Semisupervised lustering/labeled Examples Semi supervised clustering using labeled examples (S - MAI) Semi-supervised lustering AMLT / / 26
11 Semisupervised lustering/labeled Examples Seeded-KMeans Algorithm: Seeded-KMeans Input: The dataset X, the number of clusters K, a set S of labeled instances (k groups) Output: A partition of X in K groups begin ompute K initial prototypes (µ i ) using the labeled instances repeat Assign each example from X to their nearest prototype µ i Recompute the prototype µ i with the examples assigned until onvergence end (S - MAI) Semi-supervised lustering AMLT / / 26
12 Semisupervised lustering/labeled Examples onstrained-kmeans Algorithm: onstrained-kmeans Input: The dataset X, the number of clusters K, a set S of labeled instances (k groups) Output: A partition of X in K groups begin ompute K initial prototypes (µ i ) using the labeled instances repeat Maintain the examples from S in their initial classes Assign each example from X to their nearest prototype µ i Recompute the prototype µ i with the examples assigned until onvergence end (S - MAI) Semi-supervised lustering AMLT / / 26
13 Semisupervised lustering/onstraints 1 Semisupervised lustering 2 Semisupervised lustering/labeled Examples 3 Semisupervised lustering/onstraints 4 Semisupervised lustering/distance Learning (S - MAI) Semi-supervised lustering AMLT / / 26
14 Semisupervised lustering/onstraints Semi supervised clustering using constraints To have labeled examples means that the number of clusters and something about the characteristic of the data is known Sometimes it is easier to have information about if two examples have to be in the same or different clusters This information can be expressed by means of constraints among examples: must links and cannot links This information can be used to bias the search and only look for models that maintain these constraints (S - MAI) Semi-supervised lustering AMLT / / 26
15 Semisupervised lustering/onstraints Semi supervised clustering using constraints (S - MAI) Semi-supervised lustering AMLT / / 26
16 Semisupervised lustering/onstraints Semi supervised clustering using constraints Basu, Bilenko, Mooney, A probabilistic framework for semi-supervised clustering, IML 2002 Algorithm based on K-means (spherical clusters based on prototypes) A set of must-link cannot-link constraints is defined over a subset of examples The quality function of the K-means algorithm is modified to bias the search A hidden markov random field is defined using the constraints (S - MAI) Semi-supervised lustering AMLT / / 26
17 Semisupervised lustering/onstraints Semi supervised clustering using constraints Data The labels of the examples can be used to define a markov random field The must-links and cannot-links define the dependence among the variables The clustering of the examples has to maximize the probability of the hidden markov random field Must link Must link annot link Hidden MRF (S - MAI) Semi-supervised lustering AMLT / / 26
18 Semisupervised lustering/onstraints Semi supervised clustering using constraints A new objective function for the K-Means algorithm is needed The main idea is to introduce a penalty term to the objective function that: Penalizes the clustering that puts examples with must-links in different clusters Penalizes the clustering that puts examples with cannot-links in the same cluster This penalty has to be proportional to the distance among the instances (S - MAI) Semi-supervised lustering AMLT / / 26
19 Semisupervised lustering/onstraints Semi supervised clustering using constraints The new objective function is: J obj = x i X D(x i, µ li ) + (x i,x j ),[l i =l j ] (x i,x j ) M,[l i l j ] w ij (ϕ Dmax ϕ D (x i, x j )) w ij ϕ D (x i, x j ) + Where D is the distortion function, M and are the sets of must and cannot links, w ij and w ij are the weight of the penalty for violating the constraints and ϕ D is an increasing function of the distance between two examples The specific function depends on the distortion measure selected (S - MAI) Semi-supervised lustering AMLT / / 26
20 Semisupervised lustering/onstraints HMRF-KMeans The initialization of the process begins by analyzing the constraints and inferring a set of examples to obtain the initial prototypes The constraints conform a graph that divide a part of the examples in different groups (connected components of the graph) With these instances K prototypes are computed and used to initialize the algorithm The algorithm minimizes the objective function using an expectation and an minimization step (S - MAI) Semi-supervised lustering AMLT / / 26
21 Semisupervised lustering/onstraints HMRF-KMeans Algorithm: HMRF-KMeans Input: The dataset X, the number of clusters K, a set of must and cannot links, a distance function D and a set of weights for violating the constraints Output: A partition of X in K groups begin ompute K initial prototypes (µ i ) using constraints repeat E-step: Reassign the labels of the examples using the prototypes (µ i ) to minimize J obj M-step: Given the cluster labels recalculate cluster centroids to minimize J obj until onvergence end (S - MAI) Semi-supervised lustering AMLT / / 26
22 Semisupervised lustering/onstraints Semi supervised clustering using constraints The constraints can also be seen as an indication of the inadequacy of the distance measure between two instances A must-link violation means that the distance function assigns a similarity less than the desired similarity A cannot-link violation means that distance function assigns a similarity larger than the desired similarity The previous algorithm can be modified to introduce weights to modify the distances among instances with constraints (an additional maximization step is needed) (S - MAI) Semi-supervised lustering AMLT / / 26
23 Semisupervised lustering/distance Learning 1 Semisupervised lustering 2 Semisupervised lustering/labeled Examples 3 Semisupervised lustering/onstraints 4 Semisupervised lustering/distance Learning (S - MAI) Semi-supervised lustering AMLT / / 26
24 Semisupervised lustering/distance Learning Semi supervised clustering with Distance Learning Other approaches learn the more adequate distance function to fulfill the constraints The constraints are used as a guide to find a distance matrix that represents the relations among examples This problem can be defined as an optimization problem that optimizes the distances among examples with respect to the constraints This methods are related to kernel methods, the goal is to learn a Kernel matrix that represents a new space where the instances have appropriate distances (S - MAI) Semi-supervised lustering AMLT / / 26
25 Semisupervised lustering/distance Learning Semi supervised clustering with Distance Learning There are several methods: Relevant omponent Analysis [Yeung, hang (2006)] (optimization of linear combination of distance kernels generated by must and cannot links) Optimization using Spectral Graphs for maintaining the constrains in the new space and the structure of the original space Learning of Mahalanobis distances: separate/approach the different dimensions to match the constraints Kernel lustering (Kernel K-means) with kernel matrix learning via regularization Probabilistic lustering (Model Based) with constraints via EM (modify the probabilities matching the constrains) (S - MAI) Semi-supervised lustering AMLT / / 26
26 Semisupervised lustering/distance Learning Semi supervised clustering with Distance Learning Original First Iteration Second Iteration Third Iteration (S - MAI) Semi-supervised lustering AMLT / / 26
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