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Usually, the underlying geometry of the data is captured by representing the data as a graph, with samples as the vertices, and the pair wise similarities between the samples as edge- weights. Several graph based algorithms such as Label propagation [39], Markov random walks [40], Graph cut algorithms [41], Spectral graph transducer [42], and Low density separation [43] proposed in the literature are based on this assumption.

The second assumption is called the cluster assumption [44]. It states that the data. Clustering is an ill-posed problem, and it is difficult to come up with a general purpose objective function that works satisfactorily with an arbitrary dataset [16]. If any side information is available, it must be exploited to obtain a more useful or relevant clustering of the data. The pairwise constraints are of two types: must-link and cannot-link constraints.

The clustering algorithm must try to assign the same label to the pair of points participating in a must-link constraint, and assign different labels to a pair of points participating in a cannot-link constraint.

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These pairwise constraints may be specified by a user to encode his preferred clustering. Pairwise constraints can also be automatically inferred from the structure of the data, without a user having to specify them. As an example, web pages that are linked to one another may be considered as participating in a must-link constraint. Feature selection can be performed for both supervised and unsupervised settings depending on the data available.

Unsupervised feature selection is difficult for the same reasons that make clustering difficult, lack of a clear objective apart from the model assumptions. Supervised feature selection has the same limitations as classification, i. Semi-supervised feature selection aims to utilize pair wise constraints in order to identify a possibly superior subset of features for the task.

Many other learning tasks, apart from classification and clustering have their semisupervised Counterparts as well e.

For example, page ranking algorithms used by search engines can utilize existing partial ranking information on the data to obtain a final ranking based on the query. Generative models are perhaps the oldest semi-supervised learning method. With large amount of unlabeled data, the mixture components can be identified; then ideally we only need one labeled example per component to fully determine the mixture distribution, see Figure 2.

Nigam et al. They showed the resulting classifiers perform better than those trained only from L. Baluja uses the same algorithm on a face orientation discrimination task. Fujino et al. One has to pay attention to a few things:. The mixture model ideally should be identifiable. If the model family is identifiable, in theory with infinite U one can learn up to a permutation of component indices. Here is an example showing the problem with unidentifiable models.

The model p x y is uniform for[Abbildung in dieser Leseprobe nicht enthalten]. Assuming with large amount of un-labeled data U we know p x is uniform in [0, 1]. We also have 2 labeled data points [Abbildung in dieser Leseprobe nicht enthalten]. With our assumptions we cannot distinguish the following two models:. Even if we known p x top are a mixture of two uniform distributions, we cannot uniquely identify the two components.

If the mixture model assumption is correct, unlabeled data is guaranteed to improve accuracy.

However if the model is wrong, unlabeled data may actually hurt accuracy. Figure 3 shows an example. This has been observed by multiple researchers. Cozman et al.

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It is thus important to carefully construct the mixture model to reflect reality. For example in text categorization a topic may contain several subtopics, and will be better modeled by multiple multinomials instead of a single one. Another solution is to down-weighing unlabeled data, which is also used by Nigam et al. For example, a is clearly not generated from two Gaussian. If we insist that each class is a single Gaussian, b will have higher probability than c.

Even if the mixture model assumption is correct, in practice mixture components are identified by the Expectation-Maximization EM algorithm EM is prone to local maxima. If a local maximum is far from the global maximum, unlabeled data may again hurt learning. Remedies include smart choice of starting point by active learning. We shall also mention that instead of using a probabilistic generative mixture model, some approaches employ various clustering algorithms to cluster the whole dataset, and then label each cluster with labeled data.

Although they can perform well if the particular clustering algorithms match the true data distribution, these approaches are hard to analyze due to their algorithmic nature. Another approach for semi-supervised learning with generative models is to convert data into a feature representation determined by the generative model. The new feature representation is then fed into a standard discriminative classifier.

First a generative mixture model is trained, one component per class. At this stage the unlabeled data can be incorporated via EM, which is the same as in previous subsections. However instead of directly using the generative model for classification, each labeled example is converted into a fixed-length Fisher score vector, i.

These Fisher score vectors are then used in a discriminative classifier like an SVM, which empirically has high accuracy. Self-training is a commonly used technique for semi-supervised learning. In self-training a classifier is first trained with the small amount of labeled data. The classifier is then used to classify the unlabeled data. Typically the most confident unlabeled points, together with their predicted labels, are added to the training set.

The classifier is re- trained and the procedure repeated. Note the classifier uses its own predictions to teach itself. The procedure is also called self-teaching or bootstrapping not to be confused with the statistical procedure with the same name. One can imagine that a classification mistake can reinforce itself.