Abstract We introduce the implicitly constrained least squares (ICLS) classifier, a novel semi-supervised version of the least squares classifier. This classifier minimizes the squared loss on the labeled data among the set of parameters implied by all possible labelings of the unlabeled data. Unlike other discriminative semi-supervised methods, this approach does not introduce explicit additional assumptions into the objective function, but leverages implicit assumptions already present in the choice of the supervised least squares classifier. This method can be formulated as a quadratic programming problem and its solution can be found using a simple gradient descent procedure. We prove that, in a limited 1-dimensional setting, this approach never leads to performance worse than the supervised classifier. Experimental results show that also in the general multidimensional case performance improvements can be expected, both in terms of the squared loss that is intrinsic to the classifier and in terms of the expected classification error.
One of the advantages of this procedure is that it can be applied to any supervised classifier. It has also shown practical success in some application domains, particularly document classification @cite_27 @cite_5 . Unfortunately, the process of self-training can also lead to severely decreased performance, compared to the supervised solution @cite_37 @cite_42 . One can imagine that once an object is incorrectly labeled and added to the training set, its incorrect label may be reinforced, leading the solution away from the optimum. Self-learning is closely related to expectation maximization (EM) based approaches @cite_29 . Indeed, expectation maximization suffers from the same issues as self-learning @cite_35 . In we compare the proposed approach to self-learning for the least squares classifier.
@cite_35 @cite_37 @cite_29 @cite_42 @cite_27 @cite_5
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This paper shows that the accuracy of learned text classifiers can be improved by augmenting a small number of labeled training documents with a large pool of unlabeled documents. This is important because in many text classification problems obtaining training labels is expensive, while large quantities of unlabeled documents are readily available. We introduce an algorithm for learning from labeled and unlabeled documents based on the combination of Expectation-Maximization (EM) and a naive Bayes classifier. The algorithm first trains a classifier using the available labeled documents, and probabilistically labels the unlabeled documents. It then trains a new classifier using the labels for all the documents, and iterates to convergence. This basic EM procedure works well when the data conform to the generative assumptions of the model. However these assumptions are often violated in practice, and poor performance can result. We present two extensions to the algorithm that improve classification accuracy under these conditions: (1) a weighting factor to modulate the contribution of the unlabeled data, and (2) the use of multiple mixture components per class. Experimental results, obtained using text from three different real-world tasks, show that the use of unlabeled data reduces classification error by up to 30 . It sometimes happens (for instance in case control studies) that a classifier is trained on a data set that does not reflect the true a priori probabilities of the target classes on real-world data. This may have a negative effect on the classification accuracy obtained on the real-world data set, especially when the classifier's decisions are based on the a posteriori probabilities of class membership. Indeed, in this case, the trained classifier provides estimates of the a posteriori probabilities that are not valid for this real-world data set (they rely on the a priori probabilities of the training set). Applying the classifier as is (without correcting its outputs with respect to these new conditions) on this new data set may thus be suboptimal. In this note, we present a simple iterative procedure for adjusting the outputs of the trained classifier with respect to these new a priori probabilities without having to refit the model, even when these probabilities are not known in advance. As a by-product, estimates of the new a priori probabilities are also obtained. This iterative algorithm is a straightforward instance of the expectation-maximization (EM) algorithm and is shown to maximize the likelihood of the new data. Thereafter, we discuss a statistical test that can be applied to decide if the a priori class probabilities have changed from the training set to the real-world data. The procedure is illustrated on different classification problems involving a multilayer neural network, and comparisons with a standard procedure for a priori probability estimation are provided. Our original method, based on the EM algorithm, is shown to be superior to the standard one for a priori probability estimation. Experimental results also indicate that the classifier with adjusted outputs always performs better than the original one in terms of classification accuracy, when the a priori probability conditions differ from the training set to the real-world data. The gain in classification accuracy can be significant. We consider a semi-supervised approach to the problem of track classification in dense three-dimensional range data. This problem involves the classification of objects that have been segmented and tracked without the use of a class-specific tracker. This paper is an extended version of our previous work. We propose a method based on the expectation-maximization algorithm: iteratively (1) train a classifier, and (2) extract useful training examples from unlabeled data by exploiting tracking information. We evaluate our method on a large multiclass problem in dense range data collected from natural street scenes. When given only three hand-labeled training tracks of each object class, the final accuracy of the semi-supervised algorithm is comparable to that of the fully supervised equivalent which uses two orders of magnitude more. Further, we show experimentally that the accuracy of a classifier considered as a function of human labeling effort can be substantially improved using this method. Finally, we show that a simple algorithmic speedup based on incrementally updating a boosting classifier can reduce learning time by a factor of three. This paper shows that the performance of a binary classifier can be significantly improved by the processing of structured unlabeled data, i.e. data are structured if knowing the label of one example restricts the labeling of the others. We propose a novel paradigm for training a binary classifier from labeled and unlabeled examples that we call P-N learning. The learning process is guided by positive (P) and negative (N) constraints which restrict the labeling of the unlabeled set. P-N learning evaluates the classifier on the unlabeled data, identifies examples that have been classified in contradiction with structural constraints and augments the training set with the corrected samples in an iterative process. We propose a theory that formulates the conditions under which P-N learning guarantees improvement of the initial classifier and validate it on synthetic and real data. P-N learning is applied to the problem of on-line learning of object detector during tracking. We show that an accurate object detector can be learned from a single example and an unlabeled video sequence where the object may occur. The algorithm is compared with related approaches and state-of-the-art is achieved on a variety of objects (faces, pedestrians, cars, motorbikes and animals).