Learning vector quantization for proximity data

Hofmann D. Learning vector quantization for proximity data. Bielefeld: Universität Bielefeld; 2016.Prototype-based classifiers such as learning vector quantization (LVQ) often display intuitive and flexible classification and learning rules. However, classical techniques are restricted to vectorial data only, and hence not suited for more complex data structures. Therefore, a few extensions of diverse LVQ variants to more general data which are characterized based on pairwise similarities or dissimilarities only have been proposed recently in the literature. In this contribution, we propose a novel extension of LVQ to similarity data which is based on the kernelization of an underlying probabilistic model: kernel robust soft LVQ (KRSLVQ). Relying on the notion of a pseudo-Euclidean embedding of proximity data, we put this specific approach as well as existing alternatives into a general framework which characterizes different fundamental possibilities how to extend LVQ towards proximity data: the main characteristics are given by the choice of the cost function, the interface to the data in terms of similarities or dissimilarities, and the way in which optimization takes place. In particular the latter strategy highlights the difference of popular kernel approaches versus so-called relational approaches. While KRSLVQ and alternatives lead to state of the art results, these extensions have two drawbacks as compared to their vectorial counterparts: (i) a quadratic training complexity is encountered due to the dependency of the methods on the full proximity matrix; (ii) prototypes are no longer given by vectors but they are represented in terms of an implicit linear combination of data, i.e. interpretability of the prototypes is lost. We investigate different techniques to deal with these challenges: We consider a speed-up of training by means of low rank approximations of the Gram matrix by its Nyström approximation. In benchmarks, this strategy is successful if the considered data are intrinsically low-dimensional. We propose a quick check to efficiently test this property prior to training. We extend KRSLVQ by sparse approximations of the prototypes: instead of the full coefficient vectors, few exemplars which represent the prototypes can be directly inspected by practitioners in the same way as data. We compare different paradigms based on which to infer a sparse approximation: sparsity priors while training, geometric approaches including orthogonal matching pursuit and core techniques, and heuristic approximations based on the coefficients or proximities. We demonstrate the performance of these LVQ techniques for benchmark data, reaching state of the art results. We discuss the behavior of the methods to enhance performance and interpretability as concerns quality, sparsity, and representativity, and we propose different measures how to quantitatively evaluate the performance of the approaches. We would like to point out that we had the possibility to present our findings in international publication organs including three journal articles [6, 9, 2], four conference papers [8, 5, 7, 1] and two workshop contributions [4, 3]. References [1] A. Gisbrecht, D. Hofmann, and B. Hammer. Discriminative dimensionality reduction mappings. Advances in Intelligent Data Analysis, 7619: 126–138, 2012. [2] B. Hammer, D. Hofmann, F.-M. Schleif, and X. Zhu. Learning vector quantization for (dis-)similarities. Neurocomputing, 131: 43–51, 2014. [3] D. Hofmann. Sparse approximations for kernel robust soft lvq. Mittweida Workshop on Computational Intelligence, 2013. [4] D. Hofmann, A. Gisbrecht, and B. Hammer. Discriminative probabilistic prototype based models in kernel space. New Challenges in Neural Computation, TR Machine Learning Reports, 2012. [5] D. Hofmann, A. Gisbrecht, and B. Hammer. Efficient approximations of kernel robust soft lvq. Workshop on Self-Organizing Maps, 198: 183–192, 2012. [6] D. Hofmann, A. Gisbrecht, and B. Hammer. Efficient approximations of robust soft learning vector quantization for non-vectorial data. Neurocomputing, 147: 96–106, 2015. [7] D. Hofmann and B. Hammer. Kernel robust soft learning vector quantization. Artificial Neural Networks in Pattern Recognition, 7477: 14–23, 2012. [8] D. Hofmann and B. Hammer. Sparse approximations for kernel learning vector quantization. European Symposium on Artificial Neural Networks, 549–554, 2013. [9] D. Hofmann, F.-M. Schleif, B. Paaßen, and B. Hammer. Learning interpretable kernelized prototype-based models. Neurocomputing, 141: 84–96, 2014

Adaptive prototype-based dissimilarity learning

Zhu X. Adaptive prototype-based dissimilarity learning. Bielefeld: Universitätsbibliothek Bielefeld; 2015.In this thesis we focus on prototype-based learning techniques, namely three unsuper- vised techniques: generative topographic mapping (GTM), neural gas (NG) and affinity propagation (AP), and two supervised techniques: generalized learning vector quantiza- tion (GLVQ) and robust soft learning vector quantization (RSLVQ). We extend their abilities with respect to the following central aspects: • Applicability on dissimilarity data: Due to the increased complexity of data, in many cases data are only available in form of (dis)similarities which describe the relations between objects. Classical methods can not directly deal with this kind of data. For unsupervised methods this problem has been studied, here we transfer the same idea to the two supervised prototype-based techniques such that they can directly deal with dissimilarities without an explicit embedding into a vector space. • Quadratic complexity issue: For dealing with dissimilarity data, due to the need of the full dissimilarity matrix, the complexity becomes quadratic which is infeasible for large data sets. In this thesis we investigate two linear approximation techniques: Nyström approximation and patch processing, and integrate them into unsupervised and supervised prototype-based techniques. • Reliability of prototype-based classifiers: In practical applications, a relia- bility measure is beneficial for evaluating the classification quality expected by the end users. Here we adopt concepts from conformal prediction (CP), which provides point-wise confidence measure of the prediction, and we combine those with supervised prototype-based techniques. • Model complexity: By means of the confidence values provided by CP, the model complexity can be automatically adjusted by adding new prototypes to cover low confidence data space. • Extendability to semi-supervised problems: Besides its ability to evaluate a classifier, conformal prediction can also be considered as a classifier. This opens a way that supervised techniques can be easily extended for semi-supervised settings by means of a self-training approach

k-Means

The k-means clustering algorithm (k-means for short) provides a method offinding structure in input examples. It is also called the Lloyd–Forgy algorithm as it was independently introduced by both Stuart Lloyd and Edward Forgy. k-means, like other algorithms you will study in this part of the book, is an unsupervised learning algorithm and, as such, does not require labels associated with input examples. Recall that unsupervised learning algorithms provide a way of discovering some inherent structure in the input examples. This is in contrast with supervised learning algorithms, which require input examples and associated labels so as to fit a hypothesis function that maps input examples to one or more output variables

k-Means

The k-means clustering algorithm (k-means for short) provides a method offinding structure in input examples. It is also called the Lloyd–Forgy algorithm as it was independently introduced by both Stuart Lloyd and Edward Forgy. k-means, like other algorithms you will study in this part of the book, is an unsupervised learning algorithm and, as such, does not require labels associated with input examples. Recall that unsupervised learning algorithms provide a way of discovering some inherent structure in the input examples. This is in contrast with supervised learning algorithms, which require input examples and associated labels so as to fit a hypothesis function that maps input examples to one or more output variables