A geometric approach to Support Vector Machine (SVM) classification

Mavroforakis, Michael E.; Theodoridis, Sergios

doi:10.1109/tnn.2006.873281

Cited by 323 publications

(159 citation statements)

References 16 publications

Supporting

Mentioning

159

Contrasting

Order By: Relevance

“…Constructing arrangements is, in turn, an important building block for a variety of algorithms in the field of computational geometry and the optimality of the corresponding algorithm "relies heavily on a nontrivial combinatorial fact" [45,46]. We believe that the derivations provided in this chapter will stimulate the collaboration between both the field of machine learning and the field of computational geometry with respect to these problems (similar to the results [9,12,41,102,150] for standard support vector machines). Note that a comparable research direction exists for the k-means algorithm, which aims at obtaining good candidate solutions for its associated combinatorial optimization task (2.8).…”

Section: Discussionmentioning

confidence: 60%

“…An interesting future research direction is the question if one can obtain exact solutions with infinite accuracy in polynomial time. Note that for hard-margin unsupervised support vector machines, this is the case due to a connection between hard-margin support vector machines and convex hulls [12,70,102,116]. For the soft-margin case, a similar connection to the field of computational geometry is given via the concept of reduced convex hulls [102].…”

Section: Exact Solutions In Less Timementioning

confidence: 99%

“…Note that for hard-margin unsupervised support vector machines, this is the case due to a connection between hard-margin support vector machines and convex hulls [12,70,102,116]. For the soft-margin case, a similar connection to the field of computational geometry is given via the concept of reduced convex hulls [102]. However, only exact solutions with guaranteed (but not infinite) accuracy can be obtained this way.…”

Section: Exact Solutions In Less Timementioning

confidence: 99%

See 2 more Smart Citations

From Supervised to Unsupervised Support Vector Machines and Applications in Astronomy

Gieseke

2013

Künstl Intell

View full text Add to dashboard Cite

A common task in the field of machine learning is the classification of objects. The basis for such a task is usually a training set consisting of patterns and associated class labels. A typical example is, for instance, the automatic classification of stars and galaxies in the field of astronomy. Here, the training set could consist of images and associated labels, which indicate whether a particular image shows a star or a galaxy. For such a learning scenario, one aims at generating models that can automatically classify new, unseen images. In the field of machine learning, various classification schemes have been proposed. One of the most popular ones is the concept of support vector machines, which often yields excellent classification results given sufficient labeled data.However, for a variety of real-world tasks, the acquisition of sufficient labeled data can be quite time-consuming. In contrast to labeled training data, unlabeled one can often be obtained easily in huge quantities. Semi-and unsupervised techniques aim at taking these unlabeled patterns into account to generate appropriate models. In the literature, various ways of extending support vector machines to these scenarios have been proposed. One of these ways leads to combinatorial optimization tasks that are difficult to address.In this thesis, several optimization strategies will be developed for these tasks that (1) aim at solving them exactly or (2) aim at obtaining (possibly suboptimal) candidate solutions in an efficient kind of way. More specifically, we will derive a polynomial-time approach that can compute exact solutions for special cases of both tasks. This approach is among the first ones that provide upper runtime bounds for the tasks at hand and, thus, yield theoretical insights into their computational complexity. In addition to this exact scheme, two heuristics tackling both problems will be provided. The first one is based on least-squares variants of the original tasks whereas the second one relies on differentiable surrogates for the corresponding objective functions. While direct implementations of both heuristics are still computationally expensive, we will show how to make use of matrix operations to speed up their execution. This will result in two optimization schemes that exhibit an excellent classification and runtime performance.Despite these theoretical derivations, we will also depict possible application domains of machine learning methods in astronomy. Here, the massive amount of data given for today's and future projects renders a manual analysis impossible and necessitates the use of sophisticated techniques. In this context, we will derive an efficient way to preprocess spectroscopic data, which is based on an adaptation of support vector machines, and the benefits of semi-supervised learning schemes for appropriate learning tasks will be sketched. As a further contribution to this field, we will propose the use of so-called resilient algorithms for the automatic data analysis taking place aboard toda...

show abstract

Section: Discussionmentioning

confidence: 60%

Section: Exact Solutions In Less Timementioning

confidence: 99%

Section: Exact Solutions In Less Timementioning

confidence: 99%

See 1 more Smart Citation

From Supervised to Unsupervised Support Vector Machines and Applications in Astronomy

Gieseke

2013

Künstl Intell

View full text Add to dashboard Cite

show abstract

“…The reduction factor can be set to di®erent size with the constraint that < 1. 15 It can be seen that the smaller the , the smaller is the size of SCH ðSðX; ÞÞ. It is proved that no matter how the changes, the SCH has the same geometric shape as the original convex hull, which is the reason why we call it as Scaled Convex Hulls.…”

Section: Theory 21 Schmentioning

confidence: 99%

Drug discrimination of Near Infrared spectroscopy based on the scaled convex hull classifier

Liu

Jiang

Yang

2014

J. Innov. Opt. Health Sci.

View full text Add to dashboard Cite

Near Infrared spectroscopy (NIRS) has been widely used in the discrimination (classi¯cation) of pharmaceutical drugs. In real applications, however, the class imbalance of the drug samples, i.e., the number of one drug sample may be much larger than the number of the other drugs, deceases drastically the discrimination performance of the classi¯cation models. To address this class imbalance problem, a new computational method -the scaled convex hull (SCH)-based maximum margin classi¯er is proposed in this paper. By a suitable selection of the reduction factor of the SCHs generated by the two classes of drug samples, respectively, the maximal margin classi¯er between SCHs can be constructed which can obtain good classi¯cation performance. With an optimization of the parameters involved in the modeling by Cuckoo Search, a satis¯ed model is achieved for the classi¯cation of the drug. The experiments on spectra samples produced by a pharmaceutical company show that the proposed method is more e®ective and robust than the existing ones.

show abstract

“…In linear SVM, the optimal hyper-plane is the one that minimizes the accuracy error and maximizs the geometric margin. The geometric margin represents the minimum distance of the training samples of both classes from the separating hyper-plane [5].  Neural Network Classifiers: a neural network (NN) classifier consists of units arranged in layers.…”

Section: Introductionmentioning

confidence: 99%