Clifford Support Vector Machines Research Articles

Parallel Clifford Support Vector Machines Using the Gaussian Kernel

This work presents a parallelization method for the Clifford support vector machines, based in two characteristics of the Gaussian Kernel. The pure real-valued result and its commutativity allows us to separate the multivector data in its defining subspaces. These subspaces are independent from each other, so we can solve the problem using parallelism. The motivation is to present an easy approach that can be explained using the more common known concepts of complex numbers and quaternions, because in general there exists a lack of familiarity with geometric algebra.

Clifford Support Vector Machines for Classification, Regression, and Recurrence

This paper introduces the Clifford support vector machines (CSVM) as a generalization of the real and complex-valued support vector machines using the Clifford geometric algebra. In this framework, we handle the design of kernels involving the Clifford or geometric product. In this approach, one redefines the optimization variables as multivectors. This allows us to have a multivector as output. Therefore, we can represent multiple classes according to the dimension of the geometric algebra in which we work. We show that one can apply CSVM for classification and regression and also to build a recurrent CSVM. The CSVM is an attractive approach for the multiple input multiple output processing of high-dimensional geometric entities. We carried out comparisons between CSVM and the current approaches to solve multiclass classification and regression. We also study the performance of the recurrent CSVM with experiments involving time series. The authors believe that this paper can be of great use for researchers and practitioners interested in multiclass hypercomplex computing, particularly for applications in complex and quaternion signal and image processing, satellite control, neurocomputation, pattern recognition, computer vision, augmented virtual reality, robotics, and humanoids.

What Can Hamilton, Grassmann, Clifford and Hestenes Tell us About Perception and Action Systems

This paper shows the power of geometric algebra to handle complex tasks of perception action systems. We focus on three robot tasks: invariant theory and omnidirectional vision using conformal geometric algebra for object recognition, conformal geometric control of robot devices and robot learning using Clifford Support Vector Machines. The impact of the ideas as legacy of Hamilton, Grassmann, Clifford and Hestenes is vividly demonstrated with real robot vision tasks.

Theory and Applications of Clifford Support Vector Machines

This paper introduces the Clifford Support Vector Machines (CSVM) as a generalization of the real- and complex-valued Support Vector Machines using the Clifford geometric algebra. In this framework we handle the design of kernels involving the Clifford or geometric product for linear and nonlinear classification and regression. The major advantage of our approach is that we redefine the optimization variables as multivectors. This allows us to have a multivector as output therefore we can represent multiple classes according to the dimension of the geometric algebra in which we work. We conduct comparisons between CSVM and the most used approaches to solve multi-class classification to show that our approach is more suitable for practical use on certain type of multi-class classification problems.

Geometric preprocessing, geometric feedforward neural networks and Clifford support vector machines for visual learning

This paper shows the design and use of feed-forward neural networks and the support vector machines (SVM) in the coordinate-free mathematical system of the Clifford geometric algebra. We compare the McCulloch–Pitts neuron and the geometric neuron. An interesting case of the geometric neuron is the conformal neuron which can be used for RBF networks and SVM. The paper presents the generalization of the real- and complex-valued multilayer perceptron (MLP) to the Clifford valued multilayer perceptron. The paper studies also the multivector support vector machines (MSVM) which are SVMs for processing multivectors. For that we design kernels involving Clifford products. The resultant kernel resembles a sort of polynomial kernel using a multivector representation. In the context of SVMs an important contribution of the paper is the generalization of the real- and complex-valued SVM classifiers over the hyper-complex numbers. This Clifford valued SVM accepts multiple multivector inputs and it is a multi-class classifier. For the preprocessing the authors introduce a promising geometric method utilizing Clifford moments. This method is applied together with geometric MLPs for tasks of 2D pattern recognition. The experimental part shows applications of SVM using the conformal neuron and Clifford kernels. We include challenging applications of the Clifford SVM classifier for nonlinear separable problems. The authors believe that the use of the MLPs and SVMs in the geometric algebra framework expands their sphere of applicability for multivector learning in graded spaces.