Benchmarking least squares support vector machine classifiers.

In Support Vector Machines (SVMs), the solution of the classification problem is characterized by a ( convex) quadratic programming (QP) problem. In a modified version of SVMs, called Least Squares SVM classifiers (LS-SVMs), a least squares cost function is proposed so as to obtain a linear set of equations in the dual space. While the SVM classifier has a large margin interpretation, the LS-SVM formulation is related in this paper to a ridge regression approach for classification with binary targets and to Fisher's linear discriminant analysis in the feature space. Multiclass categorization problems are represented by a set of binary classifiers using different output coding schemes. While regularization is used to control the effective number of parameters of the LS-SVM classifier, the sparseness property of SVMs is lost due to the choice of the 2-norm. Sparseness can be imposed in a second stage by gradually pruning the support value spectrum and optimizing the hyperparameters during the sparse approximation procedure. In this paper, twenty public domain benchmark datasets are used to evaluate the test set performance of LS-SVM classifiers with linear, polynomial and radial basis function (RBF) kernels. Both the SVM and LS-SVM classifier with RBF kernel in combination with standard cross-validation procedures for hyperparameter selection achieve comparable test set performances. These SVM and LS-SVM performances are consistently very good when compared to a variety of methods described in the literature including decision tree based algorithms, statistical algorithms and instance based learning methods. We show on ten UCI datasets that the LS-SVM sparse approximation procedure can be successfully applied.least squares support vector machines; multiclass support vector machines; sparse approximation; discriminant-analysis; sparse approximation; learning algorithms; classification; framework; kernels; time; SISTA;

TVL₁shape approximation from scattered 3D data

With the emergence in 3D sensors such as laser scanners and 3D reconstruction from cameras, large 3D point clouds can now be sampled from physical objects within a scene. The raw 3D samples delivered by these sensors however, contain only a limited degree of information about the environment the objects exist in, which means that further geometrical high-level modelling is essential. In addition, issues like sparse data measurements, noise, missing samples due to occlusion, and the inherently huge datasets involved in such representations makes this task extremely challenging. This paper addresses these issues by presenting a new 3D shape modelling framework for samples acquired from 3D sensor. Motivated by the success of nonlinear kernel-based approximation techniques in the statistics domain, existing methods using radial basis functions are applied to 3D object shape approximation. The task is framed as an optimization problem and is extended using non-smooth L1 total variation regularization. Appropriate convex energy functionals are constructed and solved by applying the Alternating Direction Method of Multipliers approach, which is then extended using Gauss-Seidel iterations. This significantly lowers the computational complexity involved in generating 3D shape from 3D samples, while both numerical and qualitative analysis confirms the superior shape modelling performance of this new framework compared with existing 3D shape reconstruction techniques

TVL₁ Planarity Regularization for 3D Shape Approximation

The modern emergence of automation in many industries has given impetus to extensive research into mobile robotics. Novel perception technologies now enable cars to drive autonomously, tractors to till a field automatically and underwater robots to construct pipelines. An essential requirement to facilitate both perception and autonomous navigation is the analysis of the 3D environment using sensors like laser scanners or stereo cameras. 3D sensors generate a very large number of 3D data points when sampling object shapes within an environment, but crucially do not provide any intrinsic information about the environment which the robots operate within. This work focuses on the fundamental task of 3D shape reconstruction and modelling from 3D point clouds. The novelty lies in the representation of surfaces by algebraic functions having limited support, which enables the extraction of smooth consistent implicit shapes from noisy samples with a heterogeneous density. The minimization of total variation of second differential degree makes it possible to enforce planar surfaces which often occur in man-made environments. Applying the new technique means that less accurate, low-cost 3D sensors can be employed without sacrificing the 3D shape reconstruction accuracy

RBF approximation of large datasets by partition of unity and local stabilization

We present an algorithm to approximate large dataset by Radial Basis Function (RBF) techniques. The method couples a fast domain decomposition procedure with a localized stabilization method. The resulting algorithm can efficiently deal with large problems and it is robust with respect to the typical instability of kernel methods