On the Noise Model of Support Vector Machine Regression

Support Vector Machines Regression (SVMR) is a regression technique which has been recently introduced by V. Vapnik and his collaborators (Vapnik, 1995; Vapnik, Golowich and Smola, 1996). In SVMR the goodness of fit is measured not by the usual quadratic loss function (the mean square error), but by a different loss function called Vapnik"s $epsilon$- insensitive loss function, which is similar to the "robust" loss functions introduced by Huber (Huber, 1981). The quadratic loss function is well justified under the assumption of Gaussian additive noise. However, the noise model underlying the choice of Vapnik's loss function is less clear. In this paper the use of Vapnik's loss function is shown to be equivalent to a model of additive and Gaussian noise, where the variance and mean of the Gaussian are random variables. The probability distributions for the variance and mean will be stated explicitly. While this work is presented in the framework of SVMR, it can be extended to justify non-quadratic loss functions in any Maximum Likelihood or Maximum A Posteriori approach. It applies not only to Vapnik's loss function, but to a much broader class of loss functions.

Analysis and Implementation of Robust Grasping Behaviors

This thesis addresses the problem of developing automatic grasping capabilities for robotic hands. Using a 2-jointed and a 4-jointed nmodel of the hand, we establish the geometric conditions necessary for achieving form closure grasps of cylindrical objects. We then define and show how to construct the grasping pre-image for quasi-static (friction dominated) and zero-G (inertia dominated) motions for sensorless and sensor-driven grasps with and without arm motions. While the approach does not rely on detailed modeling, it is computationally inexpensive, reliable, and easy to implement. Example behaviors were successfully implemented on the Salisbury hand and on a planar 2-fingered, 4 degree-of-freedom hand.

Learning-Based Approach to Real Time Tracking and Analysis of Faces

This paper describes a trainable system capable of tracking faces and facialsfeatures like eyes and nostrils and estimating basic mouth features such as sdegrees of openness and smile in real time. In developing this system, we have addressed the twin issues of image representation and algorithms for learning. We have used the invariance properties of image representations based on Haar wavelets to robustly capture various facial features. Similarly, unlike previous approaches this system is entirely trained using examples and does not rely on a priori (hand-crafted) models of facial features based on optical flow or facial musculature. The system works in several stages that begin with face detection, followed by localization of facial features and estimation of mouth parameters. Each of these stages is formulated as a problem in supervised learning from examples. We apply the new and robust technique of support vector machines (SVM) for classification in the stage of skin segmentation, face detection and eye detection. Estimation of mouth parameters is modeled as a regression from a sparse subset of coefficients (basis functions) of an overcomplete dictionary of Haar wavelets.

From Regression to Classification in Support Vector Machines

We study the relation between support vector machines (SVMs) for regression (SVMR) and SVM for classification (SVMC). We show that for a given SVMC solution there exists a SVMR solution which is equivalent for a certain choice of the parameters. In particular our result is that for $epsilon$ sufficiently close to one, the optimal hyperplane and threshold for the SVMC problem with regularization parameter C_c are equal to (1-epsilon)^{- 1} times the optimal hyperplane and threshold for SVMR with regularization parameter C_r = (1-epsilon)C_c. A direct consequence of this result is that SVMC can be seen as a special case of SVMR.

On the V(subscript gamma) Dimension for Regression in Reproducing Kernel Hilbert Spaces

This paper presents a computation of the $V_gamma$ dimension for regression in bounded subspaces of Reproducing Kernel Hilbert Spaces (RKHS) for the Support Vector Machine (SVM) regression $epsilon$-insensitive loss function, and general $L_p$ loss functions. Finiteness of the RV_gamma$ dimension is shown, which also proves uniform convergence in probability for regression machines in RKHS subspaces that use the $L_epsilon$ or general $L_p$ loss functions. This paper presenta a novel proof of this result also for the case that a bias is added to the functions in the RKHS.