The tutorial starts with an overview of the concepts of VC dimension and structural risk minimization. We then describe linear Support Vector Machines (SVMs) for separable and non-separable data, working through a non-trivial example in detail. We describe a mechanical analogy, and discuss when SVM solutions are unique and when they are global. We describe how support vector training can be practically implemented, and discuss in detail the kernel mapping technique which is used to construct SVM solutions which are nonlinear in the data. We show how Support Vector machines can have very large (even infinite) VC dimension by computing the VC dimension for homogeneous polynomial and Gaussian radial basis function kernels. While very high VC dimension would normally bode ill for generalization performance, and while at present there exists no theory which shows that good generalization performance is guaranteed for SVMs, there are several arguments which support the observed high accuracy of SVMs, which we review. Results of some experiments which were inspired by these arguments are also presented. We give numerous examples and proofs of most of the key theorems. There is new material, and I hope that the reader will find that even old material is cast in a fresh light.

Abstract—Many problems in signal processing and statistical inference involve finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ℓ2) error term combined with a sparseness-inducing (ℓ1) regularization term.Basis pursuit, the least absolute shrinkage and selection operator (LASSO), waveletbased deconvolution, and compressed sensing are a few wellknown examples of this approach. This paper proposes gradient projection (GP) algorithms for the bound-constrained quadratic programming (BCQP) formulation of these problems. We test variants of this approach that select the line search parameters in different ways, including techniques based on the Barzilai-Borwein method. Computational experiments show that these GP approaches perform well in a wide range of applications, often being significantly faster (in terms of computation time) than competing methods. Although the performance of GP methods tends to degrade as the regularization term is de-emphasized, we show how they can be embedded in a continuation scheme to recover their efficient practical performance. A. Background I.

The Support Vector Machine (SVM) is a new and very promising classification technique developed by Vapnik and his group at AT&T Bell Laboratories [3, 6, 8, 24]. This new learning algorithm can be seen as an alternative training technique for Polynomial, Radial Basis Function and Multi-Layer Perceptron classifiers. The main idea behind the technique is to separate the classes with a surface that maximizes the margin between them. An interesting property of this approach is that it is an approximate implementation of the Structural Risk Minimization (SRM) induction principle [23]. The derivation of Support Vector Machines, its relationship with SRM, and its geometrical insight, are discussed in this paper. Since Structural Risk Minimization is an inductive principle that aims at minimizing a bound on the generalization error of a model, rather than minimizing the Mean Square Error over the data set (as Empirical Risk Minimization methods do), training a SVM to obtain the maximum margin classi er requires a different objective function. This objective function is then optimized by solving a large-scale quadratic programming problem with linear and box constraints. The problem is considered challenging, because the quadratic form is completely dense, so the memory

The purpose of this paper is to discuss the scope and functionality of a versatile environment for testing small and large-scale nonlinear optimization algorithms. Although many of these facilities were originally produced by the authors in conjunction with the software package LANCELOT, we believe that they will be useful in their own right and should be available to researchers for their development of optimization software. The tools are available by anonymous ftp from a number of sources and may, in many cases, be installed automatically. The scope of a major collection of test problems written in the standard input format (SIF) used by the LANCELOT software package is described. Recognising that most software was not written with the SIF in mind, we provide tools to assist in building an interface between this input format and other optimization packages. These tools already provide a link between the SIF and an number of existing packages, including MINOS and OSL. In ad...

In this paper a correspondence is derived between regularization operators used in Regularization Networks and Support Vector Kernels. We prove that the Green's Functions associated with regularization operators are suitable Support Vector Kernels with equivalent regularization properties. Moreover the paper provides an analysis of currently used Support Vector Kernels in the view of regularization theory and corresponding operators associated with the classes of both polynomial kernels and translation invariant kernels. The latter are also analyzed on periodical domains. As a by-product we show that a large number of Radial Basis Functions, namely conditionally positive definite functions, may be used as Support Vector kernels.

Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e. for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shiftvariant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shiftvariant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.

The thesis investigates the problem of how information contained in multiple, overlapping images of the same scene may be combined to produce images of superior quality. This area, generically titled frame fusion, offers the possibility of reducing noise, extending the field of view, removal of moving objects, removing blur, increasing spatial resolution and improving dynamic range. As such, this research has many applications in fields as diverse as forensic image restoration, computer generated special effects, video image compression, and digital video editing. An essential enabling step prior to performing frame fusion is image registration, by which an accurate estimate of the point-to-point mapping between views is computed. A robust and efficient algorithm is described to automatically register multiple images using only information contained within the images themselves. The accuracy of this method, and the statistical assumptions upon which it relies, are investigated empirically. Two forms of frame-fusion are investigated. The first is image mosaicing, which is the alignment of multiple images into a single composition representing part of a 3D scene.

. We introduce a new method for maximizing a concave quadratic function with bounds on the variables. The new algorithm combines conjugate gradients with gradient projection techniques, as the algorithm of Mor'e and Toraldo (SIAM J. on Optimization 1, pp. 93-113) and other well-known methods do. A new strategy for the decision of leaving the current face is introduced, that makes it possible to obtain finite convergence even for a singular Hessian and in the presence of dual degeneracy. We present numerical experiments. November 4, 1992 0() Work supported by FAPESP (Grant 90/3724/6), FINEP, CNPq and FAEP-UNICAMP. This paper appeared in SIAM Journal on Optimization 4 (1994) 177-192 1. Introduction. In this paper, we consider the problem of maximizing a concave quadratic function subject to bounds on the variables. This problem (or its equivalent one: minimizing a convex quadratic function on a box) appears frequently in applications, for instance in finite difference discretization ...

Abstract. An active set algorithm (ASA) for box constrained optimization is developed. The algorithm consists of a nonmonotone gradient projection step, an unconstrained optimization step, and a set of rules for branching between the two steps. Global convergence to a stationary point is established. For a nondegenerate stationary point, the algorithm eventually reduces to unconstrained optimization without restarts. Similarly, for a degenerate stationary point, where the strong secondorder sufficient optimality condition holds, the algorithm eventually reduces to unconstrained optimization without restarts. A specific implementation of the ASA is given which exploits the recently developed cyclic Barzilai–Borwein (CBB) algorithm for the gradient projection step and the recently developed conjugate gradient algorithm CG DESCENT for unconstrained optimization. Numerical experiments are presented using box constrained problems in the CUTEr and MINPACK-2 test problem libraries. Key words. nonmonotone gradient projection, box constrained optimization, active set algorithm,

. In this paper, we present a new method for minimizing a convex quadratic function of many variables with box constraints. The new algorithm is a modification of a method introduced recently by Friedlander and Mart'inez (SIAM J. on Optimization, february 1994). Following the lines of Mor'e and Toraldo (SIAM J. on Optimization 1, pp. 93-113), it combines an efficient unconstrained method with gradient projection techniques. The strategy for "leaving the current face" makes it possible to obtain convergence even when the Hessian is singular. Dual nondegeneracy is not assumed anywhere. The unconstrained minimization algorithm used within the faces was introduced by Barzilai and Borwein and analyzed by Raydan (IMA Journal on Numerical Analysis 13, pp. 321-326). Key words: quadratic programming, bound constrained problems, projected gradients, Barzilai-Borwein method. 0() Work supported by FAPESP (Grant 90/3724/6), FINEP, CNPq and FAEP-UNICAMP. This paper appeared in Optimization Methods ...