20 research outputs found
Multiclass Learning with Simplex Coding
In this paper we discuss a novel framework for multiclass learning, defined
by a suitable coding/decoding strategy, namely the simplex coding, that allows
to generalize to multiple classes a relaxation approach commonly used in binary
classification. In this framework, a relaxation error analysis can be developed
avoiding constraints on the considered hypotheses class. Moreover, we show that
in this setting it is possible to derive the first provably consistent
regularized method with training/tuning complexity which is independent to the
number of classes. Tools from convex analysis are introduced that can be used
beyond the scope of this paper
Multi-class SVMs: From Tighter Data-Dependent Generalization Bounds to Novel Algorithms
This paper studies the generalization performance of multi-class
classification algorithms, for which we obtain, for the first time, a
data-dependent generalization error bound with a logarithmic dependence on the
class size, substantially improving the state-of-the-art linear dependence in
the existing data-dependent generalization analysis. The theoretical analysis
motivates us to introduce a new multi-class classification machine based on
-norm regularization, where the parameter controls the complexity
of the corresponding bounds. We derive an efficient optimization algorithm
based on Fenchel duality theory. Benchmarks on several real-world datasets show
that the proposed algorithm can achieve significant accuracy gains over the
state of the art
Reinforced Angle-Based Multicategory Support Vector Machines
The Support Vector Machine (SVM) is a very popular classification tool with many successful applications. It was originally designed for binary problems with desirable theoretical properties. Although there exist various Multicategory SVM (MSVM) extensions in the literature, some challenges remain. In particular, most existing MSVMs make use of k classification functions for a k-class problem, and the corresponding optimization problems are typically handled by existing quadratic programming solvers. In this paper, we propose a new group of MSVMs, namely the Reinforced Angle-based MSVMs (RAMSVMs), using an angle-based prediction rule with k − 1 functions directly. We prove that RAMSVMs can enjoy Fisher consistency. Moreover, we show that the RAMSVM can be implemented using the very efficient coordinate descent algorithm on its dual problem. Numerical experiments demonstrate that our method is highly competitive in terms of computational speed, as well as classification prediction performance. Supplemental materials for the article are available online
Composite Multiclass Losses
We consider loss functions for multiclass prediction problems. We show when a multiclass loss can be expressed as a "proper composite loss", which is the composition of a proper loss and a link function. We extend existing results for binary losses to mu
A Unifying Framework in Vector-valued Reproducing Kernel Hilbert Spaces for Manifold Regularization and Co-Regularized Multi-view Learning
This paper presents a general vector-valued reproducing kernel Hilbert spaces
(RKHS) framework for the problem of learning an unknown functional dependency
between a structured input space and a structured output space. Our formulation
encompasses both Vector-valued Manifold Regularization and Co-regularized
Multi-view Learning, providing in particular a unifying framework linking these
two important learning approaches. In the case of the least square loss
function, we provide a closed form solution, which is obtained by solving a
system of linear equations. In the case of Support Vector Machine (SVM)
classification, our formulation generalizes in particular both the binary
Laplacian SVM to the multi-class, multi-view settings and the multi-class
Simplex Cone SVM to the semi-supervised, multi-view settings. The solution is
obtained by solving a single quadratic optimization problem, as in standard
SVM, via the Sequential Minimal Optimization (SMO) approach. Empirical results
obtained on the task of object recognition, using several challenging datasets,
demonstrate the competitiveness of our algorithms compared with other
state-of-the-art methods.Comment: 72 page
GenSVM: a generalized multiclass support vector machine
Traditional extensions of the binary support vector machine (SVM) to multiclass problems are either heuristics or require solving a large dual optimization problem. Here, a generalized multiclass SVM is proposed called GenSVM. In this method classification boundaries for a K-class problem are constructed in a (K - 1)-dimensional space using a simplex encoding. Additionally, several different weightings of the misclassification errors are incorporated in the loss function, such that it generalizes three existing multiclass SVMs through a single optimization problem. An iterative majorization algorithm is derived that solves the optimization problem without the need of a dual formulation. This algorithm has the advantage that it can use warm starts during cross validation and during a grid search, which signifficantly speeds up the training phase. Rigorous numerical experiments compare linear GenSVM with seven existing multiclass SVMs on both small and large data sets. These comparisons show that the proposed method is competitive with existing methods in both predictive accuracy and training time, and that it signiffcantly outperforms several existing methods on these criteria