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Abstract: The use of Receiver Operator Characteristic (ROC) analysis for the sake of model selection and threshold optimisation has become a standard practice for the design of two-class pattern recognition systems. Advantages include decision boundary adaptation to imbalanced misallocation costs, the ability to fix some classification errors, and performance evaluation in imprecise, ill-defined conditions where costs, or prior probabilities may vary. Extending this to the multiclass case has recently become a topic of interest. The primary challenge involved is the computational complexity, that increases to the power of the number of classes, rendering many problems intractable. In this paper the multiclass ROC is formalised, and the computational complexities exposed. A pairwise approach is proposed that approximates the multi-dimensional operating characteristic by discounting some interactions, resulting in an algorithm that is tractable, and extensible to large numbers of classes. Two additional multiclass optimisation techniques are also proposed that provide a benchmark for the pairwise algorithm. Experiments compare the various approaches in a variety of practical situations, demonstrating the efficacy of the pairwise approach. [Copyright &y& Elsevier]

Many computer vision and pattern recognition problems may be posed as the analysis of a set of dissimilarities between objects. For many types of data, these dissimilarities are not euclidean (i.e., they do not represent the distances between points in a euclidean space), and therefore cannot be isometrically embedded in a euclidean space. Examples include shape-dissimilarities, graph distances and mesh geodesic distances. In this paper, we provide a means of embedding such non-euclidean data onto surfaces of constant curvature. We aim to embed the data on a space whose radius of curvature is determined by the dissimilarity data. The space can be either of positive curvature (spherical) or of negative curvature (hyperbolic). We give an efficient method for solving the spherical and hyperbolic embedding problems on symmetric dissimilarity data. Our approach gives the radius of curvature and a method for approximating the objects as points on a hyperspherical manifold without optimisation. For objects which do not reside exactly on the manifold, we develop a optimisation-based procedure for approximate embedding on a hyperspherical manifold. We use the exponential map between the manifold and its local tangent space to solve the optimisation problem locally in the euclidean tangent space. This process is efficient enough to allow us to embed data sets of several thousand objects. We apply our method to a variety of data including time warping functions, shape similarities, graph similarity and gesture similarity data. In each case the embedding maintains the local structure of the data while placing the points in a metric space. [ABSTRACT FROM AUTHOR]