Confusion Matrix Stability Bounds for Multiclass Classification

In this paper, we provide new theoretical results on the generalization properties of learning algorithms for multiclass classification problems. The originality of our work is that we propose to use the confusion matrix of a classifier as a measure of its quality; our contribution is in the line of work which attempts to set up and study the statistical properties of new evaluation measures such as, e.g. ROC curves. In the confusion-based learning framework we propose, we claim that a targetted objective is to minimize the size of the confusion matrix C, measured through its operator norm ||C||. We derive generalization bounds on the (size of the) confusion matrix in an extended framework of uniform stability, adapted to the case of matrix valued loss. Pivotal to our study is a very recent matrix concentration inequality that generalizes McDiarmid's inequality. As an illustration of the relevance of our theoretical results, we show how two SVM learning procedures can be proved to be confusion-friendly. To the best of our knowledge, the present paper is the first that focuses on the confusion matrix from a theoretical point of view

On multi-class learning through the minimization of the confusion matrix norm

In imbalanced multi-class classification problems, the misclassification rate as an error measure may not be a relevant choice. Several methods have been developed where the performance measure retained richer information than the mere misclassification rate: misclassification costs, ROC-based information, etc. Following this idea of dealing with alternate measures of performance, we propose to address imbalanced classification problems by using a new measure to be optimized: the norm of the confusion matrix. Indeed, recent results show that using the norm of the confusion matrix as an error measure can be quite interesting due to the fine-grain informations contained in the matrix, especially in the case of imbalanced classes. Our first contribution then consists in showing that optimizing criterion based on the confusion matrix gives rise to a common background for cost-sensitive methods aimed at dealing with imbalanced classes learning problems. As our second contribution, we propose an extension of a recent multi-class boosting method --- namely AdaBoost.MM --- to the imbalanced class problem, by greedily minimizing the empirical norm of the confusion matrix. A theoretical analysis of the properties of the proposed method is presented, while experimental results illustrate the behavior of the algorithm and show the relevancy of the approach compared to other methods

Multiclass Data Segmentation using Diffuse Interface Methods on Graphs

We present two graph-based algorithms for multiclass segmentation of high-dimensional data. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs simplex, with the functional's double-well potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm is a uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, grayscale and color images, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current state-of-the-art multiclass segmentation algorithms.Comment: 14 page

Towards minimizing the energy of slack variables for binary classification

This paper presents a binary classification algorithm that is based on the minimization of the energy of slack variables, called the Mean Squared Slack (MSS). A novel kernel extension is proposed which includes the withholding of just a subset of input patterns that are misclassified during training. The later leads to a time and memory efficient system that converges in a few iterations. Two datasets are exploited for performance evaluation, namely the adult and the vertebral column dataset. Experimental results demonstrate the effectiveness of the proposed algorithm with respect to computation time and scalability. Accuracy is also high. In specific, it equals 84.951% for the adult dataset and 91.935%, for the vertebral column dataset, outperforming state-of-the-art methods. © 2012 EURASIP