A finite algorithm for finding the projection of a point onto the Canonical simplex of Rn
Journal of Optimization Theory and Applications
A variational level set approach to multiphase motion
Journal of Computational Physics
Normalized Cuts and Image Segmentation
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model
International Journal of Computer Vision
Problems of learning on manifolds
Problems of learning on manifolds
An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision
IEEE Transactions on Pattern Analysis and Machine Intelligence
Spectral clustering based on the graph p-Laplacian
ICML '09 Proceedings of the 26th Annual International Conference on Machine Learning
Convex Multi-class Image Labeling by Simplex-Constrained Total Variation
SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
The Split Bregman Method for L1-Regularized Problems
SIAM Journal on Imaging Sciences
Global Minimization for Continuous Multiphase Partitioning Problems Using a Dual Approach
International Journal of Computer Vision
De-noising by soft-thresholding
IEEE Transactions on Information Theory
A New TwIST: Two-Step Iterative Shrinkage/Thresholding Algorithms for Image Restoration
IEEE Transactions on Image Processing
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Recent advances in ℓ1 optimization for imaging problems provide promising tools to solve the fundamental high-dimensional data classification in machine learning. In this paper, we extend the main result of Szlam and Bresson (Proceedings of the 27th International Conference on Machine Learning, pp. 1039---1046, 2010), which introduced an exact ℓ1 relaxation of the Cheeger ratio cut problem for unsupervised data classification. The proposed extension deals with the multi-class transductive learning problem, which consists in learning several classes with a set of labels for each class. Learning several classes (i.e. more than two classes) simultaneously is generally a challenging problem, but the proposed method builds on strong results introduced in imaging to overcome the multi-class issue. Besides, the proposed multi-class transductive learning algorithms also benefit from recent fast ℓ1 solvers, specifically designed for the total variation norm, which plays a central role in our approach. Finally, experiments demonstrate that the proposed ℓ1 relaxation algorithms are more accurate and robust than standard ℓ2 relaxation methods s.a. spectral clustering, particularly when considering a very small number of labels for each class to be classified. For instance, the mean error of classification for the benchmark MNIST dataset of 60,000 data in $\mathbb{R}^{784}$ using the proposed ℓ1 relaxation of the multi-class Cheeger cut is 2.4聽% when only one label is considered for each class, while the error of classification for the ℓ2 relaxation method of spectral clustering is 24.7聽%.