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We propose a theoretical framework to analyze semi-supervised classificationunder the low density separation assumption in a high-dimensional regime. Inparticular, we introduce QLDS, a linear classification model, where the lowdensity separation assumption is implemented via quadratic margin maximization.The algorithm has an explicit solution with rich theoretical properties, and weshow that particular cases of our algorithm are the least-square support vectormachine in the supervised case, the spectral clustering in the fullyunsupervised regime, and a class of semi-supervised graph-based approaches. Assuch, QLDS establishes a smooth bridge between these supervised andunsupervised learning methods. Using recent advances in the random matrixtheory, we formally derive a theoretical evaluation of the classification errorin the asymptotic regime. As an application, we derive a hyperparameterselection policy that finds the best balance between the supervised and theunsupervised terms of our learning criterion. Finally, we provide extensiveillustrations of our framework, as well as an experimental study on severalbenchmarks to demonstrate that QLDS, while being computationally moreefficient, improves over cross-validation for hyperparameter selection,indicating a high promise of the usage of random matrix theory forsemi-supervised model selection.

Random Matrix Analysis to Balance between Supervised and Unsupervised Learning under the Low Density Separation Assumption