Noisy discriminant analysis with boundary assumptions

摘要

We address the problem of smooth discriminant analysis when data are collected from two samples with measurement errors. This problem turns to be an inverse problem and requires a specific treatment. In this context, we investigate consistency rates of convergence using both a margin assumption, and a complexity assumption in terms of entropy. In particular, we concentrate our attention on a boundary condition on the Bayes set and exhibits two distinct scenarii of convergence for the excess risk. For mildly ill-posed inverse problems, fast rates (i.e. faster than n(-1/2)) may occur whereas in the presence of one 'supersmooth' component for measurement errors, the excess risk is a negative power of log n.