This contribution introduces a new method, the Kernel Flexible Manifold Embedding (KFME), which is a semi-supervised graph-based approach that could map unseen examples to enhance the classification results. This method is an extension of the Flexible Manifold Embedding (FME) proposed by Nie et al , relying on a kernel-based reformulation of the FME. The KFME objective function is proven to be jointly convex and an algorithm is given to compute the optimal solution of the objective function. The KFME is more accurate than the FME when the data show a highly nonlinear structure, as it is demonstrated in the experimental study on various datasets. As KFME has 3 regularization parameters, a part of the experimental study are devoted to the evaluation of the method stability w.r.t. those parameter. The last part of the experimental study is dedicated to investigate the stability w.r.t. the graph methods. In all the experimental aspects, the KFME approach shows very good results.
This contribution shows some very interesting results both on theoretical and applicative side. A possible improvement concerns the section 2 (Related Work): it could ease the reading to explain how the graph similarity matrix S may be computed (page 5) before explaining how the Laplacian matrix is obtained. It could be a generic explanation before introducing the within and between class graph similarity matrices $S_b$ and $S_w$.
In section 2.1 (Semi-supervised Discriminant Analysis), it is confusing to use the same notation $S_w$ and $S_b$ as these matrices are not computed as the ones defined just before. Also, it is difficult to understand Equation (1) without referring to the original publication of Cai et al .
In the results discussion (Section 4.3), the explanation about the better results on test data for some of the dataset is unclear. What is the "complexity of the use datasets"?
Otherwise, there is very few minors typo that could be easily corrected. In the 3 equations on the bottom of page 7, the $S_w$, $S_b$ and $S$ are displayed with a normal font, should not they appear in bold face? On page 15, the second sentence of Section 3.4.1 reads "This latter ...", I think it may be "The latter ...". On page 17, the URL of the Yale dataset is not correct, the underscore character does not appear. On page 23, the term "outperformance" is not usual and may be the sentence could reformulated. In the bibliography, several authors have lost letters of their name, this may be due a to a problem with diacritical signs: see for example reference 14 with Rätsch, Schölkopf and Müller. Same thing goes with references 19 and 29.
 F. Nie, D. Xu, I. W.-H. Tsang, and C. Zhang. Flexible manifold embedding: A framework for semi-supervised and unsupervised dimension reduction. Image Processing, 19(7):1921-1932, 2010
 D. Cai, X. He, and J. Han. Semi-supervised discriminant analysis. ICCV, pp 1-7. 2007