  We present a general framework of semi-supervised dimensionality reduction
for manifold learning which naturally generalizes existing supervised and
unsupervised learning frameworks which apply the spectral decomposition.
Algorithms derived under our framework are able to employ both labeled and
unlabeled examples and are able to handle complex problems where data form
separate clusters of manifolds. Our framework offers simple views, explains
relationships among existing frameworks and provides further extensions which
can improve existing algorithms. Furthermore, a new semi-supervised
kernelization framework called ``KPCA trick'' is proposed to handle non-linear
problems.
