  We study non-linear data-dimension reduction. We are motivated by the
classical linear framework of Principal Component Analysis. In nonlinear case,
we introduce instead a new kernel-Principal Component Analysis, manifold and
feature space transforms. Our results extend earlier work for probabilistic
Karhunen-Lo\`eve transforms on compression of wavelet images. Our object is
algorithms for optimization, selection of efficient bases, or components, which
serve to minimize entropy and error; and hence to improve digital
representation of images, and hence of optimal storage, and transmission. We
prove several new theorems for data-dimension reduction. Moreover, with the use
of frames in Hilbert space, and a new Hilbert-Schmidt analysis, we identify
when a choice of Gaussian kernel is optimal.
