# Denoising and pre-images in Kernel PCA

In "Pattern Recognition and Machine Learning" by Bishop, the following problem about Kernel PCA is laid out : In linear PCA, we can approximate data points by projecting them onto the $$L < D$$-dimensional subspace, defined by:

$$\hat{x_n} = \sum_{i=1}^{L} (x_n^T u_i) u_i$$

where $$u_i$$ are the Eigenvectors. From my understanding this is used for de-noising, for example.

Bishop then explains that this is not possible for kernel PCA, since it is not guarented that a vector the pre-image $$x$$ of $$\phi(x)$$ lies on the nonlinear D-dimensional manifold. In particular, he writes:

In kernel PCA, this [projection] will in general not be possible. To see this, note that the mapping $$\phi(x)$$ maps the $$D$$-dimensional $$x$$ space into a $$D$$-dimensional manifold in the $$M$$-dimensional feature space $$\phi$$. [...] However, the projection of points in feature space onto the linear PCA subspace in that space will typically not lie on the nonlinear $$D$$-dimensional manifold and so will not have a corresponding pre-image in data space.

What nonlinear $$D$$-dimensional manifold is meant here? From my understanding, $$\phi(.)$$ maps from the $$D$$ dimensional space to the feature space. Thus every image $$\phi$$ has, by definition, a pre-image in the $$D$$ dimensional space.

What am I missing here?