# Intuitive explanation of GP-LVM

I am looking for an intuitive explanation of Gaussian Process Latent Variable Models (GP-LVM).

Here is what I understand:

• GP-LVM is a nonlinear dimensionality-reduction method. I think "nonlinear" means that there is no linear relationship between the reduced dimensions that we end up with. PCA is a linear dimensionality-reduction method: We end up with principal components, which are the projection of the original data onto the principal axes. There is a linear relationship between principal components of different principal axes as we can always transform one into the other through scaling and addition. Does this sound right?
• GP-LVM assumes there are "latent" variables in a high-dimensional dataset. These are the "important" variables of the dataset as we can obtain all the other data through a nonlinear combination of the latent variables.
• Neil Lawrence gives an introduction in this presentation. He is trying to make a point with these rotated digits on slide 18. Can anyone explain to me what this is about? I understand that when being given a handwritten digit, i.e. 64 x 57 pixel values that are either 0 or 1, when we take random samples, we will never end up seeing the original digit. "Sampling" the image translates into taking random samples and arranging them on a 64 x 57 figure, doesn't it? Now he is rotating the digits a bit and plots principal components. Does this mean he has rotated the digit X times and for each rotated image he plots PC2 and PC3 ? Why doesn't he plot PC1 ? Very confused here.
• Finally, and this is where an intuitive explanation may reach its limits: How are Gaussian processes involved in predicting the original data from the latent variables? Also: How do we find the latent variables ?