I'm having trouble understanding the main algorithmic idea in this paper. Is the idea that there are some observed samples {z} which are some geometric transformation of some latent data {x}, z = T_alpha x? Somehow, k examplars are chosen from these unobserved {x} as centers. Then, conditional probability distributions P((alpha_t, k) | (alpha_{t-1}, k_{t-1})), analagous to Kalman filters. If this is what the idea is, then my main confusion is how you know x and alpha given z. It seems like you need these for the training set and the test set.