Learning is the gateway to understanding intelligence and to reproducing it in machines. A classical example of learning algorithms is provided by regularization in Reproducing Kernel Hilbert Spaces. The corresponding architecture however is different from the deep hierarchies found in the brain. I will sketch a new attempt (with S. Smale) to develop a mathematics for hierarchical kernel machines – centered around the notion of a recursively defined “derived kernel” – and directly suggested by the neuroscience of the visual cortex.