  In this paper, we propose a theory which unifies kernel learning and symbolic
algebraic methods. We show that both worlds are inherently dual to each other,
and we use this duality to combine the structure-awareness of algebraic methods
with the efficiency and generality of kernels. The main idea lies in relating
polynomial rings to feature space, and ideals to manifolds, then exploiting
this generative-discriminative duality on kernel matrices. We illustrate this
by proposing two algorithms, IPCA and AVICA, for simultaneous manifold and
feature learning, and test their accuracy on synthetic and real world data.
