Dual-to-kernel learning with ideals
This work addresses the integration of algebraic structure with kernel methods for machine learning practitioners, but appears incremental as it builds on existing paradigms.
The paper tackled the problem of unifying kernel learning and symbolic algebraic methods by showing their inherent duality, and used this to combine structural awareness with efficiency, resulting in two algorithms (IPCA and AVICA) for simultaneous manifold and feature learning tested on synthetic and real-world data.
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.