6 research outputs found
Dual-to-kernel learning with ideals
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.Comment: 15 pages, 1 figur
The Algebraic Approach to Phase Retrieval and Explicit Inversion at the Identifiability Threshold
We study phase retrieval from magnitude measurements of an unknown signal as
an algebraic estimation problem. Indeed, phase retrieval from rank-one and more
general linear measurements can be treated in an algebraic way. It is verified
that a certain number of generic rank-one or generic linear measurements are
sufficient to enable signal reconstruction for generic signals, and slightly
more generic measurements yield reconstructability for all signals. Our results
solve a few open problems stated in the recent literature. Furthermore, we show
how the algebraic estimation problem can be solved by a closed-form algebraic
estimation technique, termed ideal regression, providing non-asymptotic success
guarantees