9,537 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
Polynomial and rational solutions of holonomic systems
The aim of this paper is to give two new algorithms, which are elimination
free, to find polynomial and rational solutions for a given holonomic system
associated to a set of linear differential operators in the Weyl algebra D =
k where k is a subfield of the complex numbers.Comment: 20 page
Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for DCTs and DSTs
This paper presents a systematic methodology based on the algebraic theory of
signal processing to classify and derive fast algorithms for linear transforms.
Instead of manipulating the entries of transform matrices, our approach derives
the algorithms by stepwise decomposition of the associated signal models, or
polynomial algebras. This decomposition is based on two generic methods or
algebraic principles that generalize the well-known Cooley-Tukey FFT and make
the algorithms' derivations concise and transparent. Application to the 16
discrete cosine and sine transforms yields a large class of fast algorithms,
many of which have not been found before.Comment: 31 pages, more information at http://www.ece.cmu.edu/~smar
Convex and Network Flow Optimization for Structured Sparsity
We consider a class of learning problems regularized by a structured
sparsity-inducing norm defined as the sum of l_2- or l_infinity-norms over
groups of variables. Whereas much effort has been put in developing fast
optimization techniques when the groups are disjoint or embedded in a
hierarchy, we address here the case of general overlapping groups. To this end,
we present two different strategies: On the one hand, we show that the proximal
operator associated with a sum of l_infinity-norms can be computed exactly in
polynomial time by solving a quadratic min-cost flow problem, allowing the use
of accelerated proximal gradient methods. On the other hand, we use proximal
splitting techniques, and address an equivalent formulation with
non-overlapping groups, but in higher dimension and with additional
constraints. We propose efficient and scalable algorithms exploiting these two
strategies, which are significantly faster than alternative approaches. We
illustrate these methods with several problems such as CUR matrix
factorization, multi-task learning of tree-structured dictionaries, background
subtraction in video sequences, image denoising with wavelets, and topographic
dictionary learning of natural image patches.Comment: to appear in the Journal of Machine Learning Research (JMLR
Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization
The purpose of this note is to survey a methodology to solve systems of
polynomial equations and inequalities. The techniques we discuss use the
algebra of multivariate polynomials with coefficients over a field to create
large-scale linear algebra or semidefinite programming relaxations of many
kinds of feasibility or optimization questions. We are particularly interested
in problems arising in combinatorial optimization.Comment: 28 pages, survey pape
Change of basis for m-primary ideals in one and two variables
Following recent work by van der Hoeven and Lecerf (ISSAC 2017), we discuss
the complexity of linear mappings, called untangling and tangling by those
authors, that arise in the context of computations with univariate polynomials.
We give a slightly faster tangling algorithm and discuss new applications of
these techniques. We show how to extend these ideas to bivariate settings, and
use them to give bounds on the arithmetic complexity of certain algebras.Comment: In Proceedings ISSAC'19, ACM, New York, USA. See proceedings version
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