The application of an oblique-projected Landweber method to a model of supervised learning

Björn Johansson, Tommy Elfving, Vladimir Kozlov, Yair Censor, Per Erik Forssén, Gösta Granlund

Research output: Contribution to journalArticlepeer-review


This paper brings together a novel information representation model for use in signal processing and computer vision problems, with a particular algorithmic development of the Landweber iterative algorithm. The information representation model allows a representation of multiple values for a variable as well as an expression for confidence. Both properties are important for effective computation using multi-level models, where a choice between models will be implementable as part of the optimization process. It is shown that in this way the algorithm can deal with a class of high-dimensional, sparse, and constrained least-squares problems, which arise in various computer vision learning tasks, such as object recognition and object pose estimation. While the algorithm has been applied to the solution of such problems, it has so far been used heuristically. In this paper we describe the properties and some of the peculiarities of the channel representation and optimization, and put them on firm mathematical ground. We consider the optimization a convexly constrained weighted least-squares problem and propose for its solution a projected Landweber method which employs oblique projections onto the closed convex constraint set. We formulate the problem, present the algorithm and work out its convergence properties, including a rate-of-convergence result. The results are put in perspective with currently available projected Landweber methods. An application to supervised learning is described, and the method is evaluated in an experiment involving function approximation, as well as application to transient signals.

Original languageEnglish
Pages (from-to)892-909
Number of pages18
JournalMathematical and Computer Modelling
Issue number7-8
StatePublished - Apr 2006

Bibliographical note

Funding Information:
We thank two anonymous referees for their constructive comments which helped us revise the paper. We thank Professor Lars Eldén for calling our attention to the paper [7] . This work was partially supported by the Swedish Research Council through a grant for the project A New Structure for Signal Processing and Learning, and by the Swedish Foundation for Strategic Research through the project VISIT (VIsual Information Technology). The work of Y. Censor is supported by grant No. 2003275 from the United States–Israel Binational Science Foundation (BSF) and by NIH grant No. HL70472. It was done in part at the Center for Computational Mathematics and Scientific Computation (CCMSC) of the University of Haifa and was supported there by grant 522/04 of the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities.


  • Channel representation
  • Nonnegative constraint
  • Preconditioner
  • Projected Landweber
  • Supervised learning

ASJC Scopus subject areas

  • Modeling and Simulation
  • Computer Science Applications


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