Minimizing the Absolute Value Energy Function: An Application to Geometrical Constraint-Solving

  • Nami Kin
  • Yoshiaki Takai
  • Tosiyasu L. Kunii
Conference paper
Part of the Focus on Computer Graphics book series (FOCUS COMPUTER)


This paper proposes a new connectionist model to minimize the absolute value energy function based on the Boltzmann machine. In conventional models, n th-order constraints (nonlinear optimization problems) are solved by minimizing the 2n thorder energy function, which brings forth explosive increase of the number of units and connections among them. By using the absolute value energy function, n th-order constraints can be directly represented by the n th-order energy function. Hence, a very great reduction of the network complexity becomes possible. Our model is particularly suited for applications, such as geometrical constraint-solving, in which nonlinearity plays a definitive role.


Energy Function Computer Graphic Connectionist Model Unordered Pair Boltzmann Machine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    E. Aarts and J. Korst. Simulated Annealing and Boltzmann Machines. John Wiley Sons, 1989.Google Scholar
  2. [2]
    J. A. Feldman and D. H. Ballard. Connectionist Models and Their Properties. Cognitive Science, 6: 205–254, 1982.CrossRefGoogle Scholar
  3. [3]
    G. E. Hinton, T. J. Sejnowski, and D. H. Ackley. Boltzmann Machines: Constraint Satisfaction Networks that Learn. Technical Report CMU-CS-84–119, 1984.Google Scholar
  4. [4]
    J. J. Hopfield. Neural Networks and Physical Systems with Emergent Collective Computational Abilities. Proceedings of National Academy of Sciences, 79: 2554 2558, 1982.MathSciNetGoogle Scholar
  5. [5]
    T. Kamada. On Visualization of Abstract Objects and Relations. PhD thesis, Dept. of Information Science, Faculty of Science, The University of Tokyo, 1989.Google Scholar
  6. [6]
    N. Kin, T. Noma, and T. L. Kunii. Picture Editor: A 2D Picture Editing System Based on Geometric Constructions and Constraints. In R. A. Earnshaw and B. Wyvill, editors, New Advances in Computer Graphics: Proceedings of CG International ‘89, pages 193–207. Springer-Verlag, 1989.Google Scholar
  7. [7]
    N. Kin, Y. Takai, and T. L. Kunii. A Connectionist Approach to Geometrical Constraint-Solving. In T. L. Kunii, editor, Modeling in Computer Graphics: Proceedings of IFIP WG 5.10 Working Conference, pages 367–380. Springer-Verlag, 1991.Google Scholar
  8. [8]
    W. Leler. Constraint Programming Languages: Their Specification and Generation. Addison-Wesley, 1988.Google Scholar
  9. [9]
    H. Sato. A Parallel Method for the Extended Boltzmann Machine Based on the Absolute Energy Function. Graduation thesis, Dept. of Information Science, Faculty of Science, The University of Tokyo. in preparation.Google Scholar
  10. [10]
    M. Takeda and J. W. Goodman. Neural Networks for Computation: Number Representations and Programming Complexity. Applied Optics, 25 (18): 3033–3046, 1986.CrossRefGoogle Scholar
  11. [11]
    D. W. Tank and J. J. Hopfield. Simple Neural Optimization Network: An A/D Converter, Signal Decision Circuit, and a Linear Programming Circuit. IEEE Transactions on Circuits and Systems, 33 (5): 533–541, 1986.CrossRefGoogle Scholar

Copyright information

© EUROGRAPHICS The European Association for Computer Graphics 1992

Authors and Affiliations

  • Nami Kin
  • Yoshiaki Takai
  • Tosiyasu L. Kunii

There are no affiliations available

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