On the representation of multi-input systems: Computational properties of polynomial algorithms

• Abelson, H.: Computational geometry of linear threshold functions. Inf. Control 34, 66–92 (1977)

  Google Scholar 

• Abelson, H.: Towards a theory of local and global in computation. Theor. Comput. Sci. 6, 41–67 (1978)

  Google Scholar 

• d'Alessandro, P., Isidori, A., Ruberti, A.: Realization and structure theory of bilinear dynamical systems. SIAM. J. Control 12, 980–996 (1974)

  Google Scholar 

• Arnold, V.I.: Representation of continuous functions of three varibles by the superposition of continuous functions of two variables. Am. Math. Soc. Trans., Ser. 2, 28 (1963)

  Google Scholar 

• Barrett, J.F.: The use of functionals in the analysis of nonlinear systems. J. Electron. Control 15, 567–615 (1963)

  Google Scholar 

• Barrett, J.F.: The series reversion method for solving forced nonlinear differential equations. Mathematica Balkanica 4.5, 43–60 (1974)

  Google Scholar 

• Bedrosian, E., Rice, S.O.: The output properties of Volterra systems. Proc. IEEE 59, 1688–1707 (1971)

  Google Scholar 

• Berger, M.S.: Nonlinearity and functional analysis. Lectures on nonlinear problems in mathematical analysis. New York, San Francisco, London: Academic Press 1977

  Google Scholar 

• Brent, R.P., Kung, H.T.: Fast algorithms for manipulating formal power series. J. ACM 25, 581–595 (1978)

  Google Scholar 

• Brilliant, M.B.: Theory of the analysis of nonlinear systems. Tech. Rep. 345 Res. Lab. El. M.I.T. (1959)

• Bülthoff, H., Poggio, T., Reichardt, W.: Figure-ground discrimination by relative movement in the visual system of the fly. Part II. Movement and relative movement algorithms. To be submitted to Biol. Cybernetics (1980)

• Coleman, B.D.: Finitely convergent learning programs for the separation of sets of shaded figures. Biol. Cybernetics 25, 57–60 (1976)

  Google Scholar 

• Coleman, B.D., Mizel, V.J.: Generalization of the perceptron convergence theorem. Brain Theory Newsletter 1, 67–68 (1976)

  Google Scholar 

• DeSantis, R.M., Porter, W.A.: On the analysis of feedback systems with a polynomic plant. Int. J. Control 21, 159–175 (1975)

  Google Scholar 

• Halme, A.: Polynomial operators for nonlinear systems analysis. Dissertation, Helsinki 1972. Acta Polytechnica Scand. M24 (1972)

• Halme, A., Orava, J., Blomberg, H.: Polynomial operators in nonlinear systems theory. Int. J. Syst. Sci. 2, 25–47 (1971)

  Google Scholar 

• Harper, T.R., Rugh, W.J.: Structural features of factorable Volterra systems. IEEE Trans. Autom. Control 17, 226–228 (1976)

  Google Scholar 

• Holtzman, J.M.: Nonlinear system theory. A functional analysis approach. Englewood Cliffs, N.J.: Prentice-Hall 1970

  Google Scholar 

• Julesz, B.: Experiments in the visual perception of texture. Sci. Am. 232, 34–44 (1975)

  Google Scholar 

• Kohonen, T.: Associative memory. A system-theoretical approach. Berlin, Heidelberg, New York: Springer 1977

  Google Scholar 

• Kolmogorov, A.N.: On the representation of continuous functions of several variables by superpositions of continuous functions of one variable and addition (in russian). Dokl. Akad. Nauk SSSR 114, 679–681 (1957); (Am. Math. Soc. Transl. 28, 55–59 (1963)

  Google Scholar 

• Marmarelis, P.Z., Marmarelis, V.Z.: Analysis of physiological systems. New York: Plenum Press 1978

  Google Scholar 

• McCulloch, W.S., Pitts, W.H.: A logical calculus of ideas immanent in nervous activity. Bull. Math. Biophys 5, 115–133 (1943); New York, London: Plenum Press 1978

  Google Scholar 

• Minsky, M., Papert, S.: Perceptrons. An introduction to computational geometry. Cambridge, Mass., London: MIT Press 1969

  Google Scholar 

• Palm, G.: On representation and approximation of nonlinear systems. Biol. Cybernetics 31, 119–124 (1978)

  Google Scholar 

• Palm, G.: On representation and approximation of nonlinear systems. Part II. Biol. Cybernetics 34, 49–52 (1979)

  Google Scholar 

• Palm, G., Poggio, T.: The Volterra representation and the Wiener expansion: validity and pitfalls. SIAM J. Appl. Math. 33, 195–216 (1977a)

  Google Scholar 

• Palm, G., Poggio, T.: Wiener-like system identification in physiology. J. Math. Biol. 4, 375–381 (1977b)

  Google Scholar 

• Palm, G., Poggio, T.: Stochastic identification methods for nonlinear systems: an extension of the Wiener theory. SIAM J. Appl. Math 34, 524–534 (1978)

  Google Scholar 

• Palm, G., Poggio, T.: Discrete system theory (in preparation 1980)

• Poggio, T.: On optimal nonlinear associative recall. Biol. Cybernetics 19, 201–209 (1975a)

  Google Scholar 

• Poggio, T.: On optimal discrete estimation. In: Proceedings of the 1st Symp. on Testing and Identification of Nonlinear Systems, March 17–20, 1975. McCann, G.D., Marmarelis, P.Z. (eds.), pp. 30–37. California Inst. of Technology 1975b

• Poggio, T., Reichardt, W.: Visual control of orientation behaviour in the fly. Q. Rev. Biophys. 9, 377–438 (1976)

  Google Scholar 

• Poggio, T., Reichardt, W.: Characterization of nonlinear interactions in the fly's visual system. In: Theoretical developments in neurobiology. Neuroscience research program bulletin. Cambridge, Mass., London: MIT Press 1980 (in press)

  Google Scholar 

• Poggio, T., Torre, V.: A Volterra representation for some neuron models. Biol. Cybernetics 27, 113–124 (1977)

  Google Scholar 

• Poggio, T., Torre, V.: A new approach to synaptic interactions. In: Theoretical approaches to complex systems. Proceedings, Tübingen, June 11–12, 1977. Heim, R., Palm, G. (eds.), pp. 89–115 (1978)

• Prenter, P.H.: A Weierstrass theorem for real separable Hilbert spaces. J. Approx. Theory 3, 341–351 (1970)

  Google Scholar 

• Prenter, P.M.: On polynomial operators and equations. In: Nonlinear functional analysis and applications, pp. 361–398. New York, London: Academic Press 1971

  Google Scholar 

• Reichardt, W., Poggio, T.: Figure-ground discrimination by relative movement in the visual system of the fly. Part I. Experimental results. Biol. Cybernetics 35, 81–100 (1979)

  Google Scholar 

• Torre, V., Poggio, T.: A synaptic mechanism possibly underlying directional selectivity to motion. Proc. R. Soc. London, Ser. B 202, 409–416 (1978)

  Google Scholar 

• Uesaka, Y.: Analog perceptrons: on additive representation of functions. Inf. Control 19, 41–65 (1971)

  Google Scholar 

• Uesaka, Y.: Analog perceptron: its decomposition and order. Inf. Control 27, 199–217 (1975)

  Google Scholar 

Download references