Neural networks with quadratic VC dimension. Koiran, P. & Sontag, E. J. Comput. System Sci., 54(1, part 2):190–198, Academic Press, Inc., Orlando, FL, USA, 1997. (1st Annual Dagstuhl Seminar on Neural Computing, 1994)
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This paper shows that neural networks which use continuous activation functions have VC dimension at least as large as the square of the number of weights w. This result settles the open question of whether whether the well-known O(w log w) bound, known for hard-threshold nets, also held for more general sigmoidal nets. Implications for the number of samples needed for valid generalization are discussed.
@ARTICLE{MR1463259,
   AUTHOR       = {P. Koiran and E.D. Sontag},
   JOURNAL      = {J. Comput. System Sci.},
   TITLE        = {Neural networks with quadratic VC dimension},
   YEAR         = {1997},
   OPTMONTH     = {},
   NOTE         = {(1st Annual Dagstuhl Seminar on Neural Computing, 1994)},
   NUMBER       = {1, part 2},
   PAGES        = {190--198},
   VOLUME       = {54},
   ADDRESS      = {Orlando, FL, USA},
   KEYWORDS     = {neural networks, VC dimension},
   PUBLISHER    = {Academic Press, Inc.},
   PDF          = {../../FTPDIR/quadratic-vc.pdf},
   ABSTRACT     = { This paper shows that neural networks which use 
      continuous activation functions have VC dimension at least as large 
      as the square of the number of weights w. This result settles the 
      open question of whether whether the well-known O(w log w) bound, 
      known for hard-threshold nets, also held for more general sigmoidal 
      nets. Implications for the number of samples needed for valid 
      generalization are discussed. },
   DOI          = {http://dx.doi.org/10.1006/jcss.1997.1479}
}
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