## Limitations and Future Trends in Neural ComputationThis book reports critical analyses on complexity issues in the continuum setting and on generalization to new examples, which are two basic milestones in learning from examples in connectionist models. The problem of loading the weights of neural networks, which is often framed as continuous optimization, has been the target of many criticisms, since the potential solution of any learning problem is severely limited by the presence of local minimal in the error function. The maturity of the field requires to convert the quest for a general solution to all learning problems into the understanding of which learning problems are likely to be solved efficiently. Likewise, the notion of efficient solution needs to be formalized so as to provide useful comparisons with the traditional theory of computational complexity in the discrete setting. The book covers these topics focussing also on recent developments in computational mathematics, where interesting notions of computational complexity emerge in the continuum setting. |

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### Contents

The Complexity of Computing with Continuous Time Devices | 23 |

EnergyBased Computation with Symmetric Hopfield Nets | 45 |

Computational Complexity and the Elusiveness of Glohal Optima | 71 |

Impact of Neural Networks on Signal Processing and Communications | 95 |

Empirical Risk | 115 |

Leaming Highdimensional Data | 141 |

The Curse of Dimensionality and the Blessing of Multiply Hyhrid | 163 |

TeraOPS Stored | 177 |

Reliahility of ManSystem Interaction and Theory of Neural Networks | 216 |

245 | |

### Common terms and phrases

adaptive ahout ahove algorithm analog analysis approach approximation architecture artificial neural networks attractor cell clustering clustermg CNN's comhination computational complexity computational power consider continuous,time convergence corresponding curse of dimensionality decoupling defined hy descrihed dimension dimensional discrete distrihution dynamical systems eigenvalues energy environment error estimation example exponential feedhack filter fixed point functional hlocks Gaussian Gaussian functions glohal graph hased hasic hecause heen hehavior hias high,dimensional hinary Hopfield nets Hopfield networks hoth hound IEEE initial conditions input isomorphism iterations leaming linear Lyapunov exponent mathematical matrix MAX CUT maximal maximum clique method multilayer perceptron Neural Computation neurons nodes noise nonlinear numher ohjects ohservations ohtained optimization prohlem output parameters pattem polynomial possihle prediction prohahility prohlem projection recurrent replicator equations rohot Roska simulation solution space stahle statistical suhgraph suhtree symmetric network theoretical theory translation Turing units values variahles variance vector field weights