1. B. DasGupta, H.T. Siegelmann, and E.D. Sontag. On the complexity of training neural networks with continuous activation functions. IEEE Trans. Neural Networks, 6:1490-1504, 1995. [PDF] Keyword(s): machine learning, neural networks, analog computing, theory of computing, neural networks, computational complexity, machine learning. Abstract:

   Blum and Rivest showed that any possible neural net learning algorithm based on fixed architectures faces severe computational barriers. This paper extends their NP-completeness result, which applied only to nets based on hard threshold activations, to nets that employ a particular continuous activation. In view of neural network practice, this is a more relevant result to understanding the limitations of backpropagation and related techniques.




2. H. T. Siegelmann and E.D. Sontag. On the computational power of neural nets. J. Comput. System Sci., 50(1):132-150, 1995. [PDF] [doi:http://dx.doi.org/10.1006/jcss.1995.1013] Keyword(s): machine learning, neural networks, recurrent neural networks, machine learning, analog computing, theory of computing, neural networks, computational complexity, super-Turing computation. Abstract:

   This paper deals with finite size networks which consist of interconnections of synchronously evolving processors. Each processor updates its state by applying a "sigmoidal" function to a rational-coefficient linear combination of the previous states of all units. We prove that one may simulate all Turing Machines by such nets. In particular, one can simulate any multi-stack Turing Machine in real time, and there is a net made up of 886 processors which computes a universal partial-recursive function. Products (high order nets) are not required, contrary to what had been stated in the literature. Non-deterministic Turing Machines can be simulated by non-deterministic rational nets, also in real time. The simulation result has many consequences regarding the decidability, or more generally the complexity, of questions about recursive nets.




3. B. DasGupta, H.T. Siegelmann, and E.D. Sontag. On the Intractability of Loading Neural Networks. In V. P. Roychowdhury, Siu K. Y., and Orlitsky A., editors, Theoretical Advances in Neural Computation and Learning, pages 357-389. Kluwer Academic Publishers, 1994. [PDF] Keyword(s): analog computing, neural networks, computational complexity, machine learning.



4. H. T. Siegelmann and E.D. Sontag. Analog computation via neural networks. Theoret. Comput. Sci., 131(2):331-360, 1994. [PDF] [doi:http://dx.doi.org/10.1016/0304-3975(94)90178-3] Keyword(s): analog computing, neural networks, computational complexity, super-Turing computation, recurrent neural networks, neural networks, computational complexity. Abstract:

   We consider recurrent networks with real-valued weights. If allowed exponential time for computation, they turn out to have unbounded power. However, under polynomial-time constraints there are limits on their capabilities, though being more powerful than Turing Machines. Moreover, there is a precise correspondence between nets and standard non-uniform circuits with equivalent resources, and as a consequence one has lower bound constraints on what they can compute. We note that these networks are not likely to solve polynomially NP-hard problems, as the equality "P=NP" in our model implies the almost complete collapse of the standard polynomial hierarchy. We show that a large class of different networks and dynamical system models have no more computational power than this neural (first-order) model with real weights. The results suggest the following Church-like Thesis of Time-bounded Analog Computing: "Any reasonable analog computer will have no more power (up to polynomial time) than first-order recurrent networks."




5. H. T. Siegelmann and E.D. Sontag. Turing computability with neural nets. Appl. Math. Lett., 4(6):77-80, 1991. [PDF] Keyword(s): machine learning, neural networks, computational complexity, recurrent neural networks. Abstract:

   This paper shows the existence of a finite neural network, made up of sigmoidal neurons, which simulates a universal Turing machine. It is composed of less than 100,000 synchronously evolving processors, interconnected linearly. High-order connections are not required. (Note: this paper was placed here by special request. The results in this paper have been by now improved considerably: see the JCSS pape which among other aspects provides a polynomial time simulation. This paper, based on a unary encoding, results in an exponential slowdown).

1. B. DasGupta, H. T. Siegelmann, and E.D. Sontag. On a learnability question associated to neural networks with continuous activations (extended abstract). In COLT '94: Proceedings of the seventh annual conference on Computational learning theory, New York, NY, USA, pages 47-56, 1994. ACM Press. [doi:http://doi.acm.org/10.1145/180139.181009] Keyword(s): machine learning, analog computing, neural networks, computational complexity.



2. J. L. Balcázar, R. Gavaldà, H. T. Siegelmann, and E.D. Sontag. Some structural complexity aspects of neural computation. In Proceedings of the Eighth Annual Structure in Complexity Theory Conference (San Diego, CA, 1993), Los Alamitos, CA, pages 253-265, 1993. IEEE Comput. Soc. Press. [PDF] Keyword(s): machine learning, analog computing, neural networks, computational complexity, super-Turing computation, theory of computing and complexity. Abstract:

   Recent work by H.T. Siegelmann and E.D. Sontag (1992) has demonstrated that polynomial time on linear saturated recurrent neural networks equals polynomial time on standard computational models: Turing machines if the weights of the net are rationals, and nonuniform circuits if the weights are real. Here, further connections between the languages recognized by such neural nets and other complexity classes are developed. Connections to space-bounded classes, simulation of parallel computational models such as Vector Machines, and a discussion of the characterizations of various nonuniform classes in terms of Kolmogorov complexity are presented.




3. H.T. Siegelmann and E.D. Sontag. Analog computation via neural networks. In Proc. 2nd Israel Symposium on Theory of Computing and Systems (ISTCS93), IEEE Computer Society Press, 1993, 1993. Keyword(s): analog computing, neural networks, computational complexity, super-Turing computation, recurrent neural networks.



4. H.T. Siegelmann and E.D. Sontag. On the computational power of neural nets. In COLT '92: Proceedings of the fifth annual workshop on Computational learning theory, New York, NY, USA, pages 440-449, 1992. ACM Press. [doi:http://doi.acm.org/10.1145/130385.130432] Keyword(s): analog computing, neural networks, computational complexity, super-Turing computation, recurrent neural networks.



5. H.T. Siegelmann and E.D. Sontag. Some results on computing with neural nets. In Proc. IEEE Conf. Decision and Control, Tucson, Dec. 1992, IEEE Publications, 1992, pages 1476-1481, 1992. Keyword(s): analog computing, neural networks, computational complexity, super-Turing computation, recurrent neural networks.



6. H.T. Siegelmann, E.D. Sontag, and C.L. Giles. The Complexity of Language Recognition by Neural Networks. In Proceedings of the IFIP 12th World Computer Congress on Algorithms, Software, Architecture - Information Processing '92, Volume 1, pages 329-335, 1992. North-Holland. Keyword(s): machine learning, neural networks, computational complexity, machine learning, recurrent neural networks, theory of computing and complexity.

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