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- Pascal Koiran, Eduardo D. Sontag
- NIPS
- 1995

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 a long-standing open… (More)

- Pascal Koiran
- Theor. Comput. Sci.
- 1994

Abstract We provide a fairly complete picture of the complexity theory of additive real machines. This model of computation is a restriction of the real Turing machine of Blum et al. (1989), since… (More)

- Pascal Koiran
- Theor. Comput. Sci.
- 2010

In their paper on the "chasm at depth four", Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2^o(m) also admit arithmetic circuits… (More)

- Erich Kaltofen, Pascal Koiran
- ISSAC
- 2005

We present algorithms that compute the linear and quadratic factors of supersparse (lacunary) bivariate polynomials over the rational numbers in polynomial-time in the input size. In supersparse… (More)

- Pascal Koiran
- ICS
- 2010

A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic… (More)

- Pascal Koiran
- J. Complexity
- 1996

We show that if the Generalized Riemann Hypothesis is true, the problem of deciding whether a system of polynomial equations in several complex variables has a solution is in the second level of the… (More)

- Pascal Koiran
- 1997

- Pascal Koiran, Michel Cosnard, Max H. Garzon
- Theor. Comput. Sci.
- 1994

It has been known for a short time that a class of recurrent neural networks has universal computational abilities. These networks can be viewed as iterated piecewise-linear maps in a… (More)

- Vincent D. Blondel, Olivier Bournez, Pascal Koiran, Christos H. Papadimitriou, John N. Tsitsiklis
- Theor. Comput. Sci.
- 2001

In this paper we study problems such as: given a discrete time dynamical system of the form x(t + 1)= f(x(t)) where f: R-n --> R-n is a piecewise affine function, decide whether all trajectories… (More)

- Pascal Koiran
- Neural Computation
- 1994

The dynamics of discrete time, continuous state Hopfield networks is driven by an energy function. In this paper, we use this tool to prove under mild hypotheses that any trajectory converges to a… (More)