J. J. Steil. Input-Output Stability of Recurrent Neural Networks. Cuvillier Verlag, Gottingen, 1999. (Also: Phd.-Dissertation, Faculty of Technology, Bielefeld University, 1999).  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... X j w ij x j h i ; 5) where the transfer function is of the form (x) max(0; x) The standard approach to obtain a non diverging dynamics (5) in the general case of nonsymmetric weights is to choose the combined gains of the transfer function and weights sufficiently small (see Steil 1999 for a review) A simple example is the condition given by Hirsch (1989) which is based on the property that all eigenvalues of the symmetrical parts of the Jacobians of the vector field in (5) must be negative. This gives for (5) the conditions w ii 1 2 X j 6=i jw ij j jw ji j 1 for ....

Steil, J. J. (1999). Input-Output Stability of Recurrent Neural Networks. Ph. D. thesis, University of Bielefeld, Faculty of Technology.

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J. J. Steil. Input-Output Stability of Recurrent Neural Networks. Cuvillier Verlag, Gottingen, 1999. (Also: Phd.-Dissertation, Faculty of Technology, Bielefeld University, 1999).

....our setting yields a typical problem where explicit modelling is impossible and the application of neural net works is a method of choice. For this exam ple we provide a throughout input output stability analysis using recently developed methods originating in non linear feedback system theory [10, 12]. These methods allow to give stability bounds for the fixed adapted network as well as for the time varying network subject to on line learning. In Section 2 we describe the learning task, the network architecture and give (t) Figure 1: A typical part of the Roessler trajectory. some results on ....

....manipulation and does not change the real network. As all feedback parameters ki, kij, ij are bounded in positive intervals and the remaining parts of the system are linear we can apply a multivariable version of the classical Popov stability theorem for the system (6) for more details see [11, 10, 12]) Theorem 1 The system (4) is input output sta ble for all Awij(t) Aij,Aij] and all non linear time stationary feedback functions i(Xi) ki(xi)xi, ki(xi) 0, 1) if there exist diagonal matrices P = diag pi,lSij, Pij ) O, Q = diag qi, 0, 0 0 such that infRe [P( I 3wQ) 3w) R i) ....

Jochen J. Steil and Helge Ritter. Inputoutput stability of recurrent neural networks with delays using circle criteria. In Proc. lnt. 1CSC/1FAC Symp. NEURAL COMPUTATION, NC'98, pages 519 525. ICSC Academic Press, 1998.

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Jochen J. Steil. Input-Output Stability of Recurrent Neural Networks. PhD thesis, University of Bielefeld, Technical Faculty, 1999.

....mainly serves as an illustration of the results in Section 6.2.3. The circle criteria crucially depend on normal approximations of the weight matrix to be analysed and therefore they are best suited for symmetric networks which are subject to on line adaptation or under the influence of noise [SR98]. # Figure as high quality postscript. For the matrices W i , i = 3, 5, 10 the simplest approximation by the respective symmetric parts is not satisfactory, because the approximation error #W (W )# = # is too large to apply any of the graphical criteria. Also the the best ....

Jochen J. Steil and Helge Ritter. Input-output stability of recurrent neural networks with delays using circle criteria. In Proc. Int. ICSC/IFAC Symp. NEURAL COMPUTATION, NC'98, pages 519--525. ICSC Academic Press, 1998. 14

....In this section we derive stability conditions for a network with given local equilibrium. As the resulting conditions can directly be applied based on the material presented here and without detailed knowledge of their theoretical derivation we refer the interested reader for more details to [9 11]. We assume that the network is given in the form 5c = Dx ( A ) x) u(t) 1) where x 6 li is the state vector, D = diag di 0 6 li x is constant, 6 li x is the time stationary weight matrix, A (t) the time varying weight matrix, x) qoz (zz) qo, z, the ....

J.J. Steil. Input-Output Stability of Recurrent Neural Networks. Cuvillier Verlag, G6ttingen, 1999. (Also: Phd.-Dissertation, Faculty of Technology, Bielefeld University, 1999).

.... approach x = G( u (x) with negative feedback used in the non linear feedback theory ( 20] 21] 22] If we define u = W 1 u and G = G C W we can transform (3) to the conventional form, however at the cost of introducing the inverse W 1 which leads to worse stability conditions [29]. B. Sector conditions and parameterization of the feedback In the feedback path we assume that the non linear activation functions are unbiased, i.e. i (0; t) 0; 8 t and that there hold so called sector conditions 0 i (x i ; t) x i k i : 4) As illustrated in Fig. 1 this means ....

....and the other 2N entries q ij ; q ij are set to zero. The corresponding P is passive and we can apply the same loop transformations as for Theorem 2 to obtain the auxiliary G 0 r;P ; and 0 r;P . As the transformed 0 r;P is passive as well and ( K r P ) 1 exists ( 22] [29]) it is possible to apply the Theorem 2 to the modified forward operator GP : Theorem 3 (RNN Popov criterion) The system (7) is inputoutput stable for all w ij (t) 2 [ ij ; ij ] and all non linear time stationary feedback functions i (x i ) k i (x i )x i ; k i (x i ) 2 [0; k i ] if ....

Jochen J. Steil, Input-Output Stability of Recurrent Neural Networks, Cuvillier Verlag, Gottingen, 1999, (Also: Phd.-Dissertation, Faculty of Technology, Bielefeld University, 1999).

....and belongs to the sector [0; 1] i.e. it can be parametrised as tanh(x) k(x)x; k(x) 2 [0; 1] In the following we assume w.r. i 2 [0; 1] 1 In principle GC ; G can also be defined in the time domain as convolution kernels derived from the inverse Laplace transform of GC (s) G(s) [28]. STEIL: INPUT OUTPUT STABILITY OF RECURRENT NEURAL NETWORKS 3 PSfrag replacements G e G u x y x W W e W k i k ij k ij I p I (a) b) e GC R PSfrag replacements G e G u x y x W W e W k i k ij k ij I p I (a) b) e GC R Fig. 2. a) The ....

Jochen J. Steil and Helge Ritter, "Input-output stability of recurrent neural networks with delays using circle criteria," in Proc. Int. ICSC/IFAC Symp. Neural Comp. '98. 1998, pp. 519--525, ICSC Academic Press.

....thus our setting yields a typical problem where explicit modelling is impossible and the application of neural networks is a method of choice. For this example we provide a throughout input output stability analysis using recently developed methods originating in non linear feedback system theory [10, 12]. These methods allow to give stability bounds for the fixed adapted network as well as for the time varying network subject to on line learning. In Section 2 we describe the learning task, the network architecture and give 5 5 10 15 20 1000 800 600 400 200 PSfrag replacements z 1 (t) z 2 ....

....and does not change the real network. As all feedback parameters k i ; k ij ; k ij are bounded in positive intervals and the remaining parts of the system are linear we can apply a multivariable version of the classical Popov stability theorem for the system (6) for more details see [11, 10, 12]) Theorem 1 The system (4) is input output stable for all Deltaw ij (t) 2 [ Delta ij ; Delta ij ] and all non linear time stationary feedback functions i (x i ) k i (x i )x i ; k i (x i ) 2 [0; 1) if there exist diagonal matrices P = diagfp i ; p ij ; p ij g 0, Q = diagfq i ; 0; ....

Jochen J. Steil and Helge Ritter. Inputoutput stability of recurrent neural networks with delays using circle criteria. In Proc. Int. ICSC/IFAC Symp. NEURAL COMPUTATION, NC'98, pages 519--525. ICSC Academic Press, 1998.

....thus our setting yields a typical problem where explicit modelling is impossible and the application of neural networks is a method of choice. For this example we provide a throughout input output stability analysis using recently developed methods originating in non linear feedback system theory [10, 12]. These methods allow to give stability bounds for the fixed adapted network as well as for the time varying network subject to on line learning. In Section 2 we describe the learning task, the network architecture and give 5 5 10 15 20 1000 800 600 400 200 PSfrag replacements z 1 (t) z 2 ....

....and does not change the real network. As all feedback parameters k i ; k ij ; k ij are bounded in positive intervals and the remaining parts of the system are linear we can apply a multivariable version of the classical Popov stability theorem for the system (6) for more details see [11, 10, 12]) Theorem 1 The system (4) is input output stable for all Deltaw ij (t) 2 [ Delta ij ; Delta ij ] and all non linear time stationary feedback functions i (x i ) k i (x i )x i ; k i (x i ) 2 [0; 1) if there exist diagonal matrices P = diagfp i ; p ij ; p ij g 0, Q = diagfq i ; 0; ....

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Jochen J. Steil. Input-Output Stability of Recurrent Neural Networks. PhD thesis, University of Bielefeld, Technical Faculty, 1999.

....q i 0; i = 1: n in this range, whereas the following N rows correspond to the time varying k ij (t) where we have q ij = 0. Remarks: 1 If W is normal (W W = WW ) and A = I we showed previously that it is possible to derive efficient graphical tests involving the eigenvalues of W only [15]. 2 The condition (2) is a special case of (5) when DeltaW = 0. Then we get C = I and can choose P = DA; Q = A Gamma1 for any positive diagonal D, which removes the frequency dependence in (5) and yields (5) D(W Gamma A) S 0 , D( GammaW A) S 0 = 2) 3 Also the norm bound ....

J. J. Steil and H. Ritter. Input-output stability of recurrent neural networks with delays using circle criteria. In Proc. Int. ICSC/IFAC Symp. NEURAL COMPUTATION, NC'98, pages 519--525. ICSC Academic Press, 1998.

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