C L Giles and C W Omlin. Inserting rules into recurrent neural networks. In Procs. of IEEE Signal Processing Workshop, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....rules, described in the next section. We also use the term to describe the (arbitrary) limits imposed on the system, such as restricting the length of the subject to 10 words. This work adopts some of the terminology used by Giles and Omlin in Inserting rules into recurrent neural networks [108]. They characterise rules as a priori knowledge that is encoded into the system. In this work the a priori knowledge is encapsulated in the grammatic framework, and also in the prohibition tables and semi local constraints described in Section 5.5. 5.2 Defining the grammatic framework The ....

C L Giles and C W Omlin. Inserting rules into recurrent neural networks. In Procs. of IEEE Signal Processing Workshop, 1992.

....This allows to neglect the weighted contribute from all the neurons representing the states that are not involved in the transition, since the output of such neurons is supposed to be nearly zero. Rules inserted in this way are defined as hints and the value H is referred to as hint strength [53]. As shown in [52] the hint strength H has a significant influence on the training time. Wrong choices of H may lead to bad training performances, comparable to those obtained without prior knowledge. In particular, experiments in [52] reveal that too large values of H are likely to slow down ....

C. Giles and C. Omlin, "Inserting rules into recurrent neural networks," in Neural Networks for Signal Processing II, Proceedings of the 1992 IEEE workshop (Kung, Fallside, Sorenson, and Kamm, eds.), pp. 13--22, IEEE Press, 1992.

.... recurrent networks (which are essentially trainable deterministic finite automata) Various architectures have been used: simple first order recurrent networks (Elman 1990, Jordan 1988) more complex first order networks (Williams and Zipser 1989, Fahlman 1991) and second order recurrent networks (Giles et al. 1992). Elman (1992) has also applied recurrent networks to context free grammars and found that they can represent up to about three levels of recursive embedding; other authors (Kwasny and Faisal 1990, Das et al. 1993, Zeng et al. 1994) deal with context free grammars by using a neural network in ....

....network to generate representation vectors for a recursive auto associative memory (RAAM) Reilly 1991) The question arises of how to relate the internal representations of a recurrent network to conventional representations of the grammar in terms of production rules or finite automata. Giles and Omlin (1992) and Das et al. 1993) have shown how to insert rules into the network before learning begins, while Castao et al. 1995) 9 14 12 00 describe several methods for converting the network s learned internal representation system into a finite automaton. Some neural networks learn to parse ....

Giles, C.L., and Omlin, C.W., 1992, Inserting rules into recurrent neural networks.

....(to summarize all useful past information) The hard learning problem is to learn to represent context (or state information) which involves performing the proper credit assignment through time. Indeed, in practice, recurrent networks (e.g. injecting prior knowledge for grammar inference [GO92, FGMS93] and HMMs (e.g. for speech recognition [LRS83, RJ86] work quite well when the representation of context (the meaning of the state variable) is decided a priori. The hidden variable is not any more completely hidden. Learning becomes much easier. Unfortunately, this requires a very ....

....[Sin92, DH93, Sut95] and path planning systems. However, with these algorithms, one generally assumes that the state of the system is observed, whereas, in this paper we concentrate on the difficulty of learning what the state variable should represent. On the HMM side, several researchers [BDGO92, Sua94] have attempted to improve HMMs for speech recognition to better model the different types of variables, intrinsically varying at different time scales, observed in speech recognition. Again, however, the focus is on setting an a priori representation, not on learning how to represent ....

[Article contains additional citation context not shown here]

C. L. Giles and C. W Omlin. Inserting rules into recurrent neural networks. In Kung, Fallside, Sorenson, and Kamm, editors, Neural Networks for Signal Processing II, Proceedings of the 1992 IEEE workshop, pages 13--22. IEEE Press, 1992.

....to extract meaningful information from another network (recurrent neural network) 1 Introduction Considerable interest has been shown in language inference using neural networks. Recurrent networks were shown to be able to learn small regular languages (Das and Das, 1991; Watrous and Kuhn, 1992; Giles et al. 1992; Zeng et al. 1994) The recurrent nature of these networks is able to capture the dynamics of the underlying computation automaton (Das, Giles and Sun, 1992) Hidden units activations represent past histories and clusters of these activations can represent the states of the generating automaton ....

....using neural networks. Recurrent networks were shown to be able to learn small regular languages (Das and Das, 1991; Watrous and Kuhn, 1992; Giles et al. 1992; Zeng et al. 1994) The recurrent nature of these networks is able to capture the dynamics of the underlying computation automaton (Das, Giles and Sun, 1992). Hidden units activations represent past histories and clusters of these activations can represent the states of the generating automaton (Giles et al. 1992) The training of the first order recurrent neural networks that recognize finite state languages is discussed in (Elman, 1990) where the ....

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Giles, C. L., and Omlin, C. W. 1992. Inserting rules into recurrent neural network. In Proceedings of The 1992 IEEE Workshop on Neural Networks for Signal Processing, Copenhagen, Denmark.

....algorithms for this purpose, and compare them to two variants of back propagation. One way to help in the training of recurrent networks is to set their connectivity and initial weights (and even constraints on the weights) using prior knowledge. For example, this is accomplished in [8] and [11] using prior rules and sequentiality constraints. In fact, the results in this paper strongly suggest that when such prior knowledge is given, it should be used, since the learning problem itself is so difficult. However, there are many instances where many long term input output dependencies are ....

....becomes increasingly inefficient when the temporal span of the dependencies increases. Furthermore, for a given problem, there are sometimes ways to help the training by setting the network connectivity and initial weights (and even constraints on the weights) using prior knowledge (e.g. 8] [11]) For some tasks, it is also possible to present a variety of examples of the input output dependencies, including short term dependencies which are sufficient to infer similar but longer term dependencies. For example, in the Latch problem or the Parity problem, if we start by training with ....

C.L. Giles and C.W. Omlin, "Inserting Rules into Recurrent Neural Networks", Neural Networks for Signal Processing II, Proceedings of the 1992 IEEE workshop, (eds. Kung, Fallside, Sorenson and Kamm), IEEE Press, pp. 13-22.

....networks. These results confirm the importance of choosing architectures tailored to the task to be solved. The design criteria based on the RNA can be seen as an attempt to tune the network to the task. Other remarkable attempts are based on prior knowledge (e.g. 15] on the use of hints (e.g. [17]) and on dynamic network generation (e.g. 11] Finally, it is worth mentioning that some failures reported in finding optimal solutions may not be related to the presence of local minima, but to very flat plateaus that may led to numerical errors. For recurrent networks, this problem is even ....

C. L. Giles and C. W. Omlin., "Inserting Rules into Recurrent Neural Networks," Neural Network for Signal Processing II, Proc. of the 1992 IEEE Workshop, King, Fallside, Sorenson, and Kamm eds., IEEE Press, pp. 13-22, 1992.

....only form of uncertainty (undefined state transitions) is limited to the number of steps needed for performing actual state transitions . Recurrent neural networks capable of exhibiting this form of nondeterministic behavior were first introduced in [9] 10] but, as pointed out by Giles et. al [14], the proposed algorithm for setting up the weight constraints assumed quite a limited hypothesis on the kind of architectures. They point out that there exist simple finite state automata which can not be represented with first order, fully recurrent architectures without additional layers of ....

....of that information does not overcome the theoretical limitations arising from computational complexity arguments [1] there is no doubt about the actual role of learning from hints in practice. Interesting hints for solving special problems have been proposed by several researchers (e.g. 2] [14], 24] In particular, in [14] 24] the learning of Finite State Automata is proposed using hints which are placed in a very elegant way using second order recurrent networks. Towell et al. 30] 29] conceive the integration of rule based knowledge with neural networks as a three step process: ....

[Article contains additional citation context not shown here]

C.L. Giles and C.W. Omlin., "Inserting Rules into Recurrent Neural Networks", Proc. of IEEE Workshop on Signal Processing and Neural Network (to appear)

....This allows to neglect the weighted contributions from all the neurons representing the states that are not involved in the transition, since the output of such neurons is supposed to be nearly zero. Rules inserted in this way are defined as hints and the value H is referred to as hint strength [55]. As shown in [54] the hint strength H has a significant influence on the training time. Wrong choices of u q 3 Delta q 2 q 1 1 1 0 0 1 0 q 2 q 3 q 1 Figure 5: Translation of knowledge based transition rules into a finite state KBANN. H may lead to bad training performances, comparable to ....

C. Giles and C. Omlin, "Inserting rules into recurrent neural networks," in Neural Networks for Signal Processing II, Proceedings of the 1992 IEEE workshop (Kung, Fallside, Sorenson, and Kamm, eds.), pp. 13--22, IEEE Press, 1992.

....transition probabilities are not exactly 0 or 1) we can obtain a similar interpretation, except that the regions Omega ij have soft boundaries. Because of the multiplicative links, there are some analogies between our architecture and second order recurrent networks that encode discrete states [40]. A second order network with n state units and m inputs evolves according to the equation x t = f 0 n X j=1 x j;t Gamma1 W j u t 1 A (38) where W j ; j = 1; n are n by m matrices of weights. An IOHMM that uses one layered state subnetworks would evolve, instead, with the linear ....

....to the equation x t = f 0 n X j=1 x j;t Gamma1 W j u t 1 A (38) where W j ; j = 1; n are n by m matrices of weights. An IOHMM that uses one layered state subnetworks would evolve, instead, with the linear recurrence i t = n X j=1 i j;t Gamma1 f (W j u t ) 39) Following [40], a second order network can represent discrete states by one hot encoding: x i;t = 1 if the state at time t is i, and x i;t = 0 otherwise. If these encoding assumption are satisfied (again, this will happen if the state units are saturated) equations 39 and 38 are equivalent. In second order ....

C. L. Giles and C. W. Omlin, "Inserting rules into recurrent neural networks," in Neural Networks for Signal Processing II, Proceedings of the 1992 IEEE workshop (Kung, Fallside, Sorenson, and Kamm, eds.), pp. 13--22, IEEE Press, 1992.

....approach of combining HMMs with RNNs. We discuss an algorithm for directly mapping a trained HMM into a RNN architecture and derive a gradient descent learning algorithm for re ning the knowledge. 1 MOTIVATION Recently, there has been a lot of interest in combining symbolic and neural learning [2, 7, 9, 10, 11, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, 27, 28, 30, 31]. There are di erent ways in which neural and symbolic learning can be combined to solve a given learning task. An excellent collection of a variety of approaches can be found in [1] The traditional approach to using neural networks is shown in the lower part ( connectionist representation ) A ....

C. Giles and C. Omlin, \Inserting rules into recurrent neural networks," in Neural Networks for Signal Processing II, Proceedings of The 1992 IEEE Workshop (S. Kung, F. Fallside, J. A. Sorenson, and C. Kamm, eds.), pp. 13{ 22, IEEE Press, 1992.

....approach of combining HMMs with RNNs. We discuss an algorithm for directly mapping a trained HMM into a RNN architecture and derive a gradient descent learning algorithm for refining the knowledge. 1 MOTIVATION Recently, there has been a lot of interest in combining symbolic and neural learning [2, 7, 9, 10, 11, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, 27, 28, 30, 31]. There are different ways in which neural and symbolic learning can be combined to solve a given learning task. An excellent collection of a variety of approaches can be found in [1] The traditional approach to using neural networks is shown in the lower part ( connectionist representation ) ....

C. Giles and C. Omlin, "Inserting rules into recurrent neural networks," in Neural Networks for Signal Processing II, Proceedings of The 1992 IEEE Workshop (S. Kung, F. Fallside, J. A. Sorenson, and C. Kamm, eds.), pp. 13-- 22, IEEE Press, 1992.

....the neuron state activations are confined to the rails of the sigmoid. But the 10 state random DFA should have had a 4 state recurrent network. However, that did not occur; training failed to converge. It would be interesting to see if knowledge inserted into the network before or during training [3, 6, 14] aids or impedes the pruning process. ....

C. L. Giles and C. W. Omlin, "Inserting rules into recurrent neural networks," in Neural Networks for Signal Processing II, Proceedings of The 1992 IEEE Workshop (S. Kung, F. Fallside, J. A. Sorenson, and C. Kamm, eds.), pp. 13--22, IEEE Press, 1992.

....[Zeng et al. 1993] therefore, it can be difficult to make predictions about the generalization performance of trained recurrent networks. Recently, we have developed a simple method for encoding partial DFAs (state transitions) into recurrent neural networks [Giles and Omlin 1993; Omlin and Giles 1992]. We demonstrated that prior knowledge can decrease the learning time significantly compared to learning without any prior knowledge. The training time improvement was proportional to the amount of prior knowledge with which we initialized networks. Important features of our encoding algorithm ....

....network with no more than 2mn 2 m 1 3n continuous neurons and no more than m(n 2 1 m 1 5n 2 5) 1 6n weights. 4. Second Order Networks The algorithm used here to construct DFAs in networks with second order weights has also been used to encode partial prior knowledge to improve convergence time [Giles and Omlin 1992; Omlin and Giles 1992] and to perform rule correction [Giles and Omlin 1993; Omlin and Giles 1996a] 4.1. NETWORK CONSTRUCTION. We use discrete time, recurrent networks with weights W ijk to implement DFAs. A network accepts a time ordered sequence of inputs and evolves with dynamics defined by ....

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GILES, C., AND OMLIN, C. 1992. Inserting rules into recurrent neural networks. In Neural Networks for Signal Processing II, Proceedings of the 1992 IEEE Workshop (S. Kung, F. Fallside, J. A.

....of a vanishing gradient is the essential reason why gradient descent methods are not sufficiently powerful to learn long term dependencies. Several approaches have been suggested to circumvent the problem of vanishing gradients in training RNNs: presetting initial weights by using prior knowledge [6, 9], alternative optimization methods instead of gradientbased [2] reduced description of data [18, 22, 23] architectures that operate on multiple time scales [10, 11] and architectures with high order gating units[12] A class of recurrent neural networks called NARX networks can perform much ....

C.L. Giles and C.W. Omlin. Inserting rules into recurrent neural networks. In S.Y. Kung, F. Fallside, J. Aa. Sorenson, and C.A. Kamm, editors, Neural Networks for Signal Processing II, Proceedings of The 1992 IEEE Workshop, pages 13--22, Piscataway, NJ, 1992. IEEE Press.

....learned DFA states to deteriorate for long strings [32] therefore, it can be difficult to make predictions about the generalization performance of trained recurrent networks. Recently, we have developed a simple method for encoding partial DFAs (state transitions) into recurrent neural networks [12, 24]. The goal was to demonstrate that prior knowledge can decrease the learning time significantly compared to learning without any prior knowledge. The training time improvement was proportional to the amount of prior knowledge with which a network was initialized. Important features of the ....

....by Omega ) are fed through a sigmoidal discriminant function ( to compute the next network state S t 1 i . 4 SECOND ORDER NETWORKS The algorithm used here to construct DFAs in networks with second order weights has also been used to encode partial prior knowledge to improve convergence time [12, 24], and to perform rule correction [13, 22] 4.1 Network Construction We use discrete time, recurrent networks with weights W ijk to implement DFAs. A network accepts a time ordered sequence of inputs and evolves with dynamics defined by the following equations: S (t 1) i = ff i (t) 1 1 ....

C. Giles and C. Omlin, "Inserting rules into recurrent neural networks," in Neural Networks for Signal Processing II, Proceedings of The 1992 IEEE Workshop (S. Kung, F. Fallside, J. A. Sorenson, and C. Kamm, eds.), pp. 13--22, IEEE Press, 1992.

....is used to initialize a new network and simulation results in Section 7. A summary of and potential future work conclude this paper. 2 Knowledge Based Neural Networks Recently, there has been a lot of interest in knowledge based neural networks, i.e. the combining symbolic with neural learning ([4, 14, 20, 29, 40, 44, 41, 43, 39]) There are different ways in which neural and symbolic Initial Domain Theory Knowledge Extraction Knowledge Insertion Symbolic Representation Refined Domain Theory Connectionist Representation Initialized Neural Network Trained Neural Network Training on Data Random Initialization ....

....until network training converges. For the case where recurrent neural networks are trained to behave like a deterministic finite state automaton (DFA) we have shown how second order recurrent networks can be initialized with partial (symbolic) knowledge of a DFA, leading to faster convergence ([20, 34]) It is also possible to extract a symbolic representation of the learned language in the form DFAs from trained network ( 17, 18, 19] The extracted rules always outperform the trained network they were extracted from [35, 32] The focus of this work is on how symbolic knowledge can guide ....

[Article contains additional citation context not shown here]

C. Giles and C. Omlin, "Inserting rules into recurrent neural networks," in Neural Networks for Signal Processing II, Proceedings of The 1992 IEEE Workshop (S. Kung, F. Fallside, J. A. Sorenson, and C. Kamm, eds.), pp. 13--22, IEEE Press, 1992.

....learned DFA states to deteriorate for long strings [23] therefore, it can be difficult to make predictions about the generalization performance of trained recurrent networks. Recently, we have developed a simple method for encoding partial DFA s (state transitions) into recurrent neural networks [9, 18]. The goal was to demonstrate that prior knowledge can decrease the learning time significantly compared to learning without any 0y Technical Report No. 94 3, Computer Science Department, Rensselaer Polytechnic Institute, Troy, NY 12180. prior knowledge. The training time improvement was ....

....with no more than 2mn Gamma m 3n continuous neurons and no more than m(n 2 m 5n Gamma 5) 6n weights. 4 SECOND ORDER NETWORKS The algorithm used here to construct DFA s in networks with second order weights has also been used to encode partial prior knowledge to improve convergence time [9, 18], and to perform rule correction [10, 17] 4.1 Network Construction We use discrete time, recurrent networks with weights W ijk to implement DFA s. A network accepts a time ordered sequence of inputs and evolves with dynamics defined by the following equations: S (t 1) i = h(a i (t) 1 1 ....

C. Giles and C. Omlin, "Inserting rules into recurrent neural networks," in Neural Networks for Signal Processing II, Proceedings of The 1992 IEEE Workshop (S. Kung, F. Fallside, J. A. Sorenson, and C. Kamm, eds.), pp. 13--22, IEEE Press, 1992.

....learned DFA states to deteriorate for long strings [33] therefore, it can be difficult to make predictions about the generalization performance of trained recurrent networks. Recently, we have developed a simple method for encoding partial DFAs (state transitions) into recurrent neural networks [12, 25]. We demonstrated that prior knowledge can decrease the learning time significantly compared to learning without any prior knowledge. The training time improvement was proportional to the amount of prior knowledge with which we initialized networks. Important features of our encoding algorithm ....

.... ( to compute the next network state S t 1 i . and no more than m(n 2 m 5n Gamma 5) 6n weights. 4 SECOND ORDER NETWORKS The algorithm used here to construct DFAs in networks with second order weights has also been used to encode partial prior knowledge to improve convergence time [12, 25], and to perform rule correction [13, 23] 4.1 Network Construction We use discrete time, recurrent networks with weights W ijk to implement DFAs. A network accepts a time ordered sequence of inputs and evolves with dynamics defined by the following equations: S (t 1) i = ff i (t) 1 1 ....

C. Giles and C. Omlin, "Inserting rules into recurrent neural networks," in Neural Networks for Signal Processing II, Proceedings of The 1992 IEEE Workshop (S. Kung, F. Fallside, J. A. Sorenson, and C. Kamm, eds.), pp. 13--22, IEEE, 1992.

....indefinitely due to the built in feedback [29] In particular, they can be encoded [20, 18] and trained [8, 11, 16, 26, 31, 33] to behave like deterministic, sequential finite state automata. Methods for inserting prior knowledge into recurrent neural networks have been previously discussed [4, 6, 7, 12, 15, 21]. It has been demonstrated [12, 21] that prior knowledge can significantly reduce the amount of training necessary for a network to correctly classify a training set of temporal sequences. Our interpretation of rule revision consists of three stages: 1) insert all the available prior knowledge by ....

....[29] In particular, they can be encoded [20, 18] and trained [8, 11, 16, 26, 31, 33] to behave like deterministic, sequential finite state automata. Methods for inserting prior knowledge into recurrent neural networks have been previously discussed [4, 6, 7, 12, 15, 21] It has been demonstrated [12, 21] that prior knowledge can significantly reduce the amount of training necessary for a network to correctly classify a training set of temporal sequences. Our interpretation of rule revision consists of three stages: 1) insert all the available prior knowledge by programming some of the weights of ....

C. Giles and C. Omlin, "Inserting rules into recurrent neural networks," in Neural Networks for Signal Processing II, Proceedings of The 1992 IEEE Workshop (S. Kung, F. Fallside, J. A. Sorenson, and C. Kamm, eds.), (Piscataway, NJ), pp. 13--22, IEEE Press, 1992.

....is the essential reason why gradient descent methods are not sufficiently powerful to learn long term dependencies. Several approaches have been suggested to circumvent the problem of vanishing gradients in training RNNs. One possible approach is to preset initial weights by using prior knowledge [6, 9] but this is often not available in many applications. Another approach is to use alternative optimization methods instead of gradient based methods [2] But, those algorithms can perform as poorly as gradient methods, or require far more computational resources. Alternatively, the input data can ....

C.L. Giles and C.W. Omlin. Inserting rules into recurrent neural networks. In S.Y. Kung, F. Fallside, J. Aa. Sorenson, and C.A. Kamm, editors, Neural Networks for Signal Processing II, Proceedings of The 1992 IEEE Workshop, pages 13--22, Piscataway, NJ, 1992. IEEE Press.

....state q 0 is a rejecting state and 1 otherwise. All weights that are not set to GammaH , GammaH=2 or H are set to zero. B. Learning with Prior Knowledge Empirical studies have shown that partial prior knowledge of a DFA (states and transitions) can significantly improve the training time [23]. Recurrent networks can even perform rule revision, i.e. refine incomplete initial rules [17, 36, 37, 57] and correct incorrect prior knowledge through learning from data [48] C. Stability of Designed Networks The following theorem asserts that time discrete, continuous recurrent neural ....

C. Giles and C. Omlin, "Inserting rules into recurrent neural networks," in Neural Networks for Signal Processing II, Proceedings of The 1992 IEEE Workshop (S. Kung, F. Fallside, J. A. Sorenson, and C. Kamm, eds.), (Piscataway, NJ), pp. 13--22, IEEE Press, 1992.

....Performance, Model Selection, Occam s Razor. To appear in Neural Networks. Neural Network Rule Extraction 2 1 INTRODUCTION There has been much interest in the integration and extraction of knowledge from neural networks (Towell et al. 1990; Frasconi et al. 1991; Hanson Burr 1991; Giles et al. 1992; Watrous Kuhn 1992; Fu 1994) One reason is that the lack of understanding of the rules that underly neural network performance has limited their use in some application domains. Another is that some intelligent processing naturally requires the use of symbolic and rule based knowledge. ....

....processing naturally requires the use of symbolic and rule based knowledge. Recurrent neural networks with discrete time inputs readily lend themselves to certain types of knowledge encoding and extraction in particular the ordered triple of a discrete Markov fstate; input next stateg process (Giles et al. 1992; Omlin Giles, 1992) What this paper addresses is the validity and usefulness of extracting this information based on a representation of the neurons activation. 1.1 Background Recently, the training of recurrent neural networks that recognize finite state languages has been discussed by ....

[Article contains additional citation context not shown here]

Giles, C.L., Omlin, C.W. (1992). Inserting Rules into Recurrent Neural Networks. In S. Kung, F.

....growth. For the third criterion, we propose that just as the network is about to grow, the network preserve as much previously acquired knowledge as possible. Previous work where rules are encoded directly into the recurrent networks have shown that prior knowledge does improve the learning speed [14, 13]. RCC accomplishes this by freezing all previous weight values after a new neuron is added. Another way to do this is to set the new weights to very small values or zero; thus causing the newly added neurons to initially have little or no effect on training. 3 Simple Dynamically Driven Recurrent ....

C. Giles and C. Omlin, "Inserting rules into recurrent neural networks," in Neural Networks for Signal Processing II, Proceedings of The 1992 IEEE Workshop (S. Kung, F. Fallside, J. A. Sorenson, and C. Kamm, eds.), (Piscataway, NJ), pp. 13--22, IEEE Press, 1992.

....is the essential reason why gradient descent methods are not sufficiently powerful to learn long term dependencies. Several approaches have been suggested to circumvent the problem of vanishing gradients in training RNNs. One possible approach is to preset initial weights by using prior knowledge [6, 9] but this is often not available in many applications. Another approach is to use alternative optimization methods instead of gradient based methods [2] But, those algorithms can perform as poorly as gradient methods, or require far more computational resources. Alternatively, the input data can ....

C.L. Giles and C.W. Omlin. Inserting rules into recurrent neural networks. In S.Y. Kung, F. Fallside, J. Aa. Sorenson, and C.A. Kamm, editors, Neural Networks for Signal Processing II, Proceedings of The 1992 IEEE Workshop, pages 13--22, Piscataway, NJ, 1992. IEEE Press.

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