The TNM staging system has been used since the early 1960's to predict breast cancer patient outcome. In an attempt to increase prognostic accuracy, many putative prognostic factors have been identified. Because the TNM stage model can not accommodate these new factors, the proliferation of factors in breast cancer has lead to clinical confusion. What is required is a new computerized prognostic system that can test putative prognostic factors and integrate the predictive factors with the TNM variables in order to increase prognostic accuracy. Using the area under the curve of the receiver operating characteristic, we compare the accuracy of the following predictive models in terms of five year breast cancer-specific survival: pTNM staging system, principal component analysis, classification and regression trees, logistic regression, cascade correlation neural network, conjugate gradient descent neural, probabilistic neural network, and backpropagation neural network. Several statistical models are significantly more ac- 1064 Harry B. Burke, David B. Rosen, Philip H. Goodman

Language inference and automata induction using recurrent neural networks has gained considerable interest in the recent years. Nevertheless, success of these models has been mostly limited to regular languages. Additional information in form of a priori knowledge has proved important and at times necessary for learning complex languages (Abu-Mostafa 1990; AI-Mashouq and Reed, 1991; Omlin and Giles, 1992; Towell, 1990). They have demonstrated that partial information incorporated in a connectionist model guides the learning process through constraints for efficient learning and better generalization. 'Ve have previously shown that the NNPDA model can learn Deterministic Context 65 66 Das, Giles, and Sun

Language inference and automata induction using recurrent neural networks has gained considerable interest in the recent years. Nevertheless, success of these models hasbeen mostly limited to regular languages. Additional information in form of a priori knowledge has proved important and at times necessary for learning complex languages(Abu-Mostafa 1990; AI-Mashouq and Reed, 1991; Omlin and Giles, 1992; Towell, 1990). They have demonstrated that partial information incorporated in a connectionist model guides the learning process through constraints for efficient learning and better generalization. 'Ve have previously shown that the NNPDA model can learn Deterministic Context 65 66 Das, Giles, and Sun Output

A routing scheme that uses a neural network has been developed that can aid in establishing point-to-point communication routes through multistage interconnection networks (MINs). The neural network is a network of the type that was examined by Hopfield (Hopfield, 1984 and 1985). In this work, the problem of establishing routes through random MINs (RMINs) in a shared-memory, distributed computing system is addressed. The performance of the neural network routing scheme is compared to two more traditional approaches - exhaustive search routing and greedy routing. The results suggest that a neural network router may be competitive for certain RMIN s. 1 INTRODUCTION A neural network has been developed that can aid in establishing point-topoint communication routes through multistage interconnection networks (MINs) (Goudreau and Giles, 1991).

In this work, the problem of establishing routes through random MINs (RMINs) in a shared-memory, distributed computing system is addressed.

J.B. Pollack, Implications of Recursive Distributed Representations, Advances in Neural Information Systems 1, D.S. Touretzky (ed), Morgan Kaufmann, San Mateo, Ca, p. 527 (1989).

J.B. Pollack, Implications of Recursive Distributed Representations, Advances in Neural InformationSystems 1, D.S. Touretzky (ed), Morgan Kaufmann, San Mateo, Ca, p. 527 (1989).

By requiring each unit to satisfy a set of constraints on the interconnection weights, a particular structure is imposed on the network. A network built using such an architecture maintains its invariant performance independent of the values the weights assume, of the learning rules used, and of the form of the nonlinearities in the network. The invariance exhibited by a firstorder networkis usually of a trivial sort, e.g., responding only to the average input in the case of translation invariance, whereas higher-order networks can perform useful functions and still exhibit the invariance. We derive the weight constraints for translation, rotation, scale, and several combinations of these transformations, and report results of simulation studies. INTRODUCTION A persistent difficulty for pattern recognition systems is the requirement that patterns or objects be recognized independent of irrelevant parameters or distortions such as orientation (position, rotation, aspect), scale or size, background or context, doppler shift, time of occurrence, or signal duration.

ENCODING GEOMETRIC INVARIANCES IN HIGHER-ORDER NEURAL NETWORKS C.L. Giles Air Force Office of Scientific Research, Bolling AFB, DC 20332 R.D. Griffin Naval Research Laboratory, Washington, DC 20375-5000 T. Maxwell Sachs-Freeman Associates, Landover, MD 20785 ABSTRACT We describe a method of constructing higher-order neural networks that respond invariantly under geometric transformations on the input space. By requiring each unit to satisfy a set of constraints on the interconnection weights, a particular structure is imposed on the network. A network built using such an architecture maintains its invariant performance independent of the values the weights assume, of the learning rules used, and of the form of the nonlinearities in the network. The invariance exhibited by a firstorder network is usually of a trivial sort, e.g., responding only to the average input in the case of translation invariance, whereas higher-order networks can perform useful functions and still exhibit the invariance. We derive the weight constraints for translation, rotation, scale, and several combinations of these transformations, and report results of simulation studies.