In many machine learning applications, one has access, not only to training data, but also to some high-level a priori knowledge about the desired behavior of the system. For example, it is known in advance that the output of a character recognizer should be invariant with respect to small spatial distortions of the input images (translations, rotations, scale changes, etcetera). We have implemented a scheme that allows a network to learn the derivative of its outputs with respect to distortion operators of our choosing. This not only reduces the learning time and the amount of training data, but also provides a powerful language for specifying what generalizations we wish the network to perform. 1 INTRODUCTION In machine learning, one very often knows more about the function to be learned than just the training data. An interesting case is when certain directional derivatives of the desired function are known at certain points.

In many machine learning applications, one has access, not only to training data, but also to some high-level a priori knowledge about the desired behavior ofthe system. For example, it is known in advance that the output of a character recognizer should be invariant with respect to small spatial distortionsof the input images (translations, rotations, scale changes, etcetera). We have implemented a scheme that allows a network to learn the derivative ofits outputs with respect to distortion operators of our choosing. This not only reduces the learning time and the amount of training data, but also provides a powerful language for specifying what generalizations we wish the network to perform. 1 INTRODUCTION In machine learning, one very often knows more about the function to be learned than just the training data. An interesting case is when certain directional derivatives ofthe desired function are known at certain points.

In many machine learning applications, one has access, not only to training data, but also to some high-level a priori knowledge about the desired behavior of the system. For example, it is known in advance that the output of a character recognizer should be invariant with respect to small spatial distortions of the input images (translations, rotations, scale changes, etcetera). We have implemented a scheme that allows a network to learn the derivative of its outputs with respect to distortion operators of our choosing. This not only reduces the learning time and the amount of training data, but also provides a powerful language for specifying what generalizations we wish the network to perform. 1 INTRODUCTION In machine learning, one very often knows more about the function to be learned than just the training data. An interesting case is when certain directional derivatives of the desired function are known at certain points.

Fully recurrent (asymmetrical) networks can be thought of as dynamic systems. The dynamics can be shaped to perform content addressable memories, recognize sequences, or generate trajectories. Unfortunately several problems can arise: First, the convergence in the state space is not guaranteed. Second, the learned fixed points or trajectories are not necessarily stable. Finally, there might exist spurious fixed points and/or spurious "attracting" trajectories that do not correspond to any patterns.

Bernard Victorri ELSAP Universite de Caen 14032 Caen Cedex France Fully recurrent (asymmetrical) networks can be thought of as dynamic systems. The dynamics can be shaped to perform content addressable memories, recognize sequences, or generate trajectories. Unfortunately several problems can arise: First, the convergence in the state space is not guaranteed. Second, the learned fixed points or trajectories are not necessarily stable. Finally, there might exist spurious fixed points and/or spurious "attracting" trajectories that do not correspond to any patterns.

Fully recurrent (asymmetrical) networks can be thought of as dynamic systems. The dynamics can be shaped to perform content addressable memories, recognize sequences, or generate trajectories. Unfortunately several problems can arise: First, the convergence in the state space is not guaranteed. Second, the learned fixed points or trajectories are not necessarily stable. Finally, there might exist spurious fixed points and/or spurious "attracting" trajectories that do not correspond to any patterns.