We develop a mean field theory for sigmoid belief networks based on ideas from statistical mechanics. Our mean field theory provides a tractable approximation to the true probability distribution in these networks; it also yields a lower bound on the likelihood of evidence. We demonstrate the utility of this framework on a benchmark problem in statistical pattern recognition---the classification of handwritten digits.

Ideally pattern recognition machines provide constant output when the inputs are transformed under a group 9 of desired invariances. These invariances can be achieved by enhancing the training data to include examples of inputs transformed by elements of g, while leaving the corresponding targets unchanged. Alternatively the cost function for training can include a regularization term that penalizes changes in the output when the input is transformed under the group. This paper relates the two approaches, showing precisely the sense in which the regularized cost function approximates the result of adding transformed (or distorted) examples to the training data. The cost function for the enhanced training set is equivalent to the sum of the original cost function plus a regularizer. For unbiased models, the regularizer reduces to the intuitively obvious choice - a term that penalizes changes in the output when the inputs are transformed under the group. For infinitesimal transformations, the coefficient of the regularization term reduces to the variance of the distortions introduced into the training data. This correspondence provides a simple bridge between the two approaches.

This paper discusses the linearly weighted combination of estimators in which the weighting functions are dependent on the input. We show that the weighting functions can be derived either by evaluating the input dependent variance of each estimator or by estimating how likely it is that a given estimator has seen data in the region of the input space close to the input pattern. The latter solution is closely related to the mixture of experts approach and we show how learning rules for the mixture of experts can be derived from the theory about learning with missing features. The presented approaches are modular since the weighting functions can easily be modified (no retraining) if more estimators are added. Furthermore, it is easy to incorporate estimators which were not derived from data such as expert systems or algorithms.

The efficiency of image search can be greatly improved by using a coarse-to-fine search strategy with a multi-resolution image representation. However, if the resolution is so low that the objects have few distinguishing features, search becomes difficult. We show that the performance of search at such low resolutions can be improved by using context information, i.e., objects visible at low-resolution which are not the objects of interest but are associated with them. The networks can be given explicit context information as inputs, or they can learn to detect the context objects, in which case the user does not have to be aware of their existence. We also use Integrated Feature Pyramids, which represent high-frequency information at low resolutions. The use of multiresolution search techniques allows us to combine information about the appearance of the objects on many scales in an efficient way. A natural fOlm of exemplar selection also arises from these techniques. We illustrate these ideas by training hierarchical systems of neural networks to find clusters of buildings in aerial photographs of farmland.

Ideally pattern recognition machines provide constant output when the inputs are transformed under a group 9 of desired invariances. These invariances can be achieved by enhancing the training data to include examples of inputs transformed by elements of g, while leaving the corresponding targets unchanged. Alternatively the cost function for training can include a regularization term that penalizes changes in the output when the input is transformed under the group. This paper relates the two approaches, showing precisely the sense in which the regularized cost function approximates the result of adding transformed (or distorted) examples to the training data. The cost function for the enhanced training set is equivalent to the sum of the original cost function plus a regularizer. For unbiased models, the regularizer reduces to the intuitively obvious choice - a term that penalizes changes in the output when the inputs are transformed under the group. For infinitesimal transformations, the coefficient of the regularization term reduces to the variance of the distortions introduced into the training data. This correspondence provides a simple bridge between the two approaches.

The efficiency of image search can be greatly improved by using a coarse-to-fine search strategy with a multi-resolution image representation. However, if the resolution is so low that the objects have few distinguishing features, search becomes difficult. We show that the performance of search at such low resolutions can be improved by using context information, i.e., objects visible at low-resolution which are not the objects of interest but are associated with them. The networks can be given explicit context information as inputs, or they can learn to detect the context objects, in which case the user does not have to be aware of their existence. We also use Integrated Feature Pyramids, which represent high-frequency information at low resolutions. The use of multiresolution search techniques allows us to combine information about the appearance of the objects on many scales in an efficient way. A natural fOlm of exemplar selection also arises from these techniques. We illustrate these ideas by training hierarchical systems of neural networks to find clusters of buildings in aerial photographs of farmland.

This paper discusses the linearly weighted combination of estimators in which the weighting functions are dependent on the input. We show that the weighting functions can be derived either by evaluating the input dependent variance of each estimator or by estimating how likely it is that a given estimator has seen data in the region of the input space close to the input pattern. The latter solution is closely related to the mixture of experts approach and we show how learning rules for the mixture of experts can be derived from the theory about learning with missing features. The presented approaches are modular since the weighting functions can easily be modified (no retraining) if more estimators are added. Furthermore, it is easy to incorporate estimators which were not derived from data such as expert systems or algorithms.

Volker Tresp*and Michiaki Taniguchi Siemens AG, Central Research Otto-Hahn-Ring 6 81730 Miinchen, Germany Abstract This paper discusses the linearly weighted combination of estimators inwhich the weighting functions are dependent on the input. We show that the weighting functions can be derived either by evaluating the input dependent variance of each estimator or by estimating how likely it is that a given estimator has seen data in the region of the input space close to the input pattern. The latter solutionis closely related to the mixture of experts approach and we show how learning rules for the mixture of experts can be derived from the theory about learning with missing features. The presented approaches are modular since the weighting functions can easily be modified (no retraining) if more estimators are added. Furthermore,it is easy to incorporate estimators which were not derived from data such as expert systems or algorithms. 1 Introduction Instead of modeling the global dependency between input x E D and output y E using a single estimator, it is often very useful to decompose a complex mapping -'\.t the time of the research for this paper, a visiting researcher at the Center for Biological and Computational Learning, MIT.

The efficiency of image search can be greatly improved by using a coarse-to-fine search strategy with a multi-resolution image representation. However,if the resolution is so low that the objects have few distinguishing features,search becomes difficult. We show that the performance of search at such low resolutions can be improved by using context information, i.e., objects visible at low-resolution which are not the objects of interest but are associated with them. The networks can be given explicit context information as inputs, or they can learn to detect the context objects, in which case the user does not have to be aware of their existence. We also use Integrated Feature Pyramids, which represent high-frequencyinformation at low resolutions. The use of multiresolution searchtechniques allows us to combine information about the appearance of the objects on many scales in an efficient way. A natural fOlm of exemplar selection also arises from these techniques. We illustrate theseideas by training hierarchical systems of neural networks to find clusters of buildings in aerial photographs of farmland.

We propose a trajectory planning and control theory for continuous movements such as connected cursive handwriting and continuous natural speech. Its hardware is based on our previously proposed forward-inverse-relaxation neural network (Wada & Kawato, 1993). Computationally, its optimization principle is the minimum torquechange criterion. Regarding the representation level, hard constraints satisfied by a trajectory are represented as a set of via-points extracted from a handwritten character. Accordingly, we propose a via-point estimation algorithm that estimates via-points by repeating the trajectory formation of a character and the via-point extraction from the character. In experiments, good quantitative agreement is found between human handwriting data and the trajectories generated by the theory. Finally, we propose a recognition schema based on the movement generation. We show a result in which the recognition schema is applied to the handwritten character recognition and can be extended to the phoneme timing estimation of natural speech. 1 INTRODUCTION In reaching movements, trajectory formation is an ill-posed problem because the hand can move along an infinite number of possible trajectories from the starting to the target point.