We present a general formulation for a network of stochastic di(cid:173) rectional units. This formulation is an extension of the Boltzmann machine in which the units are not binary, but take on values in a cyclic range, between 0 and 271' radians. The state of each unit in a Directional-Unit Boltzmann Machine (DUBM) is described by a complex variable, where the phase component specifies a direction; the weights are also complex variables. The conditional distribution of a unit's stochastic state is a circular version of the Gaussian probability distribution, known as the von Mises distribution. In a mean-field approxima(cid:173) tion to a stochastic DUBM, the phase component of a unit's state represents its mean direction, and the magnitude component spec(cid:173) ifies the degree of certainty associated with this direction.

University of Toronto University of Toronto University of Colorado Toronto, ONT M5S lA4 Toronto, ONT M5S lA4 Boulder, CO 80309-0430 Abstract We present a general formulation for a network of stochastic directional units. This formulation is an extension of the Boltzmann machine in which the units are not binary, but take on values in a cyclic range, between 0 and 271' radians. The conditional distribution of a unit's stochastic state is a circular version of the Gaussian probability distribution, known as the von Mises distribution. This combination of a value and a certainty provides additional representational power in a unit. Many kinds of information can naturally be represented in terms of angular, or directional, variables.

University of Toronto University of Toronto University of Colorado Toronto, ONT M5S lA4 Toronto, ONT M5S lA4 Boulder, CO 80309-0430 Abstract We present a general formulation for a network of stochastic directional units. This formulation is an extension of the Boltzmann machine in which the units are not binary, but take on values in a cyclic range, between 0 and 271' radians. The conditional distribution of a unit's stochastic state is a circular version of the Gaussian probability distribution, known as the von Mises distribution. This combination of a value and a certainty provides additional representational power in a unit. Many kinds of information can naturally be represented in terms of angular, or directional, variables.

University of Colorado Boulder, CO 80309-0430 Abstract We present a general formulation for a network of stochastic directional units.This formulation is an extension of the Boltzmann machine in which the units are not binary, but take on values in a cyclic range, between 0 and 271' radians. The conditional distribution of a unit's stochastic state is a circular version of the Gaussian probability distribution, known as the von Mises distribution. This combination of a value and a certainty provides additional representational powerin a unit. Many kinds of information can naturally be represented in terms of angular, or directional, variables. A circular range forms a suitable representation for explicitly directional information, such as wind direction, as well as for information where the underlying range is periodic, such as days of the week or months of the year.