This paper discusses recent results concerning the dynamics of the Hopfield-Tank travelling salesman problem (TSP) model (Euclidean). Ayier's subspace approach is used to ensure feasible solutions. A fuzzy approach to "reading" the state of the system is introduced. This consists of computing, at each iteration, the center of mass of the positive activations in each row of the neural array and sorting them to obtain a "fuzzy" tour at each iteration. The fuzzy approach exposes the inner workings of the system in a unique and surprising way. The system goes through three phases: centroid, monotonic, and nearest-city. These phases are directly related to city position statistics, such as the centroid and principal axis of the city coordinates. We provide a 50 city example.

This article describes a Hopfield-Tank model of the Euclidean traveling salesman problem (using Aiyer's subspace approach) that incorporates a “fuzzy” interpretation of the rows of the activation array. This fuzzy approach is used to explain the network's behavior. The fuzzy interpretation consists of computing the center of mass of the positive activations in each row. This produces real numbers along the time line, one for each city, and defines a tour in a natural way (referred to as a “fuzzy tour”). A fuzzy tour is computed at each iteration, and examination of such tours exposes fine features of the network dynamics. For example, these tours demonstrate that the network goes through (at least) three phases: “centroid,” “monotonic,” and “nearest-city.” The centroid (initial) phase of the network produces an approximation to the centroid tour (i.e., the tour that orders the cities by central angle with respect to the centroid of the cities). The second phase produces an approximation to the monotonic tour (i.e., the tour that orders the cities by their projection onto the principal axis of the cities). The third phase produces an approximation to a nearest-city tour, that is, a tour whose length is between the tour lengths of the nearest-city-best (best starting city) and nearest-city-worst (worst starting city). Simulations are presented for up to 125 uniformly randomly generated cities.

This paper presents a significant improvement to the traditional neural approach to the assignment problem (AP). The technique is based on identifying the feasible space (F) with a linear subspace of R(n(2)), and then analyzing the orthogonal projection onto F. The formula for the orthogonal projection is shown to be simple and easy to integrate into the traditional neural model. This projection concept was first developed by Wolfe et al. (1993), but here we show that the projection can be computed in a much simpler way, and that the addition of a "clip" operator at the boundaries of the cube can improve the results by an order of magnitude in both accuracy and run time. It is proven that the array of numbers that define an AP can be projected onto F without loss of information and the network can be constrained to operate exclusively in F until a neuron is saturated (i.e., reaches the maximum or minimum activation). Two "clip" options are presented and compared. Statistical results are presented for randomly generated APs of sizes n=10 to n=50. The statistics confirm the theory.

We introduce a generalization of mutually inhibitory networks called homogeneous networks. Such networks have symmetric connection strength matrices that are circulant (one-dimensional case) or block circulant with circulant blocks (two-dimensional case). Fourier harmonics provide universal eigenvectors, and we apply them to several homogeneous examples: k-wta, k-cluster, on/center off/surround, and the assignment problem. We also analyze one nonhomogeneous case: the subset-sum problem. We present the results of 10000 trials on a 50-node k-cluster problem and 100 trials on a 25-node subset-sum problem.

A family of symmetric neural networks that solve a simple version of the assignment problem (AP) is analyzed. The authors analyze the suboptimal performance of these networks and compare the results to optimal answers obtained by linear programming techniques. They then use the interactive activation model to define the network dynamics-a model that is closely related to the Hopfield-Tank model. A systematic analysis of hypercube corner stability and eigenspaces of the connection strength matrix leads to network parameters that give feasible solutions 100% of the time and to a projection algorithm that significantly improves performance. Two formulations of the problem are discussed: (i) nearest corner: encode the assignment numbers as initial activations, and (ii) lowest energy corner: encode the assignment numbers as external inputs.

The perspective view of three noncollinear points whose image-to-object correspondence is known is studied. Such measurements are known to be ambiguous, resulting in as many as four possible solutions to the perspective three-point problem. Although there can be four solutions, it is quite often the case that there are triangle configurations that cause one, two, three, or four solutions. The results also provide a justification for the common wisdom that there are usually two solutions.>

A special class of mutually inhibitory networks is analyzed, and parameters for reliable K-winner performance are presented. The network dynamics are modeled using interactive activation, and results are compared with the sigmoid model. For equal external inputs, network parameters that select the units with the larger initial activations (the network converges to the nearest stable state) are derived. Conversely, for equal initial activations, networks that select the units with larger external inputs (the network converges to the lowest energy stable state) are derived. When initial activations are mixed with external inputs, anomalous behavior results. These discrepancies are analyzed with several examples. Restrictions on initial states are derived which ensure accurate K-winner performance when unequal external inputs are used.

A model-based image understanding and spatial reasoning system (SRS) has been integrated with a robot planner to provide a testbed for autonomously sensing and responding to complex but well-structured environments. The entire system operates in such a way that the robot planner requests state information for entities on a task panel from the SRS. State information may consist of the location or orientation of a latch or door on the panel or it may reflect a particular function, such as whether a latch is open or closed. Examples which illustrate the operation of the entire system with actual robotic and vision hardware are provided. The advantages and disadvantages of active/passive markers and teleoperation within the context of the integrated system are also discussed.>

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