Generalization of learning by synchronous waves: from perceptual organization to invariant organization

Consideration of Eq. 7 indicates that the system will have more difficulty discriminating the target representation, \( T_{\lambda } (R) \), from the presence of additional noise compared to discriminating \( T_{\lambda } (R) \) from another representation which is a deformed version of \( T_{\lambda } (R) \). This is because \( v_{j} (p) \) will tend to be larger for \( j \in N \) in the case when \( M \cup N \), \( M \in D(R) \) is presented, compared to when \( j \in O \), \( O \in Q(R)\backslash D(R) \). More specifically, v _j (p) will tend to be higher for noise points (j∈N compared to j∈O\M∩O) and higher for points in the representation (j∈M compared to j∈M∩O). This is true even when M and O differ by one point only, that is, one point is displaced within the target representation. In short, adding noise increases activation levels compared to displacing points within the target representation. For this reason we used additional noise rather than deformations of the target to assess recognition performance.

The 2nd order synapses, as implemented in the present network, bring about two pseudo-problems which, however, do not reflect upon the properties of the generalization mechanism presently of interest. First, 2nd order synapses are not able to distinguish between representations that are rotated by π in relation to each other. Every correctly reinforced 2nd order synapse will also contribute to the ‘recognition’ of a pi-rotated version of the representation. If 3rd order synapses were used in the network, these spurious generalizations would not occur. The second pseudo-problem concerns the implementation of additional noise points in the representation. The generalization mechanism treats pairs of points with the same vector length and direction as identical, since they are translations of each other. That is, any representation that contains points a and b, gives rise a set of connections (with updated gains) of the form B(T _λ{a,b}), each element of which corresponds to the connection vector, ab. For this reason, the noise points, c, added to each representation, while otherwise random, were chosen such that they did not create connection vectors that were translations of the set of the existing set of connection vectors within the pattern. That is, ab  ≠ ca and ab  ≠ ac for each a and b in the n-point representation and for each noise point, c. Use of higher order synapses, while computationally more intensive, greatly reduces this effect, allowing for arbitrary noise points.