## Tolerance to Distortion

An initial set of simulations was conducted to determine how robust the ABSURDIST algorithm was to noise and how well the algorithm scaled to different sized systems. As such, we ran a 11 x 6 factorial combination of simulations, with 11 levels of added noise and 6 different numbers of elements per system. Noise was infused into the algorithm by varying the displacement between corresponding points across systems. The points in System A were set by randomly selecting dimension values from a uniform random distribution with a range from 0 to 1000. System B points were copied from System A, and Gaussian noise with standard deviations of 0-1% was added to the points of B. The number of points per system was 3, 4, 5, 6, 10, or 15. Correspondences were computed after 1,000 iterations of equation (1). a was set to 0 (no external information was used to determine correspondences), i was set to 0.4. For each combination of noise and number of items, 1,000 separate randomized starting configurations were tested. The results from this simulation are shown in Figure 12.2, which plots the percentage of simulations in which each of the proper correspondences between systems is recovered. For example, for 15-item systems, the figure plots the percentage of time that all 15 correspondences are recovered. The graph shows that performance gradually deteriorates with added noise, but that the algorithm is robust to modest amounts of noise. Relative to the ABSURDIST algorithm described by Goldstone and

0 0.1 0.2 0.3 0.« 0.5 0.6 0,7 0.8 0.9 1 Percent Noise figure 12.2. Probability of correctly translating every element in one system to every element in a second system, as a function of the number of items per system, and the amount of noise with which the elements of the second system are displaced relative to their positions in the first system.

0 0.1 0.2 0.3 0.« 0.5 0.6 0,7 0.8 0.9 1 Percent Noise figure 12.2. Probability of correctly translating every element in one system to every element in a second system, as a function of the number of items per system, and the amount of noise with which the elements of the second system are displaced relative to their positions in the first system.

Rogosky (2002) which lacked adaptive tuning of summed correspondence unit activation, the results in Figure 12.2 show about twice the noise tolerance and only one fifth of the iterations needed.

More surprisingly, Figure 12.2 also shows that for small levels of noise the algorithm's ability to recover true correspondences increases as a function of the number of elements in each system. Up to 0.2% noise, the highest probability of recovering all correct mappings is achieved for the largest, 15-item system. The reason for this is that as the number of elements in a system increases, the similarity relations between those elements provide increasingly strong constraints that serve to uniquely identify each element. The advantage of finding translations as the number of points in a system increases is all the more impressive when one considers chance performance. If one generated random translations that were constrained to allow only one-to-one correspondences, then the probability of generating a completely correct translation would be 1/n! when aligning systems that each have n items. Thus, with 0.7% noise, the 92% rate of recovering all 3 correspondences for a 3-item system is about 5.5 times above chance performance of 16.67%. However, with the same amount of noise, the 16% rate of recovering all of the correspondences for a 15-item system is remarkably higher than the chance rate of 7.6 x 10-13. Thus, at least in our highly simplified domain, we have support for Lenat and Feigenbaum's (1991) argument that establishing meanings on the basis of within-system relations becomes more efficient, not harder, as the size of the system increases.

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