Stability, capacity, and statistical dynamics of second-order bidirectional associative memory

The stability, capacity and statistical dynamics of second-order bidirectional associative memory (BAM) are presented here. We first use an example to illustrate that the state of second-order BAR I may converge to limited cycles. When error in the retrieved pairs is not allowed, a lower bound of memory capacity is derived. That is O(min(n ²/(log n),p²/(log p))) where n and p are the dimensions of the library pairs. Since the state of second-order BAM may converge to limited cycles, the conventional method cannot be used to estimate its memory capacity when small errors in the retrieval pairs are allowed. Hence, the statistical dynamics of second-order BAM is introduced: starting with an initial state close to the library pairs, how the confidence interval of the number of errors changes during recalling. From the dynamics, the attraction basin, memory capacity, and final error in the retrieval pairs can be estimated. Also, some numerical results are given. Finally, an extension of the results to higher-order BAM is discussed