Quadratic Hadamard Memories 1: Adaptive Stochastic Content-Addressable Memory
1- A novel associative memory is d^fecussed whl\ch overcomes the early saturation problem of Hopfi^ld memories,\without resorting to dilute state vectors or/nonlocal learnling rules. / V The mem. ""y uses a Bidirectional Linear Transformer (BLT) which transfer., the bipolar inpiit vector x into a vfector u, which is a lint ; combination of Hadamard vectors. Tlfie matrix of the BLT is of Hebbian form, equal to the sum of outer products of stored vectors "q^ and Hadamard vectors h . The_ <*.
... ard vectors are considered to serve as labels for the scored vectors. The BLT is followed by a Dominant Label Selector (DLS), which finds the dominant Hadamard component in the linear combination u, and returns the associated Hadamard vector to the BLT, to be processed in the BLT backstroke. This backstroke produces the stored vector closest to the input x. The maximum number of stored vectors that can be perfectly retrieved by associative recall is equal to the dimension N of the BLT and DLS. The present report deals with the DLS, which may be seen as an associative memory which stores N orthogonal bipolar vectors, the Hadamard vectors. A DLS architecture has been found which gives perfect associative recall of these stored vectors. The method involves a quadratic activation which, on account of a group property of Hadamard vectors, requires no more physical connections than a fully connected Hopfield memory of the same dimension. /" / J (^ The dynamics of this "quadratic Hadamard memory" is investigated in the asynchronous discrete model. Stability is assured, and it is shown in a long but simple proof tnat th? stable states of the memory are the Hadamard states and no others. Computer simulations performed for dimension N=16 are in agreement with the theory developed.