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Neural Networks, Acyclic Orientations of the Hypercube, and Sets of Orthogonal Vectors

Recent models in the theory of neural networks suggest the possibility of constructing new combinatorial invariants by associating to families of subsets of an n-set orientations of the hypercube via “energy functions.” Here, we restrict ourselves to the Hopfield model and its higher-order versions with families of orthogonal binary vectors and homogeneous polynomial functions, corresponding to sums of outerproducts of degree d, and investigate several properties of the corresponding orientations. In particular, the stability of the vectors in the family and of those in the orthogonal space is analyzed. The existence of significant differences of behavior according to the congruences modulo 4 of d is shown.