Characterization of set of vectors represented by lattices

In this paper, the dynamics of weights of perceptrons are investigated based on the perceptron training algorithm. In particular, an invariant set of the weights of the perceptrons is defined and its properties are studied. By using these properties, a set of vectors that can be represented by integer combinations of a given set of lattices is characterized. If 1) the lattices are employed as training feature vectors of the perceptrons, 2) the threshold of the perceptrons is set to be zero, 3) an initial weight of the perceptrons is in an invariant set, 4) the corresponding desirable outputs and the initial weight of the perceptrons are chosen in such a way that the perceptrons exhibit chaotic behaviors, and 5) a vector is in the invariant set, then the vector can actually be represented as an integer combination of the lattices, and the sum of the half difference between the desirable downsampled outputs and the true downsampled outputs of the perceptron are the corresponding lattice code.