Shadow-Cuts Minimization/Maximization and Complex Hopfield Neural Networks

In this article, we continue our very recent work by extending it to the complex case. Having been inspired by the real Hopfield neural network (HNN) results, our investigations here yield various novel results, some of which are as follows. First, extending the “biased pseudo-cut” concept to the complex HNN (CHNN) case, we introduce a “shadow-cut” that is defined as the sum of intercluster phased edges. Second, while the discrete-time real HNN strictly minimizes the “biased pseudo-cut” in each neuron state change, the CHNN “tends” to minimize the shadow-cut (as the CHNN energy function is minimized). Third, these definitions pose a novel L-phased graph clustering (partitioning) problem in which the sum of the shadow-cuts is minimized (or maximized) for the Hermitian complex and the directed graphs whose edges are (possibly arbitrary positive/negative) complex numbers. Finally, combining the CHNN and the pioneering algorithm GADIA of Babadi and Tarokh and their modified versions, we propose simple indirect algorithms to solve the defined shadow-cuts minimization/maximization problem. The proposed algorithms naturally include the CHNN as well as the GADIA as its special cases. The computer simulations confirm the findings.