Optimization of Real, Hermitian Quadratic Forms: Real, Complex Hopfield-Amari Neural Network
This work addresses theoretical optimization problems in neural network dynamics, but it appears incremental as it extends existing proofs and concepts to complex domains.
The paper tackles the optimization of quadratic forms in Hopfield-Amari neural networks by providing proofs for local/global minima states and generalizing results to complex-valued networks, including structured forms like Toeplitz.
In this research paper, the problem of optimization of quadratic forms associated with the dynamics of Hopfield-Amari neural network is considered. An elegant (and short) proof of the states at which local/global minima of quadratic form are attained is provided. A theorem associated with local/global minimization of quadratic energy function using the Hopfield-Amari neural network is discussed. The results are generalized to a "Complex Hopfield neural network" dynamics over the complex hypercube (using a "complex signum function"). It is also reasoned through two theorems that there is no loss of generality in assuming the threshold vector to be a zero vector in the case of real as well as a "Complex Hopfield neural network". Some structured quadratic forms like Toeplitz form and Complex Toeplitz form are discussed.