The study of Newton's method in complex-valued neural networks faces many difficulties. In this paper, we derive Newton's method backpropagation algorithms for complex-valued holomorphic multilayer perceptrons, and investigate the convergence of the one-step Newton steplength algorithm for the minimization of real-valued complex functions via Newton's method. To provide experimental support for the use of holomorphic activation functions, we perform a comparison of using sigmoidal functions versus their Taylor polynomial approximations as activation functions by using the algorithms developed in this paper and the known gradient descent backpropagation algorithm. Our experiments indicate that the Newton's method based algorithms, combined with the use of polynomial activation functions, provide significant improvement in the number of training iterations required over the existing algorithms.

To model biological systems using networks, it is desirable to allow more than two levels of expression for the nodes and to allow the introduction of parameters. Various modeling and simulation methods addressing these needs using Boolean models, both synchronous and asynchronous, have been proposed in the literature. However, analytical study of these more general Boolean networks models is lagging. This paper aims to develop a concise theory for these different Boolean logic-based modeling methods. Boolean models for networks where each node can have more than two levels of expression and Boolean models with parameters are defined algebraically with examples provided. Certain classes of random asynchronous Boolean networks and deterministic moduli asynchronous Boolean networks are investigated in detail using the setting introduced in this paper. The derived theorems provide a clear picture for the attractor structures of these asynchronous Boolean networks.

MOTIVATION: Linking the topology of a complex network to its long-term behavior is a basic problem in network theory, which has been on the focus of many recent research publications. To obtain a suitable Boolean model for a biological system, one must analyze the initial model and compare it with other experimental evidence, and if necessary, make adjustments by changing the topology of the wiring diagram. However, our knowledge on how to link the topology of a network to its long-term behavior is very limited due to the complexity of the problem. Since the need to consider complex biological networks has become ever greater, develop both theoretical foundation and algorithms for model selection and analysis has been brought to the forefront of biological network study. RESULTS: This article proposes a novel method to study intrinsically the relationship between experimental data and the possible Boolean networks, which can be used to model the underlying system. Simple and easy to use criteria for a Boolean network to have both a given network topology and a given set of stable states are derived. These criteria can be used to guide the selection of a Boolean network model for the system, as well as to gain information on the intrinsic properties, such as the robustness and the evolvability, of the system. A Boolean model for the fruit fly Drosophila melanogaster is used to explain the method.

The restricted isometry property (RIP) introduced by Candes and Tao is a fundamental property in compressed sensing theory. It says that if a sampling matrix satisfies the RIP of certain order proportional to the sparsity of the signal, then the original signal can be reconstructed even if the sampling matrix provides a sample vector which is much smaller in size than the original signal. This short note addresses the problem of how a linear transformation will affect the RIP. This problem arises from the consideration of extending the sensing matrix and the use of compressed sensing in different bases. As an application, the result is applied to the redundant dictionary setting in compressed sensing.