Stabilization and reconstructibility of Boolean networks and compressive power of autoencoders

Abstract

Gene regulatory networks play a significant role in every process of life, and many gene regulatory network modeling methods have been introduced to reveal biological and medical problems. Among these models, Boolean networks (BNs) are one of the simplest models, but can capture most of the dynamic features of gene regulatory networks, making them widely studied. On the other hand, artificial neural network, a technology based on brain and neural system research, is a subset of machine learning and the core of deep learning algorithm. Among the various types of artificial neural networks, autoencoders are attracting a lot of attention for their power to generate new objects, such as image data. In this thesis, two key problems in BNs, namely, stabilization and reconstructibility are first studied, and then the compressive power of autoencoders is investigated. Stabilization and reconstruction of sampled-data Boolean control networks (BCNs) under noisy sampling interval are considered for the first time. By transforming the sampled-data BCN under noisy sampling interval into a PBN, some necessary and sufficient conditions for global stochastic stability of the sampled-data BCN under two types of noisy sampling intervals are obtained. Moreover, the reconstruction problem of sampled-data BCNs under noisy sampling interval can be well solved as a linear programming problem. Then a novel method for the global stochastic stability analysis of aperiodic sampled-data BCNs is introduced. The sampling instants of aperiodic sampled-data control (ASDC) are uncertain and only the activation frequencies of the sampling interval are known. Using the semi-tensor product (STP) of matrices, a BCN under ASDC can be transformed into a BN with stochastic delays. Then by using the Lyapunov function and augmented method, a sufficient condition for the global stochastic stability of BCNs under ASDC is provided. The optimal state estimation issue of BCNs with stochastic disturbances coming from measurements with random delay is then studied. A method is put forward to compute the conditional probability distribution vector of the state through some input and output observations, and then the state of a BCN can be estimated by minimizing the conditional mean squared deviation. Subsequently, the problem of synchronization design for both BNs and singular BNs is investigated. A method for designing all possible synchronized response BNs is proposed for a given drive BN. For a special class of drive singular BNs, the problem of designing a response singular BN to achieve synchronization is studied. The results for constructing synchronized response (singular) BNs are then utilized to design state observers for (singular) BNs. Finally, under the condition that the input and output vectors must be the same, relations between the dimensions of the compressed vectors and the depth and width (the number of nodes in a layer) of autoencoders using linear and ReLU activation functions are studied. Some new results about autoencoders with linear and ReLU activation functions are obtained.