In this paper, we consider the simplest version of a linear neural network (LNN). Assuming that for training (constructing an optimal weight matrix $Q$) we have a set of training pairs, i.e. we know the input data \begin{ equation} G=\left\{ g^{ \left(1\right)},g^{ \left(2\right)},\cdots,g^{ \left(K\right)}\right\}, \end{ equation} as well as the correct answers to these input data \begin{ equation} H=\left\{ h^{ \left(1\right)},h^{ \left(2\right)},\cdots,h^{ \left(K\right)}\right\}. \end{ equation} We will study the possibilities of constructing a weight matrix $Q$ of a neural network that will give correct answers to arbitrary input data based on the connection of the specified problem with a system of linear algebraic equations (SLAE). Consider a class of neural networks in which each neuron has only one output signal and performs linear operations. We will show how such LNEs are reduced to SLAEs. Since the questions $G$ and the correct answers $H$ are known to us, the desired weight matrix $Q$ must satisfy the equations \begin{ equation} Qg^{ \left(k\right)}=h^{ \left(k\right)}, k=1,2,\cdots,K. \end{ equation} It is required to restore $Q$. In the general case, the matrix $Q$ is rectangular $Q=Q_{ MN}=\left\{ q_{ mn}\right\}$, $m$ is the row number, and $g^{ \left(k\right)}\in\mathbb{ R}^N$, $h^{ \left(k\right)}\in\mathbb{ R}^M$. Let $G_{ NK}$ be a matrix composed of columns $g^{ \left(1\right)}, g^{ \left(2\right)},\cdots,g^{ \left(k\right)}$, and $H_{ MK}$ be a matrix composed of columns $h^{ \left(1\right)},h^{ \left(2\right)},\cdots,h^{ \left(k\right)}$. Then, with respect to $Q_{ MN}$, we obtain a matrix SLAE. \begin{ equation} Q_{ MN}G_{ NK}=H_{ MK}. \end{ equation} This paper will present methods for regularizing the constructed system.
Categories: math.NA cs.NA