Time Complexity of In-Memory Solution of Linear Systems

In-memory computing (IMC) with cross-point resistive memory arrays has been shown to accelerate data-centric computations, such as the training and inference of deep neural networks, due to the high parallelism endowed by physical rules in the electrical circuits. By connecting cross-point arrays with negative feedback amplifiers, it is possible to solve linear algebraic problems, such as linear systems and matrix eigenvectors in just one step. Based on the theory of feedback circuits, we study the dynamics of the solution of linear systems within a memory array, showing that the time complexity of the solution is free of any direct dependence on the problem size <inline-formula> <tex-math notation="LaTeX">${N}$ </tex-math></inline-formula>, rather it is governed by the minimal eigenvalue of an associated matrix of the coefficient matrix. We show that when the linear system is modeled by a covariance matrix, the time complexity is <inline-formula> <tex-math notation="LaTeX">${O}$ </tex-math></inline-formula>(log<inline-formula> <tex-math notation="LaTeX">${N}$ </tex-math></inline-formula>) or <inline-formula> <tex-math notation="LaTeX">${O}$ </tex-math></inline-formula>(1). In the case of sparse positive-definite linear systems, the time complexity is solely determined by the minimal eigenvalue of the coefficient matrix. These results demonstrate the high speed of the circuit for solving linear systems in a wide range of applications, thus supporting IMC as a strong candidate for future big data and machine learning accelerators.