Time Complexity of In-Memory Matrix-Vector Multiplication

Abstract

Matrix-vector multiplication (MVM) is the core operation of many important algorithms. Crosspoint resistive memory array enables naturally calculating MVM in one operation, thus representing a highly promising computing accelerator for various applications. To evaluate computing performance as well as scalability of in-memory MVM, the fundamental issue of time complexity of the circuit shall be elaborated. Based on the most common MVM circuit that uses transimpedance amplifiers to read out current product in crosspoint array, we analyze its dynamic response and the corresponding time complexity. The result shows that the computing time is governed by the maximal row sum of the implemented matrix, which leads to an explicit time complexity for a specific dataset, e.g., O( N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> ) and O(ln N) for discrete cosine transformation and Toeplitz matrix, respectively. By changing accordingly feedback conductance of transimpedance amplifier for different matrix sizes, it is possible to reduce the time complexity to O(1). Impact of non-ideal factors of the circuit on computing time is also studied. This work provides an insight into the performance and its improvement of MVM computation for efficient in-memory computing accelerators.