TS-EFA: Resource-efficient High-precision Approximation of Exponential Functions Based on Template-scaling Method

Abstract

Spiking neural network (SNN) utilizes a number of temporal kernels that follow exponential functions with characteristic time-constants <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tau$</tex> . The digital-hardware implementation of SNN-referred to as digital neuromorphic processor-suffers from the heavy workload caused by the exponential function approximation. The challenge is to reconcile the approximation accuracy with hardware resource cost optimally. To this end, we propose an exponential function approximation (EFA) method that reconciles its approximation precision with circuit overhead and calculation speed. This EFA is based on a template-scaling (TS) method; a segment of a full exponential function is taken as a template, and the template is repeatedly scaled to approximate the entire function. Therefore, we refer to our EFA as TS-EFA. The TS-EFA needs two lookup tables (LUT): template and scaling LUTs. The former is allocated to the template, whereas the latter is allocated to the scaling factors for the total bins. For experimental verification, we implemented the TS-EFA in a Xilinx Virtex-7 field-programmable gate array at 500 MHz clock speed. Two types of TS-EFA modules were considered: (i) module with a single time-constant and (ii) multiple time-constants. The module (i) successfully approximates the exponential function with a maximum absolute error of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$1.3\times 10^{-5}$</tex> and a latency of four clock cycles. The module (ii) can be shared among different temporal kernels with different time-constants unlike the module (i). This module performs the approximation with the identical precision but an additional latency of four clock cycles, i.e., total eight clock cycles.