Simplicial computation: A methodology to compute vector–vector multiplications with reduced complexity

• Published in: International journal of circuit theory and applications 2021-11, Vol.49 (11), p.3766-3788
• Main Authors: Julian, P., Andreou, A. G., Villemur, M., Rodriguez, N.
• Format: Article
• Language: English
• Subjects: Algorithms; linear combination; Linear functions; Measurement; Multiplication; Parameters; piecewise‐linear functions; vector–vector multiplication
• ISSN: 0098-9886; 1097-007X
• DOI: 10.1002/cta.3128

Summary In this paper, we propose the use of the simplicial algorithm, originally proposed to implement piecewise‐linear functions, to compute a digital vector–vector multiplication (VVM) without multiplications. The main contributions of the proposed methodology are (a) an improved error propagation with respect to parameter quantization; (b) a more efficient digital implementation with respect to area, energy, and speed in the case of large number of inputs; and (c) an intrinsic capability to produce multivariable nonlinear processing. We show that when quantization of inputs and parameters are considered, the simplicial method achieves the same accuracy with fewer representation bits for the parameters, assuming the same quantization for the inputs. Actually, in the particular case of a large number of inputs, the simplicial method needs half the number of parameter bits of a linear combination plus one. In this paper, we propose the use of the simplicial algorithm, originally proposed to implement piecewise‐linear functions, to compute a digital vector–vector multiplication (VVM) without multiplication operations. We show that when quantization of inputs and parameters are considered, the simplicial method achieves the same accuracy with fewer representation bits for the parameters, assuming the same quantization for the inputs. Actually, in the particular case of a large number of inputs, the simplicial method needs half the number of parameter bits of a linear combination plus one. In addition, we show that the simplicial method requires less energy to compute under an equal number of parameter and input representation bits.