On the Computational Complexities of Complex-valued Neural Networks

TL;DR

The paper addresses the need for a unified view of computational complexities across complex-valued neural networks (CVNNs) to enable low-power deployment. It derives both quantitative (closed-form real-multiplication counts) and asymptotic complexities for CVFNN, SCFNN, MLMVN, C-RBF, FC-RBF, and PT-RBF architectures as functions of inputs $P$, outputs $R$, layers $L$, and neurons per hidden layer $N$, treating activations via lookup tables. It provides detailed tables of operation counts for shallow and deep networks and analyzes scaling regimes, showing linear, quadratic, and cubic growth depending on architecture and parameter regimes. The authors apply these results to MIMO channel estimation/decoding, FBMC/OQAM channel estimation, beamforming, and OFDM tasks, concluding that C-RBF often offers lower complexity while PT-RBF can be more resource-intensive in deep networks, thereby guiding design choices for low-power CVNN deployments. Overall, the work offers practical guidance for estimating FLOPs and power budgets, enabling hardware-software co-design of CVNNs in telecommunications and related domains.

Abstract

Complex-valued neural networks (CVNNs) are nonlinear filters used in the digital signal processing of complex-domain data. Compared with real-valued neural networks~(RVNNs), CVNNs can directly handle complex-valued input and output signals due to their complex domain parameters and activation functions. With the trend toward low-power systems, computational complexity analysis has become essential for measuring an algorithm's power consumption. Therefore, this paper presents both the quantitative and asymptotic computational complexities of CVNNs. This is a crucial tool in deciding which algorithm to implement. The mathematical operations are described in terms of the number of real-valued multiplications, as these are the most demanding operations. To determine which CVNN can be implemented in a low-power system, quantitative computational complexities can be used to accurately estimate the number of floating-point operations. We have also investigated the computational complexities of CVNNs discussed in some studies presented in the literature.