Complex-valued neural networks potentially possess better representations and performance than real-valued counterparts when dealing with some complicated tasks such as acoustic analysis, radar image classification, etc. Despite empirical successes, it remains unknown theoretically when and to what extent complex-valued neural networks outperform real-valued ones. We take one step in this direction by comparing the learnability of real-valued neurons and complex-valued neurons via gradient descent. We theoretically show that a complex-valued neuron can learn functions expressed by any one real-valued neuron and any one complex-valued neuron with convergence rates $O(t^{-3})$ and $O(t^{-1})$ where $t$ is the iteration index of gradient descent, respectively, whereas a two-layer real-valued neural network with finite width cannot learn a single non-degenerate complex-valued neuron. We prove that a complex-valued neuron learns a real-valued neuron with rate $\Omega (t^{-3})$, exponentially slower than the linear convergence rate of learning one real-valued neuron using a real-valued neuron. We then reparameterize the phase parameter of the complex-valued neuron and prove that a reparameterized complex-valued neuron can efficiently learn a real-valued neuron with a linear convergence rate. We further verify and extend these results via simulation experiments in more general settings.

Random forests are classical ensemble algorithms that construct multiple randomized decision trees and aggregate their predictions using naive averaging. Zhou and Feng [51] further propose a deep forest algorithm with multi-layer forests, which outperforms random forests in various tasks. The performance of deep forests is related to three hyperparameters in practice: depth, width, and tree size, but little has been known about its theoretical explanation. This work provides the first upper and lower bounds on the approximation complexity of deep forests concerning the three hyperparameters. Our results confirm the distinctive role of depth, which can exponentially enhance the expressiveness of deep forests compared with width and tree size. Experiments validate these theoretical findings. The detailed proof and code are available in the full version [31].

Recent years have emerged a surge of interest in spiking neural networks (SNNs). The performance of SNNs hinges not only on searching apposite architectures and connection weights, similar to conventional artificial neural networks, but also on the meticulous configuration of their intrinsic structures. However, there has been a dearth of comprehensive studies examining the impact of intrinsic structures; thus developers often feel challenging to apply a standardized configuration of SNNs across diverse datasets or tasks. This work delves deep into the intrinsic structures of SNNs. Initially, we draw two key conclusions: (1) the membrane time hyper-parameter is intimately linked to the eigenvalues of the integration operation, dictating the functional topology of spiking dynamics; (2) various hyper-parameters of the firing-reset mechanism govern the overall firing capacity of an SNN, mitigating the injection ratio or sampling density of input data. These findings elucidate why the efficacy of SNNs hinges heavily on the configuration of intrinsic structures and lead to a recommendation that enhancing the adaptability of these structures contributes to improving the overall performance and applicability of SNNs. Inspired by this recognition, we propose two feasible approaches to enhance SNN learning, involving developing self-connection architectures and stochastic spiking neurons to augment the adaptability of the integration operation and firing-reset mechanism, respectively. We theoretically prove that (1) both methods promote the expressive property for universal approximation, (2) the incorporation of self-connection architectures fosters ample solutions and structural stability for SNNs approximating adaptive dynamical systems, (3) the stochastic spiking neurons maintain generalization bounds with an exponential reduction in Rademacher complexity. Empirical experiments conducted on various real-world datasets affirm the effectiveness of our proposed methods.

Neural network models generally involve two important components, i.e., network architecture and neuron model. Although there are abundant studies about network architectures, only a few neuron models have been developed, such as the MP neuron model developed in 1943 and the spiking neuron model developed in the 1950s. Recently, a new bio-plausible neuron model, flexible transmitter (FT) model (Zhang and Zhou, 2021), has been proposed. It exhibits promising behaviors, particularly on temporal-spatial signals, even when simply embedded into the common feedforward network architecture. This article attempts to understand the properties of the FT network (FTNet) theoretically. Under mild assumptions, we show that: 1) FTNet is a universal approximator; 2) the approximation complexity of FTNet can be exponentially smaller than those of commonly used real-valued neural networks with feedforward/recurrent architectures and is of the same order in the worst case; and 3) any local minimum of FTNet is the global minimum, implying that it is possible to identify global minima by local search algorithms.

Complex-valued neural networks potentially possess better representations and performance than real-valued counterparts when dealing with some complicated tasks such as acoustic analysis, radar image classification, etc. Despite empirical successes, it remains unknown theoretically when and to what extent complex-valued neural networks outperform real-valued ones. We take one step in this direction by comparing the learnability of real-valued neurons and complex-valued neurons via gradient descent. We show that a complex-valued neuron can efficiently learn functions expressed by any one real-valued neuron and any one complex-valued neuron with convergence rate $O(t^{-3})$ and $O(t^{-1})$ where $t$ is the iteration index of gradient descent, respectively, whereas a two-layer real-valued neural network with finite width cannot learn a single non-degenerate complex-valued neuron. We prove that a complex-valued neuron learns a real-valued neuron with rate $\Omega (t^{-3})$, exponentially slower than the $O(\mathrm{e}^{- c t})$ rate of learning one real-valued neuron using a real-valued neuron with a constant $c$. We further verify and extend these results via simulation experiments in more general settings.