Shijun Zhang

Shijun Zhang, Jianfeng Lu, Hongkai Zhao

This paper explores the expressive power of deep neural networks for a diverse range of activation functions. An activation function set $\mathscr{A}$ is defined to encompass the majority of commonly used activation functions, such as $\mathtt{ReLU}$, $\mathtt{LeakyReLU}$, $\mathtt{ReLU}^2$, $\mathtt{ELU}$, $\mathtt{CELU}$, $\mathtt{SELU}$, $\mathtt{Softplus}$, $\mathtt{GELU}$, $\mathtt{SiLU}$, $\mathtt{Swish}$, $\mathtt{Mish}$, $\mathtt{Sigmoid}$, $\mathtt{Tanh}$, $\mathtt{Arctan}$, $\mathtt{Softsign}$, $\mathtt{dSiLU}$, and $\mathtt{SRS}$. We demonstrate that for any activation function $\varrho\in \mathscr{A}$, a $\mathtt{ReLU}$ network of width $N$ and depth $L$ can be approximated to arbitrary precision by a $\varrho$-activated network of width $3N$ and depth $2L$ on any bounded set. This finding enables the extension of most approximation results achieved with $\mathtt{ReLU}$ networks to a wide variety of other activation functions, albeit with slightly increased constants. Significantly, we establish that the (width,$\,$depth) scaling factors can be further reduced from $(3,2)$ to $(1,1)$ if $\varrho$ falls within a specific subset of $\mathscr{A}$. This subset includes activation functions such as $\mathtt{ELU}$, $\mathtt{CELU}$, $\mathtt{SELU}$, $\mathtt{Softplus}$, $\mathtt{GELU}$, $\mathtt{SiLU}$, $\mathtt{Swish}$, and $\mathtt{Mish}$.

Zuowei Shen, Haizhao Yang, Shijun Zhang

This paper proposes a new neural network architecture by introducing an additional dimension called height beyond width and depth. Neural network architectures with height, width, and depth as hyper-parameters are called three-dimensional architectures. It is shown that neural networks with three-dimensional architectures are significantly more expressive than the ones with two-dimensional architectures (those with only width and depth as hyper-parameters), e.g., standard fully connected networks. The new network architecture is constructed recursively via a nested structure, and hence we call a network with the new architecture nested network (NestNet). A NestNet of height $s$ is built with each hidden neuron activated by a NestNet of height $\le s-1$. When $s=1$, a NestNet degenerates to a standard network with a two-dimensional architecture. It is proved by construction that height-$s$ ReLU NestNets with $\mathcal{O}(n)$ parameters can approximate $1$-Lipschitz continuous functions on $[0,1]^d$ with an error $\mathcal{O}(n^{-(s+1)/d})$, while the optimal approximation error of standard ReLU networks with $\mathcal{O}(n)$ parameters is $\mathcal{O}(n^{-2/d})$. Furthermore, such a result is extended to generic continuous functions on $[0,1]^d$ with the approximation error characterized by the modulus of continuity. Finally, we use numerical experimentation to show the advantages of the super-approximation power of ReLU NestNets.

Shijun Zhang, Hongkai Zhao, Yimin Zhong et al.

In this work, we present a comprehensive study combining mathematical and computational analysis to explain why a two-layer neural network struggles to handle high frequencies in both approximation and learning, especially when machine precision, numerical noise, and computational cost are significant factors in practice. Specifically, we investigate the following fundamental computational issues: (1) the minimal numerical error achievable under finite precision, (2) the computational cost required to attain a given accuracy, and (3) the stability of the method with respect to perturbations. The core of our analysis lies in the conditioning of the representation and its learning dynamics. Explicit answers to these questions are provided, along with supporting numerical evidence.

Shijun Zhang, Jianfeng Lu, Hongkai Zhao

This paper explores the expressive power of deep neural networks through the framework of function compositions. We demonstrate that the repeated compositions of a single fixed-size ReLU network exhibit surprising expressive power, despite the limited expressive capabilities of the individual network itself. Specifically, we prove by construction that $\mathcal{L}_2\circ \boldsymbol{g}^{\circ r}\circ \boldsymbol{\mathcal{L}}_1$ can approximate $1$-Lipschitz continuous functions on $[0,1]^d$ with an error $\mathcal{O}(r^{-1/d})$, where $\boldsymbol{g}$ is realized by a fixed-size ReLU network, $\boldsymbol{\mathcal{L}}_1$ and $\mathcal{L}_2$ are two affine linear maps matching the dimensions, and $\boldsymbol{g}^{\circ r}$ denotes the $r$-times composition of $\boldsymbol{g}$. Furthermore, we extend such a result to generic continuous functions on $[0,1]^d$ with the approximation error characterized by the modulus of continuity. Our results reveal that a continuous-depth network generated via a dynamical system has immense approximation power even if its dynamics function is time-independent and realized by a fixed-size ReLU network.

Fenglei Fan, Juntong Fan, Dayang Wang et al.

The rapid growth of large models' size has far outpaced that of computing resources. To bridge this gap, encouraged by the parsimonious relationship between genotype and phenotype in the brain's growth and development, we propose the so-called Hyper-Compression that turns the model compression into the issue of parameter representation via a hyperfunction. Specifically, it is known that the trajectory of some low-dimensional dynamic systems can fill the high-dimensional space eventually. Thus, Hyper-Compression, using these dynamic systems as the hyperfunctions, represents the parameters of the target network by their corresponding composition number or trajectory length. This suggests a novel mechanism for model compression, substantially different from the existing pruning, quantization, distillation, and decomposition. Along this direction, we methodologically identify a suitable dynamic system with the irrational winding as the hyperfunction and theoretically derive its associated error bound. Next, guided by our theoretical insights, we propose several engineering twists to make the Hyper-Compression pragmatic and effective. Lastly, systematic and comprehensive experiments on \textcolor{black}{NLP models such as LLaMA and Qwen series and vision models} confirm that Hyper-Compression enjoys the following \textbf{PNAS} merits: 1) \textbf{P}referable compression ratio; 2) \textbf{N}o post-hoc retraining; 3) \textbf{A}ffordable inference time; and 4) \textbf{S}hort compression time. It compresses LLaMA2-7B in an hour and achieves close-to-int4-quantization performance, without retraining and with a performance drop of less than 1\%. We have open-sourced our code in https://github.com/Juntongkuki/Hyper-Compression.git for free download and evaluation.

Qianchao Wang, Shijun Zhang, Dong Zeng et al.

In this paper, we propose a new super-expressive activation function called the Parametric Elementary Universal Activation Function (PEUAF). We demonstrate the effectiveness of PEUAF through systematic and comprehensive experiments on various industrial and image datasets, including CIFAR10, Tiny-ImageNet, and ImageNet. Moreover, we significantly generalize the family of super-expressive activation functions, whose existence has been demonstrated in several recent works by showing that any continuous function can be approximated to any desired accuracy by a fixed-size network with a specific super-expressive activation function. Specifically, our work addresses two major bottlenecks in impeding the development of super-expressive activation functions: the limited identification of super-expressive functions, which raises doubts about their broad applicability, and their often peculiar forms, which lead to skepticism regarding their scalability and practicality in real-world applications.

Shijun Zhang, Zuowei Shen, Yuesheng Xu

We study multigrade deep learning (MGDL) as a principled framework for structured error refinement in deep neural networks. While the approximation power of neural networks is now relatively well understood, training very deep architectures remains challenging due to highly non-convex and often ill-conditioned optimization landscapes. In contrast, for relatively shallow networks, most notably one-hidden-layer $\texttt{ReLU}$ models, training admits convex reformulations with global guarantees, motivating learning paradigms that improve stability while scaling to depth. MGDL builds upon this insight by training deep networks grade by grade: previously learned grades are frozen, and each new residual block is trained solely to reduce the remaining approximation error, yielding an interpretable and stable hierarchical refinement process. We develop an operator-theoretic foundation for MGDL and prove that, for any continuous target function, there exists a fixed-width multigrade $\texttt{ReLU}$ scheme whose residuals decrease strictly across grades and converge uniformly to zero. To the best of our knowledge, this work provides the first rigorous theoretical guarantee that grade-wise training yields provable vanishing approximation error in deep networks. Numerical experiments further illustrate the theoretical results.

Hanyu Pei, Jing-Xiao Liao, Qibin Zhao et al.

Drawing inspiration from our human brain that designs different neurons for different tasks, recent advances in deep learning have explored modifying a network's neurons to develop so-called task-driven neurons. Prototyping task-driven neurons (referred to as NeuronSeek) employs symbolic regression (SR) to discover the optimal neuron formulation and construct a network from these optimized neurons. Along this direction, this work replaces symbolic regression with tensor decomposition (TD) to discover optimal neuronal formulations, offering enhanced stability and faster convergence. Furthermore, we establish theoretical guarantees that modifying the aggregation functions with common activation functions can empower a network with a fixed number of parameters to approximate any continuous function with an arbitrarily small error, providing a rigorous mathematical foundation for the NeuronSeek framework. Extensive empirical evaluations demonstrate that our NeuronSeek-TD framework not only achieves superior stability, but also is competitive relative to the state-of-the-art models across diverse benchmarks. The code is available at https://github.com/HanyuPei22/NeuronSeek.

ZeYu Li, ShiJun Zhang, TieYong Zeng et al.

Universal approximation theory offers a foundational framework to verify neural network expressiveness, enabling principled utilization in real-world applications. However, most existing theoretical constructions are established by uniformly dividing the input space into tiny hypercubes without considering the local regularity of the target function. In this work, we investigate the universal approximation capabilities of ReLU networks from a view of polytope decomposition, which offers a more realistic and task-oriented approach compared to current methods. To achieve this, we develop an explicit kernel polynomial method to derive an universal approximation of continuous functions, which is characterized not only by the refined Totik-Ditzian-type modulus of continuity, but also by polytopical domain decomposition. Then, a ReLU network is constructed to approximate the kernel polynomial in each subdomain separately. Furthermore, we find that polytope decomposition makes our approximation more efficient and flexible than existing methods in many cases, especially near singular points of the objective function. Lastly, we extend our approach to analytic functions to reach a higher approximation rate.

Shijun Zhang, Zuowei Shen, Yuesheng Xu

Depth is widely viewed as a central contributor to the success of deep neural networks, whereas standard neural network approximation theory typically provides guarantees only for the final output and leaves the role of intermediate layers largely unclear. We address this gap by developing a quantitative framework in which depth admits a precise scale-dependent interpretation. Specifically, we design a single shared mixed-activation architecture of fixed width $2dN+d+2$ and any prescribed finite depth such that each intermediate readout $Φ_\ell$ is itself an approximant to the target function $f$. For $f\in L^p([0,1]^d)$ with $p\in [1,\infty)$, the approximation error of $Φ_\ell$ is controlled by $(2d+1)$ times the $L^p$ modulus of continuity at the geometric scale $N^{-\ell}$ for all $\ell$. The estimate reduces to the geometric rate $(2d+1)N^{-\ell}$ if $f$ is $1$-Lipschitz. Our network design is inspired by multigrade deep learning, where depth serves as a progressive refinement mechanism: each new correction targets residual information at a finer scale while the earlier correction terms remain part of the later readouts, yielding a nested architecture that supports adaptive refinement without redesigning the preceding network.

Shijun Zhang, Hongkai Zhao, Yimin Zhong et al.

The architecture of a neural network and the selection of its activation function are both fundamental to its performance. Equally vital is ensuring these two elements are well-matched, as their alignment is key to achieving effective representation and learning. In this paper, we introduce the Fourier Multi-Component and Multi-Layer Neural Network (FMMNN), a novel model that creates a strong synergy between them. We demonstrate that FMMNNs are highly effective and flexible in modeling high-frequency components. Our theoretical results demonstrate that FMMNNs have exponential expressive power for function approximation. We also analyze the optimization landscape of FMMNNs and find it to be much more favorable than that of standard fully connected neural networks, especially when dealing with high-frequency features. In addition, we propose a scaled random initialization method for the first layer's weights in FMMNNs, which significantly speeds up training and enhances overall performance. Extensive numerical experiments support our theoretical insights, showing that FMMNNs consistently outperform traditional approaches in accuracy and efficiency across various tasks.

Shijun Zhang, Hongkai Zhao, Yimin Zhong et al.

In this work, we propose a balanced multi-component and multi-layer neural network (MMNN) structure to accurately and efficiently approximate functions with complex features, in terms of both degrees of freedom and computational cost. The main idea is inspired by a multi-component approach, in which each component can be effectively approximated by a single-layer network, combined with a multi-layer decomposition strategy to capture the complexity of the target function. Although MMNNs can be viewed as a simple modification of fully connected neural networks (FCNNs) or multi-layer perceptrons (MLPs) by introducing balanced multi-component structures, they achieve a significant reduction in training parameters, a much more efficient training process, and improved accuracy compared to FCNNs or MLPs. Extensive numerical experiments demonstrate the effectiveness of MMNNs in approximating highly oscillatory functions and their ability to automatically adapt to localized features.

Zuowei Shen, Haizhao Yang, Shijun Zhang

One of the arguments to explain the success of deep learning is the powerful approximation capacity of deep neural networks. Such capacity is generally accompanied by the explosive growth of the number of parameters, which, in turn, leads to high computational costs. It is of great interest to ask whether we can achieve successful deep learning with a small number of learnable parameters adapting to the target function. From an approximation perspective, this paper shows that the number of parameters that need to be learned can be significantly smaller than people typically expect. First, we theoretically design ReLU networks with a few learnable parameters to achieve an attractive approximation. We prove by construction that, for any Lipschitz continuous function $f$ on $[0,1]^d$ with a Lipschitz constant $λ>0$, a ReLU network with $n+2$ intrinsic parameters (those depending on $f$) can approximate $f$ with an exponentially small error $5λ\sqrt{d}\,2^{-n}$. Such a result is generalized to generic continuous functions. Furthermore, we show that the idea of learning a small number of parameters to achieve a good approximation can be numerically observed. We conduct several experiments to verify that training a small part of parameters can also achieve good results for classification problems if other parameters are pre-specified or pre-trained from a related problem.

Zuowei Shen, Haizhao Yang, Shijun Zhang

This paper develops simple feed-forward neural networks that achieve the universal approximation property for all continuous functions with a fixed finite number of neurons. These neural networks are simple because they are designed with a simple, computable, and continuous activation function $σ$ leveraging a triangular-wave function and the softsign function. We first prove that $σ$-activated networks with width $36d(2d+1)$ and depth $11$ can approximate any continuous function on a $d$-dimensional hypercube within an arbitrarily small error. Hence, for supervised learning and its related regression problems, the hypothesis space generated by these networks with a size not smaller than $36d(2d+1)\times 11$ is dense in the continuous function space $C([a,b]^d)$ and therefore dense in the Lebesgue spaces $L^p([a,b]^d)$ for $p\in [1,\infty)$. Furthermore, we show that classification functions arising from image and signal classification are in the hypothesis space generated by $σ$-activated networks with width $36d(2d+1)$ and depth $12$ when there exist pairwise disjoint bounded closed subsets of $\mathbb{R}^d$ such that the samples of the same class are located in the same subset. Finally, we use numerical experimentation to show that replacing the rectified linear unit (ReLU) activation function by ours would improve the experiment results.

Zuowei Shen, Haizhao Yang, Shijun Zhang

This paper concentrates on the approximation power of deep feed-forward neural networks in terms of width and depth. It is proved by construction that ReLU networks with width $\mathcal{O}\big(\max\{d\lfloor N^{1/d}\rfloor,\, N+2\}\big)$ and depth $\mathcal{O}(L)$ can approximate a Hölder continuous function on $[0,1]^d$ with an approximation rate $\mathcal{O}\big(λ\sqrt{d} (N^2L^2\ln N)^{-α/d}\big)$, where $α\in (0,1]$ and $λ>0$ are Hölder order and constant, respectively. Such a rate is optimal up to a constant in terms of width and depth separately, while existing results are only nearly optimal without the logarithmic factor in the approximation rate. More generally, for an arbitrary continuous function $f$ on $[0,1]^d$, the approximation rate becomes $\mathcal{O}\big(\,\sqrt{d}\,ω_f\big( (N^2L^2\ln N)^{-1/d}\big)\,\big)$, where $ω_f(\cdot)$ is the modulus of continuity. We also extend our analysis to any continuous function $f$ on a bounded set. Particularly, if ReLU networks with depth $31$ and width $\mathcal{O}(N)$ are used to approximate one-dimensional Lipschitz continuous functions on $[0,1]$ with a Lipschitz constant $λ>0$, the approximation rate in terms of the total number of parameters, $W=\mathcal{O}(N^2)$, becomes $\mathcal{O}(\tfracλ{W\ln W})$, which has not been discovered in the literature for fixed-depth ReLU networks.

Zuowei Shen, Haizhao Yang, Shijun Zhang

A three-hidden-layer neural network with super approximation power is introduced. This network is built with the floor function ($\lfloor x\rfloor$), the exponential function ($2^x$), the step function ($1_{x\geq 0}$), or their compositions as the activation function in each neuron and hence we call such networks as Floor-Exponential-Step (FLES) networks. For any width hyper-parameter $N\in\mathbb{N}^+$, it is shown that FLES networks with width $\max\{d,N\}$ and three hidden layers can uniformly approximate a Hölder continuous function $f$ on $[0,1]^d$ with an exponential approximation rate $3λ(2\sqrt{d})^α 2^{-αN}$, where $α\in(0,1]$ and $λ>0$ are the Hölder order and constant, respectively. More generally for an arbitrary continuous function $f$ on $[0,1]^d$ with a modulus of continuity $ω_f(\cdot)$, the constructive approximation rate is $2ω_f(2\sqrt{d}){2^{-N}}+ω_f(2\sqrt{d}\,2^{-N})$. Moreover, we extend such a result to general bounded continuous functions on a bounded set $E\subseteq\mathbb{R}^d$. As a consequence, this new class of networks overcomes the curse of dimensionality in approximation power when the variation of $ω_f(r)$ as $r\rightarrow 0$ is moderate (e.g., $ω_f(r)\lesssim r^α$ for Hölder continuous functions), since the major term to be concerned in our approximation rate is essentially $\sqrt{d}$ times a function of $N$ independent of $d$ within the modulus of continuity. Finally, we extend our analysis to derive similar approximation results in the $L^p$-norm for $p\in[1,\infty)$ via replacing Floor-Exponential-Step activation functions by continuous activation functions.

Zuowei Shen, Haizhao Yang, Shijun Zhang

A new network with super approximation power is introduced. This network is built with Floor ($\lfloor x\rfloor$) or ReLU ($\max\{0,x\}$) activation function in each neuron and hence we call such networks Floor-ReLU networks. For any hyper-parameters $N\in\mathbb{N}^+$ and $L\in\mathbb{N}^+$, it is shown that Floor-ReLU networks with width $\max\{d,\, 5N+13\}$ and depth $64dL+3$ can uniformly approximate a Hölder function $f$ on $[0,1]^d$ with an approximation error $3λd^{α/2}N^{-α\sqrt{L}}$, where $α\in(0,1]$ and $λ$ are the Hölder order and constant, respectively. More generally for an arbitrary continuous function $f$ on $[0,1]^d$ with a modulus of continuity $ω_f(\cdot)$, the constructive approximation rate is $ω_f(\sqrt{d}\,N^{-\sqrt{L}})+2ω_f(\sqrt{d}){N^{-\sqrt{L}}}$. As a consequence, this new class of networks overcomes the curse of dimensionality in approximation power when the variation of $ω_f(r)$ as $r\to 0$ is moderate (e.g., $ω_f(r) \lesssim r^α$ for Hölder continuous functions), since the major term to be considered in our approximation rate is essentially $\sqrt{d}$ times a function of $N$ and $L$ independent of $d$ within the modulus of continuity.

Jianfeng Lu, Zuowei Shen, Haizhao Yang et al.

This paper establishes the (nearly) optimal approximation error characterization of deep rectified linear unit (ReLU) networks for smooth functions in terms of both width and depth simultaneously. To that end, we first prove that multivariate polynomials can be approximated by deep ReLU networks of width $\mathcal{O}(N)$ and depth $\mathcal{O}(L)$ with an approximation error $\mathcal{O}(N^{-L})$. Through local Taylor expansions and their deep ReLU network approximations, we show that deep ReLU networks of width $\mathcal{O}(N\ln N)$ and depth $\mathcal{O}(L\ln L)$ can approximate $f\in C^s([0,1]^d)$ with a nearly optimal approximation error $\mathcal{O}(\|f\|_{C^s([0,1]^d)}N^{-2s/d}L^{-2s/d})$. Our estimate is non-asymptotic in the sense that it is valid for arbitrary width and depth specified by $N\in\mathbb{N}^+$ and $L\in\mathbb{N}^+$, respectively.

Zuowei Shen, Haizhao Yang, Shijun Zhang

This paper quantitatively characterizes the approximation power of deep feed-forward neural networks (FNNs) in terms of the number of neurons. It is shown by construction that ReLU FNNs with width $\mathcal{O}\big(\max\{d\lfloor N^{1/d}\rfloor,\, N+1\}\big)$ and depth $\mathcal{O}(L)$ can approximate an arbitrary Hölder continuous function of order $α\in (0,1]$ on $[0,1]^d$ with a nearly tight approximation rate $\mathcal{O}\big(\sqrt{d} N^{-2α/d}L^{-2α/d}\big)$ measured in $L^p$-norm for any $N,L\in \mathbb{N}^+$ and $p\in[1,\infty]$. More generally for an arbitrary continuous function $f$ on $[0,1]^d$ with a modulus of continuity $ω_f(\cdot)$, the constructive approximation rate is $\mathcal{O}\big(\sqrt{d}\,ω_f( N^{-2/d}L^{-2/d})\big)$. We also extend our analysis to $f$ on irregular domains or those localized in an $\varepsilon$-neighborhood of a $d_{\mathcal{M}}$-dimensional smooth manifold $\mathcal{M}\subseteq [0,1]^d$ with $d_{\mathcal{M}}\ll d$. Especially, in the case of an essentially low-dimensional domain, we show an approximation rate $\mathcal{O}\big(ω_f(\tfrac{\varepsilon}{1-δ}\sqrt{\tfrac{d}{d_δ}}+\varepsilon)+\sqrt{d}\,ω_f(\tfrac{\sqrt{d}}{(1-δ)\sqrt{d_δ}}N^{-2/d_δ}L^{-2/d_δ})\big)$ for ReLU FNNs to approximate $f$ in the $\varepsilon$-neighborhood, where $d_δ=\mathcal{O}\big(d_{\mathcal{M}}\tfrac{\ln (d/δ)}{δ^2}\big)$ for any $δ\in(0,1)$ as a relative error for a projection to approximate an isometry when projecting $\mathcal{M}$ to a $d_δ$-dimensional domain.

Zuowei Shen, Haizhao Yang, Shijun Zhang

Given a function dictionary $\cal D$ and an approximation budget $N\in\mathbb{N}^+$, nonlinear approximation seeks the linear combination of the best $N$ terms $\{T_n\}_{1\le n\le N}\subseteq{\cal D}$ to approximate a given function $f$ with the minimum approximation error\[\varepsilon_{L,f}:=\min_{\{g_n\}\subseteq{\mathbb{R}},\{T_n\}\subseteq{\cal D}}\|f(x)-\sum_{n=1}^N g_n T_n(x)\|.\]Motivated by recent success of deep learning, we propose dictionaries with functions in a form of compositions, i.e.,\[T(x)=T^{(L)}\circ T^{(L-1)}\circ\cdots\circ T^{(1)}(x)\]for all $T\in\cal D$, and implement $T$ using ReLU feed-forward neural networks (FNNs) with $L$ hidden layers. We further quantify the improvement of the best $N$-term approximation rate in terms of $N$ when $L$ is increased from $1$ to $2$ or $3$ to show the power of compositions. In the case when $L>3$, our analysis shows that increasing $L$ cannot improve the approximation rate in terms of $N$. In particular, for any function $f$ on $[0,1]$, regardless of its smoothness and even the continuity, if $f$ can be approximated using a dictionary when $L=1$ with the best $N$-term approximation rate $\varepsilon_{L,f}={\cal O}(N^{-η})$, we show that dictionaries with $L=2$ can improve the best $N$-term approximation rate to $\varepsilon_{L,f}={\cal O}(N^{-2η})$. We also show that for Hölder continuous functions of order $α$ on $[0,1]^d$, the application of a dictionary with $L=3$ in nonlinear approximation can achieve an essentially tight best $N$-term approximation rate $\varepsilon_{L,f}={\cal O}(N^{-2α/d})$. Finally, we show that dictionaries consisting of wide FNNs with a few hidden layers are more attractive in terms of computational efficiency than dictionaries with narrow and very deep FNNs for approximating Hölder continuous functions if the number of computer cores is larger than $N$ in parallel computing.