arXiv subjects

Zuowei Shen

Publications and source records attributed to Zuowei Shen.

At least 19 recordsLinked to original sources

arXiv2026

Forward-Evolution Error Analysis and Adaptive Design for Matrix-Valued Diffusion Models

Diffusion models learn to reverse a predefined corruption process, but sampling still requires a costly time discretization and depends on the chosen noise schedule. We study these two issues for variance-preserving diffusions with matrix-valued schedules. Our analysis transfers reverse-time discretization errors to the forward corruption law and treats two numerical schemes within a common framework. The first freezes the score and yields, through a matrix-sensitive local comparison and forward information dissipation, an ambient-dimensional step complexity with leading factor $d/\varepsilon^2$ for KL accuracy $\varepsilon^2$. The second keeps the known Gaussian drift exact and freezes the posterior mean. For data of metric-entropy dimension $k$, a forward Markov identity, an anisotropic covering estimate, and Stieltjes integration by parts give the corresponding factor $k\log k/\varepsilon^2$. In both cases, the proof identifies a local error, accumulates it through the forward evolution, and inserts the result into a common KL decomposition. The local errors further provide directional criteria for matrix schedules and an asymptotically optimal square-root adaptive grid. A high-dimensional Gaussian-mixture experiment illustrates the resulting schedule and grid improvements.

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arXiv2026

Unpaired Joint Distribution Modeling via Multi-Scale Image Representations

This paper studies the problem of learning a joint distribution from marginal observations, which is inherently ill-posed due to the ambiguity of feasible couplings. We propose LUD-MSR, a latent-variable probabilistic framework that models the joint distribution via auxiliary representations and optimizes evidence lower bounds using only marginal data. Under mild assumptions, we establish an upper bound on the distribution approximation error. This analysis reveals a trade-off in representation learning between domain consistency and information preservation. To address this trade-off, we introduce a Multi-Scale image Representation (MSR) mapping that exploits structural similarity at coarse scales while suppressing domain-specific variations. We show that MSR achieves a more favorable balance of this trade-off compared to existing approaches. Experiments on real-world denoising benchmarks, including cryo-electron microscopy (cryo-EM), demonstrate the effectiveness of the proposed framework.

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arXiv2026

Layer-wise Geometric Approximation Rates for Deep Networks

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 $\Phi_\ell$ 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 $\Phi_\ell$ 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. For every prescribed terminal depth, the construction yields a finite nested family of prefix readouts whose earlier correction terms remain embedded in later readouts. Thus the approximation may be truncated within the prescribed depth range once the desired certified accuracy is reached.

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arXiv2026

Deep learning and the rate of approximation by flows

We investigate the dependence of the approximation capacity of deep residual networks on its depth in a continuous dynamical systems setting. This can be formulated as the general problem of quantifying the minimal time-horizon required to approximate a diffeomorphism by flows driven by a given family $\mathcal F$ of vector fields. We show that this minimal time can be identified as a geodesic distance on a sub-Finsler manifold of diffeomorphisms, where the local geometry is characterised by a variational principle involving $\mathcal F$. This connects the learning efficiency of target relationships to their compatibility with the learning architectural choice. Further, the results suggest that the key approximation mechanism in deep learning, namely the approximation of functions by composition or dynamics, differs in a fundamental way from linear approximation theory, where linear spaces and norm-based rate estimates are replaced by manifolds and geodesic distances.

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arXiv2026

Multigrade Neural Network Approximation

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 nonconvex and often ill-conditioned optimization landscapes. In contrast, for relatively shallow networks, most notably certain one-hidden-layer ReLU models, training admits convex reformulations with global guarantees under appropriate settings, motivating learning paradigms that improve stability while scaling to depth. MGDL builds on this insight by training deep networks grade by grade: previously learned grades are frozen, and each newly added grade-wise subnetwork is composed on top of the previously learned grades and trained to fit the residual left by the current approximation, yielding a structured and interpretable hierarchical refinement process. We develop an operator-theoretic foundation for MGDL and prove that, for any continuous target function defined on a hypercube, there exists a fixed-width multigrade ReLU scheme whose residuals are pointwise nonincreasing in magnitude and converge uniformly to zero, with strict $L^p$-norm decay at every nontrivial grade for $p\in [1,\infty)$. To the best of our knowledge, this work provides the first rigorous constructive approximation guarantee showing that a grade-wise residual refinement scheme can achieve vanishing error in a fixed-width multigrade ReLU architecture.

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arXiv2026

Enhancing Low-resolution Image Representation Through Normalizing Flows

Low-resolution image representation is a special form of sparse representation that retains only low-frequency information while discarding high-frequency components. This property reduces storage and transmission costs and benefits various image processing tasks. However, a key challenge is to preserve essential visual content while maintaining the ability to accurately reconstruct the original images. This work proposes LR2Flow, a nonlinear framework that learns low-resolution image representations by integrating wavelet tight frame blocks with normalizing flows. We conduct a reconstruction error analysis of the proposed network, which demonstrates the necessity of designing invertible neural networks in the wavelet tight frame domain. Experimental results on various tasks, including image rescaling, compression, and denoising, demonstrate the effectiveness of the learned representations and the robustness of the proposed framework.

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arXiv2025

A unified framework for establishing the universal approximation of transformer-type architectures

We investigate the universal approximation property (UAP) of transformer-type architectures, providing a unified theoretical framework that extends prior results on residual networks to models incorporating attention mechanisms. Our work identifies token distinguishability as a fundamental requirement for UAP and introduces a general sufficient condition that applies to a broad class of architectures. Leveraging an analyticity assumption on the attention layer, we can significantly simplify the verification of this condition, providing a non-constructive approach in establishing UAP for such architectures. We demonstrate the applicability of our framework by proving UAP for transformers with various attention mechanisms, including kernel-based and sparse attention mechanisms. The corollaries of our results either generalize prior works or establish UAP for architectures not previously covered. Furthermore, our framework offers a principled foundation for designing novel transformer architectures with inherent UAP guarantees, including those with specific functional symmetries. We propose examples to illustrate these insights.

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arXiv2024

Accelerated Gradient Methods with Gradient Restart: Global Linear Convergence

Gradient restarting has been shown to improve the numerical performance of accelerated gradient methods. This paper provides a mathematical analysis to understand these advantages. First, we establish global linear convergence guarantees for both the original and gradient restarted accelerated proximal gradient method when solving strongly convex composite optimization problems. Second, through analysis of the corresponding ordinary differential equation model, we prove the continuous trajectory of the gradient restarted Nesterov's accelerated gradient method exhibits global linear convergence for quadratic convex objectives, while the non-restarted version provably lacks this property by [Su, Boyd, and Cand\'es, \textit{J. Mach. Learn. Res.}, 2016, 17(153), 1-43].

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arXiv2024

Analysis of a wavelet frame based two-scale model for enhanced edges

Image restoration is a class of important tasks that emerges from a wide range of scientific disciplines. It has been noticed that most practical images can be modeled as a composition from a sparse singularity set (edges) where the image contents or their gradients change drastically, and cartoon chunks in which a high degree of regularity is dominant. Enhancing edges while promoting regularity elsewhere has been an important criterion for successful restoration in many image classes. In this article, we present a wavelet frame based image restoration model that captures potential edges and facilitates the restoration procedure by a dedicated treatment both of singularity and of cartoon. Moreover, its geometric robustness is enhanced by exploiting subtle inter-scale information available in the coarse image. To substantiate our intuition, we prove that this model converges to one variant of the celebrated Mumford-Shah model when adequate asymptotic specifications are given.

math.FA↗

arXiv2023

Interpolation, Approximation and Controllability of Deep Neural Networks

We investigate the expressive power of deep residual neural networks idealized as continuous dynamical systems through control theory. Specifically, we consider two properties that arise from supervised learning, namely universal interpolation - the ability to match arbitrary input and target training samples - and the closely related notion of universal approximation - the ability to approximate input-target functional relationships via flow maps. Under the assumption of affine invariance of the control family, we give a characterisation of universal interpolation, showing that it holds for essentially any architecture with non-linearity. Furthermore, we elucidate the relationship between universal interpolation and universal approximation in the context of general control systems, showing that the two properties cannot be deduced from each other. At the same time, we identify conditions on the control family and the target function that ensures the equivalence of the two notions.

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arXiv2023

On the Universal Approximation Property of Deep Fully Convolutional Neural Networks

We study the approximation of shift-invariant or equivariant functions by deep fully convolutional networks from the dynamical systems perspective. We prove that deep residual fully convolutional networks and their continuous-layer counterpart can achieve universal approximation of these symmetric functions at constant channel width. Moreover, we show that the same can be achieved by non-residual variants with at least 2 channels in each layer and convolutional kernel size of at least 2. In addition, we show that these requirements are necessary, in the sense that networks with fewer channels or smaller kernels fail to be universal approximators.

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arXiv2023

Neural Network Architecture Beyond Width and Depth

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.

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arXiv2022

Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons

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.

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arXiv2022

IAE-Net: Integral Autoencoders for Discretization-Invariant Learning

Discretization invariant learning aims at learning in the infinite-dimensional function spaces with the capacity to process heterogeneous discrete representations of functions as inputs and/or outputs of a learning model. This paper proposes a novel deep learning framework based on integral autoencoders (IAE-Net) for discretization invariant learning. The basic building block of IAE-Net consists of an encoder and a decoder as integral transforms with data-driven kernels, and a fully connected neural network between the encoder and decoder. This basic building block is applied in parallel in a wide multi-channel structure, which are repeatedly composed to form a deep and densely connected neural network with skip connections as IAE-Net. IAE-Net is trained with randomized data augmentation that generates training data with heterogeneous structures to facilitate the performance of discretization invariant learning. The proposed IAE-Net is tested with various applications in predictive data science, solving forward and inverse problems in scientific computing, and signal/image processing. Compared with alternatives in the literature, IAE-Net achieves state-of-the-art performance in existing applications and creates a wide range of new applications.

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arXiv2022

Deep Neural Network Approximation of Invariant Functions through Dynamical Systems

We study the approximation of functions which are invariant with respect to certain permutations of the input indices using flow maps of dynamical systems. Such invariant functions includes the much studied translation-invariant ones involving image tasks, but also encompasses many permutation-invariant functions that finds emerging applications in science and engineering. We prove sufficient conditions for universal approximation of these functions by a controlled equivariant dynamical system, which can be viewed as a general abstraction of deep residual networks with symmetry constraints. These results not only imply the universal approximation for a variety of commonly employed neural network architectures for symmetric function approximation, but also guide the design of architectures with approximation guarantees for applications involving new symmetry requirements.

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arXiv2022

Deep Network Approximation in Terms of Intrinsic Parameters

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.

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arXiv2021

Deep Network Approximation for Smooth Functions

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.

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arXiv2021

Optimal Approximation Rate of ReLU Networks in terms of Width and Depth

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.

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