arXiv subjects

Zuowei Shen

Publications and source records attributed to Zuowei Shen.

36 records · Page 2Linked to original sources

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.

cs.LG↗

arXiv2021

Neural Network Approximation: Three Hidden Layers Are Enough

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.

cs.LG↗

arXiv2021

Deep Network Approximation Characterized by Number of Neurons

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.

math.NA↗

arXiv2020

Nonlinear Approximation via Compositions

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.

cs.LG↗

arXiv2020

Deep Network with Approximation Error Being Reciprocal of Width to Power of Square Root of Depth

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.

cs.LG↗

arXiv2020

Deep Learning via Dynamical Systems: An Approximation Perspective

We build on the dynamical systems approach to deep learning, where deep residual networks are idealized as continuous-time dynamical systems, from the approximation perspective. In particular, we establish general sufficient conditions for universal approximation using continuous-time deep residual networks, which can also be understood as approximation theories in $L^p$ using flow maps of dynamical systems. In specific cases, rates of approximation in terms of the time horizon are also established. Overall, these results reveal that composition function approximation through flow maps present a new paradigm in approximation theory and contributes to building a useful mathematical framework to investigate deep learning.

cs.LG↗

arXiv2020

Optimization in Machine Learning: A Distribution Space Approach

We present the viewpoint that optimization problems encountered in machine learning can often be interpreted as minimizing a convex functional over a function space, but with a non-convex constraint set introduced by model parameterization. This observation allows us to repose such problems via a suitable relaxation as convex optimization problems in the space of distributions over the training parameters. We derive some simple relationships between the distribution-space problem and the original problem, e.g. a distribution-space solution is at least as good as a solution in the original space. Moreover, we develop a numerical algorithm based on mixture distributions to perform approximate optimization directly in distribution space. Consistency of this approximation is established and the numerical efficacy of the proposed algorithm is illustrated on simple examples. In both theory and practice, this formulation provides an alternative approach to large-scale optimization in machine learning.

cs.LG↗

arXiv2019

A Quantitative Analysis of the Effect of Batch Normalization on Gradient Descent

Despite its empirical success and recent theoretical progress, there generally lacks a quantitative analysis of the effect of batch normalization (BN) on the convergence and stability of gradient descent. In this paper, we provide such an analysis on the simple problem of ordinary least squares (OLS). Since precise dynamical properties of gradient descent (GD) is completely known for the OLS problem, it allows us to isolate and compare the additional effects of BN. More precisely, we show that unlike GD, gradient descent with BN (BNGD) converges for arbitrary learning rates for the weights, and the convergence remains linear under mild conditions. Moreover, we quantify two different sources of acceleration of BNGD over GD -- one due to over-parameterization which improves the effective condition number and another due having a large range of learning rates giving rise to fast descent. These phenomena set BNGD apart from GD and could account for much of its robustness properties. These findings are confirmed quantitatively by numerical experiments, which further show that many of the uncovered properties of BNGD in OLS are also observed qualitatively in more complex supervised learning problems.

cs.LG↗

arXiv2017

Coherence retrieval using trace regularization

The mutual intensity and its equivalent phase-space representations quantify an optical field's state of coherence and are important tools in the study of light propagation and dynamics, but they can only be estimated indirectly from measurements through a process called coherence retrieval, otherwise known as phase-space tomography. As practical considerations often rule out the availability of a complete set of measurements, coherence retrieval is usually a challenging high-dimensional ill-posed inverse problem. In this paper, we propose a trace-regularized optimization model for coherence retrieval and a provably-convergent adaptive accelerated proximal gradient algorithm for solving the resulting problem. Applying our model and algorithm to both simulated and experimental data, we demonstrate an improvement in reconstruction quality over previous models as well as an increase in convergence speed compared to existing first-order methods.

physics.optics↗

arXiv2016

Image Restoration: A General Wavelet Frame Based Model and Its Asymptotic Analysis

Image restoration is one of the most important areas in imaging science. Mathematical tools have been widely used in image restoration, where wavelet frame based approach is one of the successful examples. In this paper, we introduce a generic wavelet frame based image restoration model, called the "general model", which includes most of the existing wavelet frame based models as special cases. Moreover, the general model also includes examples that are new to the literature. Motivated by our earlier studies [1-3], We provide an asymptotic analysis of the general model as image resolution goes to infinity, which establishes a connection between the general model in discrete setting and a new variatonal model in continuum setting. The variational model also includes some of the existing variational models as special cases, such as the total generalized variational model proposed by [4]. In the end, we introduce an algorithm solving the general model and present one numerical simulation as an example.

math.FA↗

arXiv2015

Symmetric Canonical Quincunx Tight Framelets with High Vanishing Moments and Smoothness

We propose an approach to construct a family of two-dimensional compactly supported real-valued symmetric quincunx tight framelets $\{ϕ; ψ_1,ψ_2,ψ_3\}$ in $L_2(R^2)$ with arbitrarily high orders of vanishing moments. Such symmetric quincunx tight framelets are associated with quincunx tight framelet filter banks $\{a;b_1,b_2,b_3\}$ having increasing orders of vanishing moments and enjoying the additional double canonical properties: \[ b_1(k_1,k_2)=(-1)^{1+k_1+k_2} a(1-k_1,-k_2), b_3(k_1,k_2)=(-1)^{1+k_1+k_2} b_2(1-k_1,-k_2). \] For a low-pass filter $a$ which is not a quincunx orthonormal wavelet filter, we show that a quincunx tight framelet filter bank $\{a;b_1,\ldots,b_L\}$ with $b_1$ taking the above canonical form must have $L\ge 3$ high-pass filters. Thus, our family of symmetric double canonical quincunx tight framelets has the minimum number of generators. Numerical calculation indicates that this family of symmetric double canonical quincunx tight framelets can be arbitrarily smooth. Using one-dimensional filters having linear-phase moments, in this paper we also provide a second approach to construct multiple canonical quincunx tight framelets with symmetry. In particular, the second approach yields a family of $6$-multiple canonical real-valued quincunx tight framelets in $L_2(R^2)$ and a family of double canonical complex-valued quincunx tight framelets in $L_2(R^2)$ such that both of them have symmetry and arbitrarily increasing orders of smoothness and vanishing moments. Several examples are provided to illustrate our general construction and theoretical results on canonical quincunx tight framelets in $L_2(R^2)$ with symmetry, high vanishing moments, and smoothness. Symmetric quincunx tight framelets constructed by both approaches in this paper are of particular interest for their applications in computer graphics and image processing.

cs.IT↗

arXiv2013

Adaptive low rank and sparse decomposition of video using compressive sensing

We address the problem of reconstructing and analyzing surveillance videos using compressive sensing. We develop a new method that performs video reconstruction by low rank and sparse decomposition adaptively. Background subtraction becomes part of the reconstruction. In our method, a background model is used in which the background is learned adaptively as the compressive measurements are processed. The adaptive method has low latency, and is more robust than previous methods. We will present experimental results to demonstrate the advantages of the proposed method.

cs.IT↗

arXiv2013

Surveillance Video Processing Using Compressive Sensing

A compressive sensing method combined with decomposition of a matrix formed with image frames of a surveillance video into low rank and sparse matrices is proposed to segment the background and extract moving objects in a surveillance video. The video is acquired by compressive measurements, and the measurements are used to reconstruct the video by a low rank and sparse decomposition of matrix. The low rank component represents the background, and the sparse component is used to identify moving objects in the surveillance video. The decomposition is performed by an augmented Lagrangian alternating direction method. Experiments are carried out to demonstrate that moving objects can be reliably extracted with a small amount of measurements.

cs.CV↗

arXiv2012

Cine cone beam CT reconstruction using low-rank matrix factorization: algorithm and a proof-of-princple study

Respiration-correlated CBCT, commonly called 4DCBCT, provide respiratory phase-resolved CBCT images. In many clinical applications, it is more preferable to reconstruct true 4DCBCT with the 4th dimension being time, i.e., each CBCT image is reconstructed based on the corresponding instantaneous projection. We propose in this work a novel algorithm for the reconstruction of this truly time-resolved CBCT, called cine-CBCT, by effectively utilizing the underlying temporal coherence, such as periodicity or repetition, in those cine-CBCT images. Assuming each column of the matrix $\bm{U}$ represents a CBCT image to be reconstructed and the total number of columns is the same as the number of projections, the central idea of our algorithm is that the rank of $\bm{U}$ is much smaller than the number of projections and we can use a matrix factorization form $\bm{U}=\bm{L}\bm{R}$ for $\bm{U}$. The number of columns for the matrix $\bm{L}$ constraints the rank of $\bm{U}$ and hence implicitly imposing a temporal coherence condition among all the images in cine-CBCT. The desired image properties in $\bm{L}$ and the periodicity of the breathing pattern are achieved by penalizing the sparsity of the tight wavelet frame transform of $\bm{L}$ and that of the Fourier transform of $\bm{R}$, respectively. A split Bregman method is used to solve the problem. In this paper we focus on presenting this new algorithm and showing the proof of principle using simulation studies on an NCAT phantom.

physics.med-ph↗

arXiv2012

On B-spline framelets derived from the unitary extension principle

Spline wavelet tight frames of Ron-Shen have been used widely in frame based image analysis and restorations. However, except for the tight frame property and the approximation order of the truncated series, there are few other properties of this family of spline wavelet tight frames to be known. This paper is to present a few new properties of this family that will provide further understanding of it and, hopefully, give some indications why it is efficient in image analysis and restorations. In particular, we present a recurrence formula of computing generators of higher order spline wavelet tight frames from the lower order ones. We also represent each generator of spline wavelet tight frames as certain order of derivative of some univariate box spline. With this, we further show that each generator of sufficiently high order spline wavelet tight frames is close to a right order of derivative of a properly scaled Gaussian function. This leads to the result that the wavelet system generated by a finitely many consecutive derivatives of a properly scaled Gaussian function forms a frame whose frame bounds can be almost tight.

math.FA↗

arXiv2009

A Unitary Extension Principle for Shearlet Systems

In this paper, we first introduce the concept of an adaptive MRA (AMRA) structure which is a variant of the classical MRA structure suited to the main goal of a fast flexible decomposition strategy adapted to the data at each decomposition level. We then study this novel methodology for the general case of affine-like systems, and derive a Unitary Extension Principle (UEP) for filter design. Finally, we apply our results to the directional representation system of shearlets. This leads to a comprehensive theory for fast decomposition algorithms associated with shearlet systems which encompasses tight shearlet frames with spatially compactly supported generators within such an AMRA structure. Also shearlet-like systems associated with parabolic scaling and unimodular matrices optimally close to rotation as well as 3D shearlet systems are studied within this framework.

math.NA↗

arXiv2009

A New Multiscale Representation for Shapes and Its Application to Blood Vessel Recovery

In this paper, we will first introduce a novel multiscale representation (MSR) for shapes. Based on the MSR, we will then design a surface inpainting algorithm to recover 3D geometry of blood vessels. Because of the nature of irregular morphology in vessels and organs, both phantom and real inpainting scenarios were tested using our new algorithm. Successful vessel recoveries are demonstrated with numerical estimation of the degree of arteriosclerosis and vessel occlusion.

math.AP↗

arXiv2008

A Singular Value Thresholding Algorithm for Matrix Completion

This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood as the convex relaxation of a rank minimization problem, and arises in many important applications as in the task of recovering a large matrix from a small subset of its entries (the famous Netflix problem). Off-the-shelf algorithms such as interior point methods are not directly amenable to large problems of this kind with over a million unknown entries. This paper develops a simple first-order and easy-to-implement algorithm that is extremely efficient at addressing problems in which the optimal solution has low rank. The algorithm is iterative and produces a sequence of matrices (X^k, Y^k) and at each step, mainly performs a soft-thresholding operation on the singular values of the matrix Y^k. There are two remarkable features making this attractive for low-rank matrix completion problems. The first is that the soft-thresholding operation is applied to a sparse matrix; the second is that the rank of the iterates X^k is empirically nondecreasing. Both these facts allow the algorithm to make use of very minimal storage space and keep the computational cost of each iteration low. We provide numerical examples in which 1,000 by 1,000 matrices are recovered in less than a minute on a modest desktop computer. We also demonstrate that our approach is amenable to very large scale problems by recovering matrices of rank about 10 with nearly a billion unknowns from just about 0.4% of their sampled entries. Our methods are connected with linearized Bregman iterations for l1 minimization, and we develop a framework in which one can understand these algorithms in terms of well-known Lagrange multiplier algorithms.

math.OC↗