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Researcher profile

Xiliang Lu

Xiliang Lu contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Machine Learningmath.NANumerical AnalysisComputationeess.IVInformation Theorymath.APmath.ITmath.OCNeural and Evolutionary Computingquant-ph

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Published work

10 published item(s)

preprint2026arXiv

Approximation Error Upper and Lower Bounds for Hölder Class with Transformers

We explore the expressive power of Transformers by establishing precise approximation error upper and lower bounds for Hölder class. Specifically, a new approximation upper bound is derived for the standard Transformer architecture equipped with Softmax operators, ReLU activation functions, and residual connections. We prove that a Transformer network composed of at most $\mathcal{O}(\varepsilon^{-{d_{0}}/α})$ blocks can approximate any bounded Hölder function with $d_{0}$-dimensional input and smoothness $α\in(0,1]$ under any accuracy $\varepsilon>0$. In the case of approximation lower bounds, leveraging the VC-dimension upper bound, we are the first to rigorously prove that Transformers demand for at least $Ω(\varepsilon^{-{d_{0}}/({4α})})$ blocks to achieve the $\varepsilon$ approximation accuracy. As a final step, we extend the derived results for standard Transformers to a general regression task and establish the corresponding excess risk rates demonstrating Transformers' empirical effectiveness in real-world settings.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

A rate of convergence of Physics Informed Neural Networks for the linear second order elliptic PDEs

In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition, by establishing the upper bounds on the number of training samples, depth and width of the deep neural networks to achieve desired accuracy. The error of PINNs is decomposed into approximation error and statistical error, where the approximation error is given in $C^2$ norm with $\mathrm{ReLU}^{3}$ networks (deep network with activations function $\max\{0,x^3\}$) and the statistical error is estimated by Rademacher complexity. We derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with $\mathrm{ReLU}^{3}$ network, which is of immense independent interest.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Deep Neural Networks with ReLU-Sine-Exponential Activations Break Curse of Dimensionality in Approximation on Hölder Class

In this paper, we construct neural networks with ReLU, sine and $2^x$ as activation functions. For general continuous $f$ defined on $[0,1]^d$ with continuity modulus $ω_f(\cdot)$, we construct ReLU-sine-$2^x$ networks that enjoy an approximation rate $\mathcal{O}(ω_f(\sqrt{d})\cdot2^{-M}+ω_{f}\left(\frac{\sqrt{d}}{N}\right))$, where $M,N\in \mathbb{N}^{+}$ denote the hyperparameters related to widths of the networks. As a consequence, we can construct ReLU-sine-$2^x$ network with the depth $5$ and width $\max\left\{\left\lceil2d^{3/2}\left(\frac{3μ}ε\right)^{1/α}\right\rceil,2\left\lceil\log_2\frac{3μd^{α/2}}{2ε}\right\rceil+2\right\}$ that approximates $f\in \mathcal{H}_μ^α([0,1]^d)$ within a given tolerance $ε>0$ measured in $L^p$ norm $p\in[1,\infty)$, where $\mathcal{H}_μ^α([0,1]^d)$ denotes the Hölder continuous function class defined on $[0,1]^d$ with order $α\in (0,1]$ and constant $μ> 0$. Therefore, the ReLU-sine-$2^x$ networks overcome the curse of dimensionality on $\mathcal{H}_μ^α([0,1]^d)$. In addition to its supper expressive power, functions implemented by ReLU-sine-$2^x$ networks are (generalized) differentiable, enabling us to apply SGD to train.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Efficient and practical quantum compiler towards multi-qubit systems with deep reinforcement learning

Efficient quantum compiling tactics greatly enhance the capability of quantum computers to execute complicated quantum algorithms. Due to its fundamental importance, a plethora of quantum compilers has been designed in past years. However, there are several caveats to current protocols, which are low optimality, high inference time, limited scalability, and lack of universality. To compensate for these defects, here we devise an efficient and practical quantum compiler assisted by advanced deep reinforcement learning (RL) techniques, i.e., data generation, deep Q-learning, and AQ* search. In this way, our protocol is compatible with various quantum machines and can be used to compile multi-qubit operators. We systematically evaluate the performance of our proposal in compiling quantum operators with both inverse-closed and inverse-free universal basis sets. In the task of single-qubit operator compiling, our proposal outperforms other RL-based quantum compilers in the measure of compiling sequence length and inference time. Meanwhile, the output solution is near-optimal, guaranteed by the Solovay-Kitaev theorem. Notably, for the inverse-free universal basis set, the achieved sequence length complexity is comparable with the inverse-based setting and dramatically advances previous methods. These empirical results contribute to improving the inverse-free Solovay-Kitaev theorem. In addition, for the first time, we demonstrate how to leverage RL-based quantum compilers to accomplish two-qubit operator compiling. The achieved results open an avenue for integrating RL with quantum compiling to unify efficiency and practicality and thus facilitate the exploration of quantum advantages.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Global Optimization via Schr{ö}dinger-F{ö}llmer Diffusion

We study the problem of finding global minimizers of $V(x):\mathbb{R}^d\rightarrow\mathbb{R}$ approximately via sampling from a probability distribution $μ_σ$ with density $p_σ(x)=\dfrac{\exp(-V(x)/σ)}{\int_{\mathbb R^d} \exp(-V(y)/σ) dy }$ with respect to the Lebesgue measure for $σ\in (0,1]$ small enough. We analyze a sampler based on the Euler-Maruyama discretization of the Schr{ö}dinger-F{ö}llmer diffusion processes with stochastic approximation under appropriate assumptions on the step size $s$ and the potential $V$. We prove that the output of the proposed sampler is an approximate global minimizer of $V(x)$ with high probability at cost of sampling $\mathcal{O}(d^{3})$ standard normal random variables. Numerical studies illustrate the effectiveness of the proposed method and its superiority to the Langevin method.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Imaging Anisotropic Conductivities from Current Densities

In this paper, we propose and analyze a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. It is based on a regularized output least-squares formulation with the standard $L^2(Ω)^{d,d}$ penalty, which is then discretized by the standard Galerkin finite element method. We establish the continuity and differentiability of the forward map with respect to the conductivity tensor in the $L^p(Ω)^{d,d}$-norms, the existence of minimizers and optimality systems of the regularized formulation using the concept of H-convergence. Further, we provide a detailed analysis of the discretized problem, especially the convergence of the discrete approximations with respect to the mesh size, using the discrete counterpart of H-convergence. In addition, we develop a projected Newton algorithm for solving the first-order optimality system. We present extensive two-dimensional numerical examples to show the efficiency of the proposed method.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Imaging Conductivity from Current Density Magnitude using Neural Networks

Conductivity imaging represents one of the most important tasks in medical imaging. In this work we develop a neural network based reconstruction technique for imaging the conductivity from the magnitude of the internal current density. It is achieved by formulating the problem as a relaxed weighted least-gradient problem, and then approximating its minimizer by standard fully connected feedforward neural networks. We derive bounds on two components of the generalization error, i.e., approximation error and statistical error, explicitly in terms of properties of the neural networks (e.g., depth, total number of parameters, and the bound of the network parameters). We illustrate the performance and distinct features of the approach on several numerical experiments. Numerically, it is observed that the approach enjoys remarkable robustness with respect to the presence of data noise.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Convergence Rate Analysis for Deep Ritz Method

Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) \cite{wan11} for second order elliptic equations with Neumann boundary conditions. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRM using deep networks with $\mathrm{ReLU}^2$ activation functions. In addition to providing a theoretical justification of DRM, our study also shed light on how to set the hyper-parameter of depth and width to achieve the desired convergence rate in terms of number of training samples. Technically, we derive bounds on the approximation error of deep $\mathrm{ReLU}^2$ network in $H^1$ norm and on the Rademacher complexity of the non-Lipschitz composition of gradient norm and $\mathrm{ReLU}^2$ network, both of which are of independent interest.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A Support Detection and Root Finding Approach for Learning High-dimensional Generalized Linear Models

Feature selection is important for modeling high-dimensional data, where the number of variables can be much larger than the sample size. In this paper, we develop a support detection and root finding procedure to learn the high dimensional sparse generalized linear models and denote this method by GSDAR. Based on the KKT condition for $\ell_0$-penalized maximum likelihood estimations, GSDAR generates a sequence of estimators iteratively. Under some restricted invertibility conditions on the maximum likelihood function and sparsity assumption on the target coefficients, the errors of the proposed estimate decays exponentially to the optimal order. Moreover, the oracle estimator can be recovered if the target signal is stronger than the detectable level. We conduct simulations and real data analysis to illustrate the advantages of our proposed method over several existing methods, including Lasso and MCP.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Robust Decoding from Binary Measurements with Cardinality Constraint Least Squares

The main goal of 1-bit compressive sampling is to decode $n$ dimensional signals with sparsity level $s$ from $m$ binary measurements. This is a challenging task due to the presence of nonlinearity, noises and sign flips. In this paper, the cardinality constraint least square is proposed as a desired decoder. We prove that, up to a constant $c$, with high probability, the proposed decoder achieves a minimax estimation error as long as $m \geq \mathcal{O}( s\log n)$. Computationally, we utilize a generalized Newton algorithm (GNA) to solve the cardinality constraint minimization problem with the cost of solving a least squares problem with small size at each iteration. We prove that, with high probability, the $\ell_{\infty}$ norm of the estimation error between the output of GNA and the underlying target decays to $\mathcal{O}(\sqrt{\frac{\log n }{m}}) $ after at most $\mathcal{O}(\log s)$ iterations. Moreover, the underlying support can be recovered with high probability in $\mathcal{O}(\log s)$ steps provided that the target signal is detectable. Extensive numerical simulations and comparisons with state-of-the-art methods are presented to illustrate the robustness of our proposed decoder and the efficiency of the GNA algorithm.

0 citations0 reviewsNo code linkedOpen access