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Ruowei Li

Publications and source records attributed to Ruowei Li.

10 recordsLinked to original sources
arXiv2026

Estimation of Riemannian Quantities from Noisy Data via Density Derivatives

We study the recovery of geometric structure from data generated by convolving the uniform measure on a smooth compact submanifold $M\subset\mathbb{R}^D$ with ambient Gaussian noise. Our main result is that several fundamental Riemannian quantities of $M$, including tangent spaces, the intrinsic dimension, and the second fundamental form, are identifiable from derivatives of the noisy density. We first derive uniform small-noise expansions of the data density and its derivatives in a tubular neighborhood of $M$. These expansions show that, at the population level, tangent spaces can be recovered from the density Hessian with $O(\sigma^2)$ error, while the intrinsic dimension can be estimated consistently. We further construct estimators for the second fundamental form from density derivatives, obtaining $O(d(y,M)+\sigma)$ and $O(d(y,M)+\sigma^2)$ errors for hypersurfaces and submanifolds with arbitrary codimension. At the sample level, we estimate the density and its derivatives by kernel methods in the ambient space and plug them into the population constructions, yielding uniform nonparametric rates in the ambient dimension. Finally, we show that these density-based constructions admit a geometric interpretation through density-induced ambient metrics, linking the geometry of $M$ to ambient geodesic structure.

math.ST↗
arXiv2026

Curvature-driven manifold fitting under unbounded isotropic noise

Manifold fitting aims to reconstruct a low-dimensional manifold from high-dimensional data, whose framework is established by Fefferman et al. \cite{fefferman2020reconstruction,fefferman2021reconstruction}. This paper studies the recovery of a compact $C^3$ submanifold $\mathcal{M} \subset \mathbb{R}^D$ with dimension $d 0$ and achieves a state-of-the-art Hausdorff distance of $O(\sigma^2)$ to $\mathcal{M}$. Numerical experiments confirm the quadratic decay of the reconstruction error and demonstrate the computational efficiency of the estimator $F$. Our work provides a curvature-driven framework for denoising and reconstructing manifolds with second-order accuracy.

math.ST↗
arXiv2025

Discrete Caffarelli-Kohn-Nirenberg inequalities and ground state solutions to nonlinear elliptic equations

In this paper, we prove the discrete Caffarelli-Kohn-Nirenberg inequalities on the lattice $\mathbb{Z}^{N}$ ($N\geq 1$) in a broader range of parameters than the classical continuous version [8]: \[ \parallel u\parallel_{\ell_{b}^{q}}\leq C(a,b,c,p,q,r,\theta,N)\parallel u\parallel_{D_{a}^{1,p}}^{\theta}\parallel u\parallel_{\ell_{c}^{r}}^{1-\theta},\:\forall u\in D_{a,0}^{1,p}(\mathbb{Z}^{N}) \cap \ell_c ^r(\mathbb{Z}^{N}), \] where $p,q,r>1,0\leq\theta\leq1$, $\frac{1}{p}+\frac{a}{N}>0,\frac{1}{r}+\frac{c}{N}>0,b\leq\theta a+(1-\theta)c,$$\frac{1}{q^{\ast}}+\frac{b}{N}= \theta(\frac{1}{p}+\frac{a-1}{N})+(1-\theta)(\frac{1}{r}+\frac{c}{N})$ and $q\geq q^{\ast}$. For two special cases $\theta=1,a=0$ and $a=b=c=0$, by the discrete Schwarz rearrangement established in [24], we prove the existence of extremal functions for the best constants in the supercritical case $q>q^{\ast}$. As an application, we get positive ground state solutions to the nonlinear elliptic equations.

math.AP↗
arXiv2024

Eigenvalue estimates for the poly-Laplace operator on lattice subgraphs

We introduce the discrete poly-Laplace operator on a subgraph with Dirichlet boundary condition. We obtain upper and lower bounds for the sum of the first $k$ Dirichlet eigenvalues of the poly-Laplace operators on a finite subgraph of lattice graph $\mathbb{Z}^{d}$ extending classical results of Li-Yau and Kr\"oger. Moreover, we prove that the Dirichlet $2l$-order poly-Laplace eigenvalues are at least as large as the squares of the Dirichlet $l$-order poly-Laplace eigenvalues.

math.SP↗
arXiv2024

The convergence and uniqueness of a discrete-time nonlinear Markov chain

In this paper, we prove the convergence and uniqueness of a general discrete-time nonlinear Markov chain with specific conditions. The results have important applications in discrete differential geometry. First, we prove the discrete-time Ollivier Ricci curvature flow $d_{n+1}:=(1-\alpha\kappa_{d_{n}})d_{n}$ converges to a constant curvature metric on a finite weighted graph. As shown in \cite[Theorem 5.1]{M23}, a Laplacian separation principle holds on a locally finite graph with nonnegative Ollivier curvature. We further prove that the Laplacian separation flow converges to the constant Laplacian solution and generalize the result to nonlinear $p$-Laplace operators. Moreover, our results can also be applied to study the long-time behavior in the nonlinear Dirichlet forms theory and nonlinear Perron-Frobenius theory. Finally, we define the Ollivier Ricci curvature of the nonlinear Markov chain which is consistent with the classical Ollivier Ricci curvature, sectional curvature \cite{CMS24}, coarse Ricci curvature on hypergraphs \cite{IKTU21} and the modified Ollivier Ricci curvature for $p$-Laplace. We also establish the convergence results for the nonlinear Markov chain with nonnegative Ollivier Ricci curvature.

math.DS↗
arXiv2022

The existence and convergence of solutions for the nonlinear Choquard equations on groups of polynomial growth

In this paper, we study the nonlinear Choquard equation \begin{eqnarray*} \Delta^{2}u-\Delta u+(1+\lambda a(x))u=(R_{\alpha}\ast|u|^{p})|u|^{p-2}u \end{eqnarray*} on a Cayley graph of a discrete group of polynomial growth with the homogeneous dimension $N\geq 2$, where $\alpha\in(0,N),\,p>\frac{N+\alpha}{N},\,\lambda$ is a positive parameter and $R_\alpha$ stands for the Green's function of the discrete fractional Laplacian, which has same asymptotics as the Riesz potential. Under some assumptions on $a(x)$, we establish the existence and asymptotic behavior of ground state solutions for the nonlinear Choquard equation by the method of Nehari manifold.

math.AP↗
arXiv2022

The existence of positive ground state solutions for the Choquard type equation on groups of polynomial growth

In this paper, let $G$ be a Cayley graph of a discrete group of polynomial growth with homogeneous dimension $N\geq3$. We study the Choquard type equation on $G$: \begin{equation} \Delta u+(R_{\alpha}\ast\mid u\mid^{p})\mid u\mid^{p-2}u=0, \end{equation} where $\alpha\in(0,N)$, $p>\frac{N+\alpha}{N-2}$ and $R_{\alpha}$ stands for the Green's function of the discrete fractional Laplace operator, which has same asymptotics as the Riesz potential. We prove the discrete Hardy-Littlewood-Sobolev inequality on such Cayley graphs, and by the discrete Concentration-Compactness principle we prove the existence of extremal functions for the corresponding Sobolev type inequalities in supercritical cases, which yields a positive ground state solution of the above Choquard type equation. Moreover, we obtain positive ground state solutions of Choquard type equations with $p$-Laplace, biharmonic and $p$-biharmonic operators etc.

math.AP↗
arXiv2022

Extremal functions for the second-order Sobolev inequality on groups of polynomial growth

In this paper, we prove the second-order Sobolev inequalities on Cayley graphs of groups of polynomial growth. We use the discrete Concentration-Compactness principle to prove the existence of extremal functions for best constants in supercritical cases. As applications, we get the existence of positive ground state solutions to the $p$-biharmonic equations and the Lane-Emden systems.

math.AP↗
arXiv2019

Three-dimensional Backbone Network for 3D Object Detection in Traffic Scenes

The task of detecting 3D objects in traffic scenes has a pivotal role in many real-world applications. However, the performance of 3D object detection is lower than that of 2D object detection due to the lack of powerful 3D feature extraction methods. To address this issue, this study proposes a 3D backbone network to acquire comprehensive 3D feature maps for 3D object detection. It primarily consists of sparse 3D convolutional neural network operations in the point cloud. The 3D backbone network can inherently learn 3D features from the raw data without compressing the point cloud into multiple 2D images. The sparse 3D convolutional neural network takes full advantage of the sparsity in the 3D point cloud to accelerate computation and save memory, which makes the 3D backbone network feasible in a real-world application. Empirical experiments were conducted on the KITTI benchmark and comparable results were obtained with respect to the state-of-the-art performance for 3D object detection.

cs.CV↗