arXiv subjects

Rongjie Lai

Publications and source records attributed to Rongjie Lai.

At least 55 records · Page 3Linked to original sources

arXiv2020

Generalized Unnormalized Optimal Transport and its fast algorithms

We introduce fast algorithms for generalized unnormalized optimal transport. To handle densities with different total mass, we consider a dynamic model, which mixes the $L^p$ optimal transport with $L^p$ distance. For $p=1$, we derive the corresponding $L^1$ generalized unnormalized Kantorovich formula. We further show that the problem becomes a simple $L^1$ minimization which is solved efficiently by a primal-dual algorithm. For $p=2$, we derive the $L^2$ generalized unnormalized Kantorovich formula, a new unnormalized Monge problem and the corresponding Monge-Ampère equation. Furthermore, we introduce a new unconstrained optimization formulation of the problem. The associated gradient flow is essentially related to an elliptic equation which can be solved efficiently. Here the proposed gradient descent procedure together with the Nesterov acceleration involves the Hamilton-Jacobi equation which arises from the KKT conditions. Several numerical examples are presented to illustrate the effectiveness of the proposed algorithms.

math.NA↗

arXiv2019

Shape Analysis via Functional Map Construction and Bases Pursuit

We propose a method to simultaneously compute scalar basis functions with an associated functional map for a given pair of triangle meshes. Unlike previous techniques that put emphasis on smoothness with respect to the Laplace--Beltrami operator and thus favor low-frequency eigenfunctions, we aim for a spectrum that allows for better feature matching. This change of perspective introduces many degrees of freedom into the problem which we exploit to improve the accuracy of our computed correspondences. To effectively search in this high dimensional space of solutions, we incorporate into our minimization state-of-the-art regularizers. We solve the resulting highly non-linear and non-convex problem using an iterative scheme via the Alternating Direction Method of Multipliers. At each step, our optimization involves simple to solve linear or Sylvester-type equations. In practice, our method performs well in terms of convergence, and we additionally show that it is similar to a provably convergent problem. We show the advantages of our approach by extensively testing it on multiple datasets in a few applications including shape matching, consistent quadrangulation and scalar function transfer.

cs.GR↗

arXiv2018

Low-rank Matrix Completion in a General Non-orthogonal Basis

This paper considers theoretical analysis of recovering a low rank matrix given a few expansion coefficients with respect to any basis. The current approach generalizes the existing analysis for the low-rank matrix completion problem with sampling under entry sensing or with respect to a symmetric orthonormal basis. The analysis is based on dual certificates using a dual basis approach and does not assume the restricted isometry property (RIP). We introduce a condition on the basis called the correlation condition. This condition can be computed in time $O(n^3)$ and holds for many cases of deterministic basis where RIP might not hold or is NP hard to verify. If the correlation condition holds and the underlying low rank matrix obeys the coherence condition with parameter $ν$, under additional mild assumptions, our main result shows that the true matrix can be recovered with very high probability from $O(nrν\log^2n)$ uniformly random expansion coefficients.

cs.IT↗

arXiv2018

Parallel Transport Convolution: A New Tool for Convolutional Neural Networks on Manifolds

Convolution has been playing a prominent role in various applications in science and engineering for many years. It is the most important operation in convolutional neural networks. There has been a recent growth of interests of research in generalizing convolutions on curved domains such as manifolds and graphs. However, existing approaches cannot preserve all the desirable properties of Euclidean convolutions, namely compactly supported filters, directionality, transferability across different manifolds. In this paper we develop a new generalization of the convolution operation, referred to as parallel transport convolution (PTC), on Riemannian manifolds and their discrete counterparts. PTC is designed based on the parallel transportation which is able to translate information along a manifold and to intrinsically preserve directionality. PTC allows for the construction of compactly supported filters and is also robust to manifold deformations. This enables us to preform wavelet-like operations and to define deep convolutional neural networks on curved domains.

cs.LG↗

arXiv2018

Exact Reconstruction of Euclidean Distance Geometry Problem Using Low-rank Matrix Completion

The Euclidean distance geometry problem arises in a wide variety of applications, from determining molecular conformations in computational chemistry to localization in sensor networks. When the distance information is incomplete, the problem can be formulated as a nuclear norm minimization problem. In this paper, this minimization program is recast as a matrix completion problem of a low-rank $r$ Gram matrix with respect to a suitable basis. The well known restricted isometry property can not be satisfied in this scenario. Instead, a dual basis approach is introduced to theoretically analyze the reconstruction problem. If the Gram matrix satisfies certain coherence conditions with parameter $ν$, the main result shows that the underlying configuration of $n$ points can be recovered with very high probability from $O(nrν\log^{2}(n))$ uniformly random samples. Computationally, simple and fast algorithms are designed to solve the Euclidean distance geometry problem. Numerical tests on different three dimensional data and protein molecules validate effectiveness and efficiency of the proposed algorithms.

cs.IT↗

arXiv2018

Nonisometric Surface Registration via Conformal Laplace-Beltrami Basis Pursuit

Surface registration is one of the most fundamental problems in geometry processing. Many approaches have been developed to tackle this problem in cases where the surfaces are nearly isometric. However, it is much more challenging to compute correspondence between surfaces which are intrinsically less similar. In this paper, we propose a variational model to align the Laplace-Beltrami (LB) eigensytems of two non-isometric genus zero shapes via conformal deformations. This method enables us compute to geometric meaningful point-to-point maps between non-isometric shapes. Our model is based on a novel basis pursuit scheme whereby we simultaneously compute a conformal deformation of a 'target shape' and its deformed LB eigensytem. We solve the model using an proximal alternating minimization algorithm hybridized with the augmented Lagrangian method which produces accurate correspondences given only a few landmark points. We also propose a reinitialization scheme to overcome some of the difficulties caused by the non-convexity of the variational problem. Intensive numerical experiments illustrate the effectiveness and robustness of the proposed method to handle non-isometric surfaces with large deformation with respect to both noise on the underlying manifolds and errors within the given landmarks.

math.NA↗

arXiv2018

Rational Neural Networks for Approximating Jump Discontinuities of Graph Convolution Operator

For node level graph encoding, a recent important state-of-art method is the graph convolutional networks (GCN), which nicely integrate local vertex features and graph topology in the spectral domain. However, current studies suffer from several drawbacks: (1) graph CNNs relies on Chebyshev polynomial approximation which results in oscillatory approximation at jump discontinuities; (2) Increasing the order of Chebyshev polynomial can reduce the oscillations issue, but also incurs unaffordable computational cost; (3) Chebyshev polynomials require degree $Ω$(poly(1/$ε$)) to approximate a jump signal such as $|x|$, while rational function only needs $\mathcal{O}$(poly log(1/$ε$))\cite{liang2016deep,telgarsky2017neural}. However, it's non-trivial to apply rational approximation without increasing computational complexity due to the denominator. In this paper, the superiority of rational approximation is exploited for graph signal recovering. RatioanlNet is proposed to integrate rational function and neural networks. We show that rational function of eigenvalues can be rewritten as a function of graph Laplacian, which can avoid multiplication by the eigenvector matrix. Focusing on the analysis of approximation on graph convolution operation, a graph signal regression task is formulated. Under graph signal regression task, its time complexity can be significantly reduced by graph Fourier transform. To overcome the local minimum problem of neural networks model, a relaxed Remez algorithm is utilized to initialize the weight parameters. Convergence rate of RatioanlNet and polynomial based methods on jump signal is analyzed for a theoretical guarantee. The extensive experimental results demonstrated that our approach could effectively characterize the jump discontinuities, outperforming competing methods by a substantial margin on both synthetic and real-world graphs.

cs.LG↗

arXiv2017

Triangulated Surface Denoising using High Order Regularization with Dynamic Weights

Recovering high quality surfaces from noisy triangulated surfaces is a fundamental important problem in geometry processing. Sharp features including edges and corners can not be well preserved in most existing denoising methods except the recent total variation (TV) and $\ell_0$ regularization methods. However, these two methods have suffered producing staircase artifacts in smooth regions. In this paper, we first introduce a second order regularization method for restoring a surface normal vector field, and then propose a new vertex updating scheme to recover the desired surface according to the restored surface normal field. The proposed model can preserve sharp features and simultaneously suppress the staircase effects in smooth regions which overcomes the drawback of the first order models. In addition, the new vertex updating scheme can prevent ambiguities introduced in existing vertex updating methods. Numerically, the proposed high order model is solved by the augmented Lagrangian method with a dynamic weighting strategy. Intensive numerical experiments on a variety of surfaces demonstrate the superiority of our method by visually and quantitatively.

cs.CG↗

arXiv2017

Solving Partial Differential Equations on Manifolds From Incomplete Inter-Point Distance

Solutions of partial differential equations (PDEs) on manifolds have provided important applications in different fields in science and engineering. Existing methods are majorly based on discretization of manifolds as implicit functions, triangle meshes, or point clouds, where the manifold structure is approximated by either zero level set of an implicit function or a set of points. In many applications, manifolds might be only provided as an inter-point distance matrix with possible missing values. This paper discusses a framework to discretize PDEs on manifolds represented as incomplete inter-point distance information. Without conducting a time-consuming global coordinates reconstruction, we propose a more efficient strategy by discretizing differential operators only based on point-wisely local reconstruction. Our local reconstruction model is based on the recent advances of low-rank matrix completion theory, where only a very small random portion of distance information is required. This method enables us to conduct analyses of incomplete distance data using solutions of special designed PDEs such as the Laplace-Beltrami (LB) eigen-system. As an application, we demonstrate a new way of manifold reconstruction from an incomplete distance by stitching patches using the spectrum of the LB operator. Intensive numerical experiments demonstrate the effectiveness of the proposed methods.

math.NA↗

arXiv2017

Global Optimization with Orthogonality Constraints via Stochastic Diffusion on Manifold

Orthogonality constrained optimization is widely used in applications from science and engineering. Due to the nonconvex orthogonality constraints, many numerical algorithms often can hardly achieve the global optimality. We aim at establishing an efficient scheme for finding global minimizers under one or more orthogonality constraints. The main concept is based on noisy gradient flow constructed from stochastic differential equations (SDE) on the Stiefel manifold, the differential geometric characterization of orthogonality constraints. We derive an explicit representation of SDE on the Stiefel manifold endowed with a canonical metric and propose a numerically efficient scheme to simulate this SDE based on Cayley transformation with theoretical convergence guarantee. The convergence to global optimizers is proved under second-order continuity. The effectiveness and efficiency of the proposed algorithms are demonstrated on a variety of problems including homogeneous polynomial optimization, computation of stability number, and 3D structure determination from Common Lines in Cryo-EM.

math.OC↗

arXiv2017

CT Image Reconstruction in a Low Dimensional Manifold

Regularization methods are commonly used in X-ray CT image reconstruction. Different regularization methods reflect the characterization of different prior knowledge of images. In a recent work, a new regularization method called a low-dimensional manifold model (LDMM) is investigated to characterize the low-dimensional patch manifold structure of natural images, where the manifold dimensionality characterizes structural information of an image. In this paper, we propose a CT image reconstruction method based on the prior knowledge of the low-dimensional manifold of CT image. Using the clinical raw projection data from GE clinic, we conduct comparisons for the CT image reconstruction among the proposed method, the simultaneous algebraic reconstruction technique (SART) with the total variation (TV) regularization, and the filtered back projection (FBP) method. Results show that the proposed method can successfully recover structural details of an imaging object, and achieve higher spatial and contrast resolution of the reconstructed image than counterparts of FBP and SART with TV.

physics.med-ph↗

arXiv2017

Point cloud discretization of Fokker-Planck operators for committor functions

The committor functions provide useful information to the understanding of transitions of a stochastic system between disjoint regions in phase space. In this work, we develop a point cloud discretization for Fokker-Planck operators to numerically calculate the committor function, with the assumption that the transition occurs on an intrinsically low-dimensional manifold in the ambient potentially high dimensional configurational space of the stochastic system. Numerical examples on model systems validate the effectiveness of the proposed method.

math.NA↗

arXiv2017

Optimal Impulse Control of a Simple Reparable System in a Nonreflexive Banach Space

We discuss the problem of optimal impulse control representing the preventive maintenance of a simple reparable system. The system model is governed by coupled transport and integro-differential equations in a nonreflexive Banach space. The objective of this paper is to construct nonnegative impulse control inputs at given system running times that minimize the probability of the system in failure mode. To guarantee the nonnegativity of the controlled system, we consider the control inputs to depend on the system state. This essentially leads to a bilinear control problem. We first present a rigorous proof of existence of an optimal controller and then apply the variational inequality to derive the first-order necessary conditions of optimality.

math.OC↗

arXiv2017

Manifold Based Low-rank Regularization for Image Restoration and Semi-supervised Learning

Low-rank structures play important role in recent advances of many problems in image science and data science. As a natural extension of low-rank structures for data with nonlinear structures, the concept of the low-dimensional manifold structure has been considered in many data processing problems. Inspired by this concept, we consider a manifold based low-rank regularization as a linear approximation of manifold dimension. This regularization is less restricted than the global low-rank regularization, and thus enjoy more flexibility to handle data with nonlinear structures. As applications, we demonstrate the proposed regularization to classical inverse problems in image sciences and data sciences including image inpainting, image super-resolution, X-ray computer tomography (CT) image reconstruction and semi-supervised learning. We conduct intensive numerical experiments in several image restoration problems and a semi-supervised learning problem of classifying handwritten digits using the MINST data. Our numerical tests demonstrate the effectiveness of the proposed methods and illustrate that the new regularization methods produce outstanding results by comparing with many existing methods.

cs.CV↗

arXiv2016

Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces

Let $(M,g)$ be a connected, closed, orientable Riemannian surface and denote by $λ_k(M,g)$ the $k$-th eigenvalue of the Laplace-Beltrami operator on $(M,g)$. In this paper, we consider the mapping $(M, g)\mapsto λ_k(M,g)$. We propose a computational method for finding the conformal spectrum $Λ^c_k(M,[g_0])$, which is defined by the eigenvalue optimization problem of maximizing $λ_k(M,g)$ for $k$ fixed as $g$ varies within a conformal class $[g_0]$ of fixed volume $textrm{vol}(M,g) = 1$. We also propose a computational method for the problem where $M$ is additionally allowed to vary over surfaces with fixed genus, $γ$. This is known as the topological spectrum for genus $γ$ and denoted by $Λ^t_k(γ)$. Our computations support a conjecture of N. Nadirashvili (2002) that $Λ^t_k(0) = 8 πk$, attained by a sequence of surfaces degenerating to a union of $k$ identical round spheres. Furthermore, based on our computations, we conjecture that $Λ^t_k(1) = \frac{8π^2}{\sqrt{3}} + 8π(k-1)$, attained by a sequence of surfaces degenerating into a union of an equilateral flat torus and $k-1$ identical round spheres. The values are compared to several surfaces where the Laplace-Beltrami eigenvalues are well-known, including spheres, flat tori, and embedded tori. In particular, we show that among flat tori of volume one, the $k$-th Laplace-Beltrami eigenvalue has a local maximum with value $λ_k = 4π^2 \left\lceil \frac{k}{2} \right\rceil^2 \left( \left\lceil \frac{k}{2} \right\rceil^2 - \frac{1}{4}\right)^{-\frac{1}{2}}$. Several properties are also studied computationally, including uniqueness, symmetry, and eigenvalue multiplicity.

math.DG↗

arXiv2015

Localized density matrix minimization and linear scaling algorithms

We propose a convex variational approach to compute localized density matrices for both zero temperature and finite temperature cases, by adding an entry-wise $\ell_1$ regularization to the free energy of the quantum system. Based on the fact that the density matrix decays exponential away from the diagonal for insulating system or system at finite temperature, the proposed $\ell_1$ regularized variational method provides a nice way to approximate the original quantum system. We provide theoretical analysis of the approximation behavior and also design convergence guaranteed numerical algorithms based on Bregman iteration. More importantly, the $\ell_1$ regularized system naturally leads to localized density matrices with banded structure, which enables us to develop approximating algorithms to find the localized density matrices with computation cost linearly dependent on the problem size.

math.NA↗

arXiv2014

Multi-scale Non-Rigid Point Cloud Registration Using Robust Sliced-Wasserstein Distance via Laplace-Beltrami Eigenmap

In this work, we propose computational models and algorithms for point cloud registration with non-rigid transformation. First, point clouds sampled from manifolds originally embedded in some Euclidean space $\mathbb{R}^D$ are transformed to new point clouds embedded in $\mathbb{R}^n$ by Laplace-Beltrami(LB) eigenmap using the $n$ leading eigenvalues and corresponding eigenfunctions of LB operator defined intrinsically on the manifolds. The LB eigenmap are invariant under isometric transformation of the original manifolds. Then we design computational models and algorithms for registration of the transformed point clouds in distribution/probability form based on the optimal transport theory which provides both generality and flexibility to handle general point clouds setting. Our methods use robust sliced-Wasserstein distance, which is as the average of projected Wasserstein distance along different directions, and incorporate a rigid transformation to handle ambiguities introduced by the Laplace-Beltrami eigenmap. By going from smaller $n$, which provides a quick and robust registration (based on coarse scale features) as well as a good initial guess for finer scale registration, to a larger $n$, our method provides an efficient, robust and accurate approach for multi-scale non-rigid point cloud registration.

math.NA↗

arXiv2014

Compressed Wannier modes found from an $L_1$ regularized energy functional

We propose a method for calculating Wannier functions of periodic solids directly from a modified variational principle for the energy, subject to the requirement that the Wannier functions are orthogonal to all their translations ("shift-orthogonality"). Localization is achieved by adding an $L_1$ regularization term to the energy functional. This approach results in "compressed" Wannier modes with compact support, where one parameter $μ$ controls the trade-off between the accuracy of the total energy and the size of the support of the Wannier modes. Efficient algorithms for shift-orthogonalization and solution of the variational minimization problem are demonstrated.

cond-mat.mtrl-sci↗