Rongjie Lai

Rensselaer Polytechnic Institute | RPI · Department of Mathematical Sciences

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 l...

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...

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 im...

We propose a convex variational principle to find sparse representation of low-lying eigenspace of symmetric matrices. In the context of electronic structure calculation, this corresponds to a sparse density matrix minimization algorithm with $\ell_1$ regularization. The minimization problem can be efficiently solved by a split Bergman iteration ty...

Significance Intuition suggests that many interesting phenomena in physics, chemistry, and materials science are “short-sighted”—that is, perturbation in a small spatial region only affects its immediate surroundings. In mathematical terms, near-sightedness is described by functions of finite range. As an example, the so-called Wannier functions in...

Autoencoding is a popular method in representation learning. Conventional autoencoders employ symmetric encoding-decoding procedures and a simple Euclidean latent space to detect hidden low-dimensional structures in an unsupervised way. This work introduces a chart autoencoder with an asymmetric encoding-decoding process that can incorporate additi...

Mean-field games (MFGs) are a modeling framework for systems with a large number of interacting agents. They have applications in economics, finance, and game theory. Normalizing flows (NFs) are a family of deep generative models that compute data likelihoods by using an invertible mapping, which is typically parameterized by using neural networks....

Conventional Mean-field games/control study the behavior of a large number of rational agents moving in the Euclidean spaces. In this work, we explore the mean-field games on Riemannian manifolds. We formulate the mean-field game Nash Equilibrium on manifolds. We also establish the equivalence between the PDE system and the optimality conditions of...

Massive molecular simulations of drug-target proteins have been used as a tool to understand disease mechanism and develop therapeutics. This work focuses on learning a generative neural network on a structural ensemble of a drug-target protein, e.g. SARS-CoV-2 Spike protein, obtained from computationally expensive molecular simulations. Model task...

A neural network with the widely-used ReLU activation has been shown to partition the sample space into many convex polytopes for prediction. However, the parameterized way a neural network and other machine learning models use to partition the space has imperfections, e.g., the compromised interpretability for complex models, the inflexibility in...

Convolution plays a crucial role in various applications in signal and image processing, analysis, and recognition. It is also the main building block of convolution neural networks (CNNs). Designing appropriate convolution neural networks on manifold-structured point clouds can inherit and empower recent advances of CNNs to analyzing and processin...

Inspired by diversity of biological neurons, quadratic artificial neurons can play an important role in deep learning models. The type of quadratic neurons of our interest replaces the inner-product operation in the conventional neuron with a quadratic function. Despite promising results so far achieved by networks of quadratic neurons, there are i...

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 basis that allows for better feature matching....

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 ce...

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 variationa...

We introduce fast algorithms for generalized unnormalized optimal transport. To handle densities with different total mass, we consider a dynamic model, which mixes the Lp optimal transport with Lp distance. For p=1, we derive the corresponding L1 generalized unnormalized Kantorovich formula. We further show that the problem becomes a simple L1 min...

In this paper, we propose an efficient and flexible algorithm to solve dynamic mean-field planning problems based on an accelerated proximal gradient method. Besides an easy-to-implement gradient descent step in this algorithm, a crucial projection step becomes solving an elliptic equation whose solution can be obtained by conventional methods effi...

The problem of finding the configuration of points given partial information on pairwise inter-point distances, the Euclidean distance geometry problem, appears in multiple applications. In this paper, we propose an approach that integrates structural similarity and a nonconvex distance geometry algorithm for the protein structure determination pro...

While classic studies proved that wide networks allow universal approximation, recent research and successes of deep learning demonstrate the power of the network depth. Based on a symmetric consideration, we investigate if the design of artificial neural networks should have a directional preference, and what the mechanism of interaction is betwee...

The loss landscapes of deep neural networks are not well understood due to their high nonconvexity. Empirically, the local minima of these loss functions can be connected by a learned curve in model space, along which the loss remains nearly constant; a feature known as mode connectivity. Yet, current curve finding algorithms do not consider the in...

In this work, a simple and efficient dual iterative refinement (DIR) method is proposed for dense correspondence between two nearly isometric shapes. The key idea is to use dual information, such as spatial and spectral, or local and global features, in a complementary and effective way, and extract more accurate information from current iteration...

For non-Euclidean data such as meshes of humans, a prominent task for generative models is geometric disentanglement; the separation of latent codes for intrinsic (i.e. identity) and extrinsic (i.e. pose) geometry. This work introduces a novel mesh feature, the conformal factor and normal feature (CFAN), for use in mesh convolutional autoencoders....

In this work, we introduce a novel local pairwise descriptor and then develop a simple, effective iterative method to solve the resulting quadratic assignment through sparsity control for shape correspondence between two approximate isometric surfaces. Our pairwise descriptor is based on the stiffness and mass matrix of finite element approximation...

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 si...

Auto-encoding and generative models have made tremendous successes in image and signal representation learning and generation. These models, however, generally employ the full Euclidean space or a bounded subset (such as $[0,1]^l$) as the latent space, whose flat geometry is often too simplistic to meaningfully reflect the topological structure of...

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 matchi...

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 si...

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 c...

Convolution plays a crucial role in various applications in signal and image processing , analysis and recognition. It is also the main building block of convolution neural networks (CNNs). Designing appropriate convolution neural networks on manifold-structured point clouds can inherit and empower recent advances of CNNs to analyzing and processin...

Convolution plays a crucial role in various applications in signal and image processing, analysis and recognition. It is also the main building block of convolution neural networks (CNNs). Designing appropriate convolution neural networks on manifold-structured point clouds can inherit and empower recent advances of CNNs to analyzing and processing...

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 ce...

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 variationa...

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 appr...

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...

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...

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 recas...

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 prod...

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...

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...

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...

We propose a convex variational principle to find sparse representation of low-lying eigenspace of symmetric matrices. In the context of electronic structure calculation, this corresponds to a sparse density matrix minimization algorithm with l(1) regularization. The minimization problem can be efficiently solved by a split Bregman iteration type a...

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 eigenvalu...

Let $(M,g)$ be a connected, closed, orientable Riemannian surface and denote by $\lambda_k(M,g)$ the $k$-th eigenvalue of the Laplace-Beltrami operator on $(M,g)$. In this paper, we consider the mapping $(M, g)\mapsto \lambda_k(M,g)$. We propose a computational method for finding the conformal spectrum $\lambda^c_k(M,[g_0])$, which is defined by th...

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...

In this paper we present a novel approach for the intrinsic mapping of anatomical surfaces and its application in brain mapping research. Using the Laplace-Beltrami eigensystem, we represent each surface with an isometry invariant embedding in a high dimensional space. The key idea in our system is that we realize surface deformation in the embeddi...

Surface conformal maps between genus-0 surfaces play important roles in applied mathematics and engineering, with applications in medical image analysis and computer graphics. Previous work (Gu and Yau in Commun Inf Syst 2(2):121–146, 2002) introduces a variational approach, where global conformal parameterization of genus-0 surfaces was addressed...

This paper presents a fast algorithm for projecting a given function to the set of shift orthogonal functions (i.e. set containing functions with unit $L^2$ norm that are orthogonal to their prescribed shifts). The algorithm can be parallelized easily and its computational complexity is bounded by $O(M\log(M))$, where $M$ is the number of coefficie...

This paper describes an L1 regularized variational framework for developing a spatially localized basis, compressed plane waves (CPWs), that spans the eigenspace of a differential operator, for instance, the Laplace operator. Our approach generalizes the concept of plane waves to an orthogonal real-space basis with multi-resolution capa-bilities....

Orthogonality constrained problems are widely used in science and engineering. However, it is challenging to solve these problems efficiently due to the non-convex constraints. In this paper, a splitting method based on Bregman iteration is represented to tackle the optimization problems with orthogonality constraints. With the proposed method, the...

A new image decomposition scheme, called the adaptive directional total variation (ADTV) model, is proposed to achieve effective segmentation and enhancement for latent fingerprint images in this work. The proposed model is inspired by the classical total variation models, but it differentiates itself by integrating two unique features of fingerpri...

In this work, we introduce a numerical method to approximate differential operators and integrals on point clouds sampled from a two dimensional manifold embedded in ℝ n . Global mesh structure is usually hard to construct in this case. While our method only relies on the local mesh structure at each data point, which is constructed through local t...

In this paper we develop a novel approach for computing conformal maps between anatomical surfaces with the ability of aligning anatomical features and achieving greatly reduced metric distortion. In contrast to conventional approaches that focused on conformal maps to the sphere or plane, our method computes the conformal map between surfaces in t...

One challenge in surface restoration is to design surface diffusion preserving ridges and sharp corners. In this paper, we propose a new surface restoration model based on the observation that surfaces’ implicit representations are continuous functions whose first order derivatives have discontinuities at ridges and sharp corners. Regularized by ve...

In this paper we present a novel system for the automated reconstruction of cortical surfaces from T1-weighted magnetic resonance images. At the core of our system is a unified Reeb analysis framework for the detection and removal of geometric and topological outliers on tissue boundaries. Using intrinsic Reeb analysis, our system can pinpoint the...

A key challenge in the accurate reconstruction of cortical surfaces is the automated correction of geometric and topological outliers in tissue boundaries. Conventionally these two types of errors are handled separately. In this work, we propose a unified analysis framework for the joint correction of geometric and topological outliers in cortical...

Latent fingerprint detection and segmentation play a critical role in image forensics for law enforcement. Being collected from crime scenes, a latent fingerprint is often mixed with other components such as structured noise or other fingerprints. Existing fingerprint recognition algorithms fail to work properly for latent fingerprint images, since...