arXiv subjects

Jianfeng Lu

Publications and source records attributed to Jianfeng Lu.

At least 127 records · Page 7Linked to original sources

arXiv2023

Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions

We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small $L^2$ error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in reverse KL divergence in $ε$-accuracy can be done in $\tilde O\left(\frac{d \log (1/δ)}ε\right)$ steps: 1) the variance-$δ$ Gaussian perturbation of any data distribution; 2) data distributions with $1/δ$-smooth score functions. Our analysis also provides a quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

cs.LG↗

arXiv2022

MonoSIM: Simulating Learning Behaviors of Heterogeneous Point Cloud Object Detectors for Monocular 3D Object Detection

Monocular 3D object detection is a fundamental but very important task to many applications including autonomous driving, robotic grasping and augmented reality. Existing leading methods tend to estimate the depth of the input image first, and detect the 3D object based on point cloud. This routine suffers from the inherent gap between depth estimation and object detection. Besides, the prediction error accumulation would also affect the performance. In this paper, a novel method named MonoSIM is proposed. The insight behind introducing MonoSIM is that we propose to simulate the feature learning behaviors of a point cloud based detector for monocular detector during the training period. Hence, during inference period, the learned features and prediction would be similar to the point cloud based detector as possible. To achieve it, we propose one scene-level simulation module, one RoI-level simulation module and one response-level simulation module, which are progressively used for the detector's full feature learning and prediction pipeline. We apply our method to the famous M3D-RPN detector and CaDDN detector, conducting extensive experiments on KITTI and Waymo Open datasets. Results show that our method consistently improves the performance of different monocular detectors for a large margin without changing their network architectures. Our codes will be publicly available at https://github.com/sunh18/MonoSIM}{https://github.com/sunh18/MonoSIM.

cs.CV↗

arXiv2022

A Structure-guided Effective and Temporal-lag Connectivity Network for Revealing Brain Disorder Mechanisms

Brain network provides important insights for the diagnosis of many brain disorders, and how to effectively model the brain structure has become one of the core issues in the domain of brain imaging analysis. Recently, various computational methods have been proposed to estimate the causal relationship (i.e., effective connectivity) between brain regions. Compared with traditional correlation-based methods, effective connectivity can provide the direction of information flow, which may provide additional information for the diagnosis of brain diseases. However, existing methods either ignore the fact that there is a temporal-lag in the information transmission across brain regions, or simply set the temporal-lag value between all brain regions to a fixed value. To overcome these issues, we design an effective temporal-lag neural network (termed ETLN) to simultaneously infer the causal relationships and the temporal-lag values between brain regions, which can be trained in an end-to-end manner. In addition, we also introduce three mechanisms to better guide the modeling of brain networks. The evaluation results on the Alzheimer's Disease Neuroimaging Initiative (ADNI) database demonstrate the effectiveness of the proposed method.

q-bio.NC↗

arXiv2022

On the global convergence of randomized coordinate gradient descent for non-convex optimization

In this work, we analyze the global convergence property of coordinate gradient descent with random choice of coordinates and stepsizes for non-convex optimization problems. Under generic assumptions, we prove that the algorithm iterate will almost surely escape strict saddle points of the objective function. As a result, the algorithm is guaranteed to converge to local minima if all saddle points are strict. Our proof is based on viewing coordinate descent algorithm as a nonlinear random dynamical system and a quantitative finite block analysis of its linearization around saddle points.

math.OC↗

arXiv2022

Human Joint Kinematics Diffusion-Refinement for Stochastic Motion Prediction

Stochastic human motion prediction aims to forecast multiple plausible future motions given a single pose sequence from the past. Most previous works focus on designing elaborate losses to improve the accuracy, while the diversity is typically characterized by randomly sampling a set of latent variables from the latent prior, which is then decoded into possible motions. This joint training of sampling and decoding, however, suffers from posterior collapse as the learned latent variables tend to be ignored by a strong decoder, leading to limited diversity. Alternatively, inspired by the diffusion process in nonequilibrium thermodynamics, we propose MotionDiff, a diffusion probabilistic model to treat the kinematics of human joints as heated particles, which will diffuse from original states to a noise distribution. This process offers a natural way to obtain the "whitened" latents without any trainable parameters, and human motion prediction can be regarded as the reverse diffusion process that converts the noise distribution into realistic future motions conditioned on the observed sequence. Specifically, MotionDiff consists of two parts: a spatial-temporal transformer-based diffusion network to generate diverse yet plausible motions, and a graph convolutional network to further refine the outputs. Experimental results on two datasets demonstrate that our model yields the competitive performance in terms of both accuracy and diversity.

cs.CV↗

arXiv2022

A Regularity Theory for Static Schrödinger Equations on $\mathbb{R}^d$ in Spectral Barron Spaces

Spectral Barron spaces have received considerable interest recently as it is the natural function space for approximation theory of two-layer neural networks with a dimension-free convergence rate. In this paper we study the regularity of solutions to the whole-space static Schrödinger equation in spectral Barron spaces. We prove that if the source of the equation lies in the spectral Barron space $\mathcal{B}^s(\mathbb{R}^d)$ and the potential function admitting a non-negative lower bound decomposes as a positive constant plus a function in $\mathcal{B}^s(\mathbb{R}^d)$, then the solution lies in the spectral Barron space $\mathcal{B}^{s+2}(\mathbb{R}^d)$.

math.AP↗

arXiv2022

Convergence of score-based generative modeling for general data distributions

Score-based generative modeling (SGM) has grown to be a hugely successful method for learning to generate samples from complex data distributions such as that of images and audio. It is based on evolving an SDE that transforms white noise into a sample from the learned distribution, using estimates of the score function, or gradient log-pdf. Previous convergence analyses for these methods have suffered either from strong assumptions on the data distribution or exponential dependencies, and hence fail to give efficient guarantees for the multimodal and non-smooth distributions that arise in practice and for which good empirical performance is observed. We consider a popular kind of SGM -- denoising diffusion models -- and give polynomial convergence guarantees for general data distributions, with no assumptions related to functional inequalities or smoothness. Assuming $L^2$-accurate score estimates, we obtain Wasserstein distance guarantees for any distribution of bounded support or sufficiently decaying tails, as well as TV guarantees for distributions with further smoothness assumptions.

cs.LG↗

arXiv2022

A deep learning framework for geodesics under spherical Wasserstein-Fisher-Rao metric and its application for weighted sample generation

Wasserstein-Fisher-Rao (WFR) distance is a family of metrics to gauge the discrepancy of two Radon measures, which takes into account both transportation and weight change. Spherical WFR distance is a projected version of WFR distance for probability measures so that the space of Radon measures equipped with WFR can be viewed as metric cone over the space of probability measures with spherical WFR. Compared to the case for Wasserstein distance, the understanding of geodesics under the spherical WFR is less clear and still an ongoing research focus. In this paper, we develop a deep learning framework to compute the geodesics under the spherical WFR metric, and the learned geodesics can be adopted to generate weighted samples. Our approach is based on a Benamou-Brenier type dynamic formulation for spherical WFR. To overcome the difficulty in enforcing the boundary constraint brought by the weight change, a Kullback-Leibler (KL) divergence term based on the inverse map is introduced into the cost function. Moreover, a new regularization term using the particle velocity is introduced as a substitute for the Hamilton-Jacobi equation for the potential in dynamic formula. When used for sample generation, our framework can be beneficial for applications with given weighted samples, especially in the Bayesian inference, compared to sample generation with previous flow models.

cs.LG↗

arXiv2022

Overlooked Poses Actually Make Sense: Distilling Privileged Knowledge for Human Motion Prediction

Previous works on human motion prediction follow the pattern of building a mapping relation between the sequence observed and the one to be predicted. However, due to the inherent complexity of multivariate time series data, it still remains a challenge to find the extrapolation relation between motion sequences. In this paper, we present a new prediction pattern, which introduces previously overlooked human poses, to implement the prediction task from the view of interpolation. These poses exist after the predicted sequence, and form the privileged sequence. To be specific, we first propose an InTerPolation learning Network (ITP-Network) that encodes both the observed sequence and the privileged sequence to interpolate the in-between predicted sequence, wherein the embedded Privileged-sequence-Encoder (Priv-Encoder) learns the privileged knowledge (PK) simultaneously. Then, we propose a Final Prediction Network (FP-Network) for which the privileged sequence is not observable, but is equipped with a novel PK-Simulator that distills PK learned from the previous network. This simulator takes as input the observed sequence, but approximates the behavior of Priv-Encoder, enabling FP-Network to imitate the interpolation process. Extensive experimental results demonstrate that our prediction pattern achieves state-of-the-art performance on benchmarked H3.6M, CMU-Mocap and 3DPW datasets in both short-term and long-term predictions.

cs.CV↗

arXiv2022

On the closedness and geometry of tensor network state sets

Tensor network states (TNS) are a powerful approach for the study of strongly correlated quantum matter. The curse of dimensionality is addressed by parametrizing the many-body state in terms of a network of partially contracted tensors. These tensors form a substantially reduced set of effective degrees of freedom. In practical algorithms, functionals like energy expectation values or overlaps are optimized over certain sets of TNS. Concerning algorithmic stability, it is important whether the considered sets are closed because, otherwise, the algorithms may approach a boundary point that is outside the TNS set and tensor elements diverge. We discuss the closedness and geometries of TNS sets, and we propose regularizations for optimization problems on non-closed TNS sets. We show that sets of matrix product states (MPS) with open boundary conditions, tree tensor network states (TTNS), and the multiscale entanglement renormalization ansatz (MERA) are always closed, whereas sets of translation-invariant MPS with periodic boundary conditions (PBC), heterogeneous MPS with PBC, and projected entangled-pair states (PEPS) are generally not closed. The latter is done using explicit examples like the W state, states that we call two-domain states, and fine-grained versions thereof.

quant-ph↗

arXiv2022

Quantum Orbital Minimization Method for Excited States Calculation on Quantum Computer

We propose a quantum-classical hybrid variational algorithm, the quantum orbital minimization method (qOMM), for obtaining the ground state and low-lying excited states of a Hermitian operator. Given parameterized ansatz circuits representing eigenstates, qOMM implements quantum circuits to represent the objective function in the orbital minimization method and adopts classical optimizer to minimize the objective function with respect to parameters in ansatz circuits. The objective function has orthogonality implicitly embedded, which allows qOMM to apply a different ansatz circuit to each reference state. We carry out numerical simulations that seek to find excited states of the $\text{H}_{2}$, $\text{LiH}$, and a toy model consisting of 4 hydrogen atoms arranged in a square lattice in the STO-3G basis and UCCSD ansatz circuits. Comparing the numerical results with existing excited states methods, qOMM is less prone to getting stuck in local minima and can achieve convergence with more shallow ansatz circuits.

physics.chem-ph↗

arXiv2022

Single Time-scale Actor-critic Method to Solve the Linear Quadratic Regulator with Convergence Guarantees

We propose a single time-scale actor-critic algorithm to solve the linear quadratic regulator (LQR) problem. A least squares temporal difference (LSTD) method is applied to the critic and a natural policy gradient method is used for the actor. We give a proof of convergence with sample complexity $\mathcal{O}(\varepsilon^{-1} \log(\varepsilon^{-1})^2)$. The method in the proof is applicable to general single time-scale bilevel optimization problem. We also numerically validate our theoretical results on the convergence.

math.OC↗

arXiv2022

Posterior computation with the Gibbs zig-zag sampler

An intriguing new class of piecewise deterministic Markov processes (PDMPs) has recently been proposed as an alternative to Markov chain Monte Carlo (MCMC). In order to facilitate the application to a larger class of problems, we propose a new class of PDMPs termed Gibbs zig-zag samplers, which allow parameters to be updated in blocks with a zig-zag sampler applied to certain parameters and traditional MCMC-style updates to others. We demonstrate the flexibility of this framework on posterior sampling for logistic models with shrinkage priors for high-dimensional regression and random effects and provide conditions for geometric ergodicity and the validity of a central limit theorem.

stat.CO↗

arXiv2022

Fast Algorithms of Bath Calculations in Simulations of Quantum System-Bath Dynamics

We present fast algorithms for the summation of Dyson series and the inchworm Monte Carlo method for quantum systems that are coupled with harmonic baths. The algorithms are based on evolving the integro-differential equations where the most expensive part comes from the computation of bath influence functionals. To accelerate the computation, we design fast algorithms based on reusing the bath influence functionals computed in the previous time steps to reduce the number of calculations. It is proven that the proposed fast algorithms reduce the number of such calculations by a factor of $O(N)$, where $N$ is the total number of time steps. Numerical experiments are carried out to show the efficiency of the method and to verify the theoretical results.

quant-ph↗

arXiv2022

Neural Network Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions

In this paper, we propose and study neural network based methods for solutions of high-dimensional quadratic porous medium equation (QPME). Three variational formulations of this nonlinear PDE are presented: a strong formulation and two weak formulations. For the strong formulation, the solution is directly parameterized with a neural network and optimized by minimizing the PDE residual. It can be proved that the convergence of the optimization problem guarantees the convergence of the approximate solution in the $L^1$ sense. The weak formulations are derived following Brenier, Y., 2020, which characterizes the very weak solutions of QPME. Specifically speaking, the solutions are represented with intermediate functions who are parameterized with neural networks and are trained to optimize the weak formulations. Extensive numerical tests are further carried out to investigate the pros and cons of each formulation in low and high dimensions. This is an initial exploration made along the line of solving high-dimensional nonlinear PDEs with neural network based methods, which we hope can provide some useful experience for future investigations.

math.NA↗

arXiv2022

Asymptotic analysis of diabatic surface hopping algorithm in the adiabatic and non-adiabatic limits

Surface hopping algorithms, as an important class of quantum dynamics simulation algorithms for non-adiabatic dynamics, are typically performed in the adiabatic representation, which can break down in the presence of ill-defined adiabatic potential energy surfaces (PESs) and adiabatic coupling term. Another issue of surface hopping algorithms is the difficulty in capturing the correct scaling of the transition rate in the Marcus (weak-coupling/non-adiabatic) regime. Though the first issue can be circumvented by exploiting the diabatic representation, diabatic surface hopping algorithms usually lack justification on the theoretical level. We consider the diabatic surface hopping algorithm proposed in [Fang, Lu. Multiscale Model. Simul. 16:4, 1603-1622, 2018] and provide the asymptotic analysis of the transition rate in the Marcus regime that justifies the correct scaling for the spin-boson model. We propose two conditions that guarantee the correctness for general potentials. In the opposite (strong-coupling/adiabatic) regime, we derive the asymptotic behavior of the algorithm that interestingly matches a type of mean-field description. The techniques used here may shed light on the analysis for other diabatic-based algorithms.

math.NA↗

arXiv2022

Low-rank approximation for multiscale PDEs

Historically, analysis for multiscale PDEs is largely unified while numerical schemes tend to be equation-specific. In this paper, we propose a unified framework for computing multiscale problems through random sampling. This is achieved by incorporating randomized SVD solvers and manifold learning techniques to numerically reconstruct the low-rank features of multiscale PDEs. We use multiscale radiative transfer equation and elliptic equation with rough media to showcase the application of this framework.

math.NA↗

arXiv2022

Universal approximation of symmetric and anti-symmetric functions

We consider universal approximations of symmetric and anti-symmetric functions, which are important for applications in quantum physics, as well as other scientific and engineering computations. We give constructive approximations with explicit bounds on the number of parameters with respect to the dimension and the target accuracy $ε$. While the approximation still suffers from the curse of dimensionality, to the best of our knowledge, these are the first results in the literature with explicit error bounds for functions with symmetry or anti-symmetry constraints.

math.NA↗