arXiv subjects

Jianfeng Lu

Publications and source records attributed to Jianfeng Lu.

At least 181 records · Page 10Linked to original sources

arXiv2020

Ensemble Kalman Inversion for nonlinear problems: weights, consistency, and variance bounds

Ensemble Kalman Inversion (EnKI) and Ensemble Square Root Filter (EnSRF) are popular sampling methods for obtaining a target posterior distribution. They can be seem as one step (the analysis step) in the data assimilation method Ensemble Kalman Filter. Despite their popularity, they are, however, not unbiased when the forward map is nonlinear. Important Sampling (IS), on the other hand, obtains the unbiased sampling at the expense of large variance of weights, leading to slow convergence of high moments. We propose WEnKI and WEnSRF, the weighted versions of EnKI and EnSRF in this paper. It follows the same gradient flow as that of EnKI/EnSRF with weight corrections. Compared to the classical methods, the new methods are unbiased, and compared with IS, the method has bounded weight variance. Both properties will be proved rigorously in this paper. We further discuss the stability of the underlying Fokker-Planck equation. This partially explains why EnKI, despite being inconsistent, performs well occasionally in nonlinear settings. Numerical evidence will be demonstrated at the end.

math.NA↗

arXiv2020

Neural Machine Translation with Error Correction

Neural machine translation (NMT) generates the next target token given as input the previous ground truth target tokens during training while the previous generated target tokens during inference, which causes discrepancy between training and inference as well as error propagation, and affects the translation accuracy. In this paper, we introduce an error correction mechanism into NMT, which corrects the error information in the previous generated tokens to better predict the next token. Specifically, we introduce two-stream self-attention from XLNet into NMT decoder, where the query stream is used to predict the next token, and meanwhile the content stream is used to correct the error information from the previous predicted tokens. We leverage scheduled sampling to simulate the prediction errors during training. Experiments on three IWSLT translation datasets and two WMT translation datasets demonstrate that our method achieves improvements over Transformer baseline and scheduled sampling. Further experimental analyses also verify the effectiveness of our proposed error correction mechanism to improve the translation quality.

cs.CL↗

arXiv2020

Defect resonances of truncated crystal structures

Defects in the atomic structure of crystalline materials may spawn electronic bound states, known as \emph{defect states}, which decay rapidly away from the defect. Simplified models of defect states typically assume the defect is surrounded on all sides by an infinite perfectly crystalline material. In reality the surrounding structure must be finite, and in certain contexts the structure can be small enough that edge effects are significant. In this work we investigate these edge effects and prove the following result. Suppose that a one-dimensional infinite crystalline material hosting a positive energy defect state is truncated a distance $M$ from the defect. Then, for sufficiently large $M$, there exists a resonance \emph{exponentially close} (in $M$) to the bound state eigenvalue. It follows that the truncated structure hosts a metastable state with an exponentially long lifetime. Our methods allow both the resonance frequency and associated resonant state to be computed to all orders in $e^{-M}$. We expect this result to be of particular interest in the context of photonic crystals, where defect states are used for wave-guiding and structures are relatively small. Finally, under a mild additional assumption we prove that if the defect state has negative energy then the truncated structure hosts a bound state with exponentially-close energy.

math.AP↗

arXiv2020

Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach

We propose a new method to solve eigenvalue problems for linear and semilinear second order differential operators in high dimensions based on deep neural networks. The eigenvalue problem is reformulated as a fixed point problem of the semigroup flow induced by the operator, whose solution can be represented by Feynman-Kac formula in terms of forward-backward stochastic differential equations. The method shares a similar spirit with diffusion Monte Carlo but augments a direct approximation to the eigenfunction through neural-network ansatz. The criterion of fixed point provides a natural loss function to search for parameters via optimization. Our approach is able to provide accurate eigenvalue and eigenfunction approximations in several numerical examples, including Fokker-Planck operator and the linear and nonlinear Schrödinger operators in high dimensions.

cs.LG↗

arXiv2020

ELSI -- An Open Infrastructure for Electronic Structure Solvers

Routine applications of electronic structure theory to molecules and periodic systems need to compute the electron density from given Hamiltonian and, in case of non-orthogonal basis sets, overlap matrices. System sizes can range from few to thousands or, in some examples, millions of atoms. Different discretization schemes (basis sets) and different system geometries (finite non-periodic vs. infinite periodic boundary conditions) yield matrices with different structures. The ELectronic Structure Infrastructure (ELSI) project provides an open-source software interface to facilitate the implementation and optimal use of high-performance solver libraries covering cubic scaling eigensolvers, linear scaling density-matrix-based algorithms, and other reduced scaling methods in between. In this paper, we present recent improvements and developments inside ELSI, mainly covering (1) new solvers connected to the interface, (2) matrix layout and communication adapted for parallel calculations of periodic and/or spin-polarized systems, (3) routines for density matrix extrapolation in geometry optimization and molecular dynamics calculations, and (4) general utilities such as parallel matrix I/O and JSON output. The ELSI interface has been integrated into four electronic structure code projects (DFTB+, DGDFT, FHI-aims, SIESTA), allowing us to rigorously benchmark the performance of the solvers on an equal footing. Based on results of a systematic set of large-scale benchmarks performed with Kohn-Sham density-functional theory and density-functional tight-binding theory, we identify factors that strongly affect the efficiency of the solvers, and propose a decision layer that assists with the solver selection process. Finally, we describe a reverse communication interface encoding matrix-free iterative solver strategies that are amenable, e.g., for use with planewave basis sets.

physics.comp-ph↗

arXiv2020

Estimating Normalizing Constants for Log-Concave Distributions: Algorithms and Lower Bounds

Estimating the normalizing constant of an unnormalized probability distribution has important applications in computer science, statistical physics, machine learning, and statistics. In this work, we consider the problem of estimating the normalizing constant $Z=\int_{\mathbb{R}^d} e^{-f(x)}\,\mathrm{d}x$ to within a multiplication factor of $1 \pm \varepsilon$ for a $μ$-strongly convex and $L$-smooth function $f$, given query access to $f(x)$ and $\nabla f(x)$. We give both algorithms and lowerbounds for this problem. Using an annealing algorithm combined with a multilevel Monte Carlo method based on underdamped Langevin dynamics, we show that $\widetilde{\mathcal{O}}\Bigl(\frac{d^{4/3}κ+ d^{7/6}κ^{7/6}}{\varepsilon^2}\Bigr)$ queries to $\nabla f$ are sufficient, where $κ= L / μ$ is the condition number. Moreover, we provide an information theoretic lowerbound, showing that at least $\frac{d^{1-o(1)}}{\varepsilon^{2-o(1)}}$ queries are necessary. This provides a first nontrivial lowerbound for the problem.

cs.DS↗

arXiv2020

A low-rank Schwarz method for radiative transport equation with heterogeneous scattering coefficient

Random sampling has been used to find low-rank structure and to build fast direct solvers for multiscale partial differential equations of various types. In this work, we design an accelerated Schwarz method for radiative transfer equations that makes use of approximate local solution maps constructed offline via a random sampling strategy. Numerical examples demonstrate the accuracy, robustness, and efficiency of the proposed approach.

math.NA↗

arXiv2020

A Proximal-Gradient Algorithm for Crystal Surface Evolution

As a counterpoint to recent numerical methods for crystal surface evolution, which agree well with microscopic dynamics but suffer from significant stiffness that prevents simulation on fine spatial grids, we develop a new numerical method based on the macroscopic partial differential equation, leveraging its formal structure as the gradient flow of the total variation energy, with respect to a weighted $H^{-1}$ norm. This gradient flow structure relates to several metric space gradient flows of recent interest, including 2-Wasserstein flows and their generalizations to nonlinear mobilities. We develop a novel semi-implicit time discretization of the gradient flow, inspired by the classical minimizing movements scheme (known as the JKO scheme in the 2-Wasserstein case). We then use a primal dual hybrid gradient (PDHG) method to compute each element of the semi-implicit scheme. In one dimension, we prove convergence of the PDHG method to the semi-implicit scheme, under general integrability assumptions on the mobility and its reciprocal. Finally, by taking finite difference approximations of our PDHG method, we arrive at a fully discrete numerical algorithm, with iterations that converge at a rate independent of the spatial discretization: in particular, the convergence properties do not deteriorate as we refine our spatial grid. We close with several numerical examples illustrating the properties of our method, including facet formation at local maxima, pinning at local minima, and convergence as the spatial and temporal discretizations are refined.

math.NA↗

arXiv2020

Stable Phase Retrieval from Locally Stable and Conditionally Connected Measurements

This paper is concerned with stable phase retrieval for a family of phase retrieval models we name "locally stable and conditionally connected" (LSCC) measurement schemes. For every signal $f$, we associate a corresponding weighted graph $G_f$, defined by the LSCC measurement scheme, and show that the phase retrievability of the signal $f$ is determined by the connectivity of $G_f$. We then characterize the phase retrieval stability of the signal $f$ by two measures that are commonly used in graph theory to quantify graph connectivity: the Cheeger constant of $G_f$ for real valued signals, and the algebraic connectivity of $G_f$ for complex valued signals. We use our results to study the stability of two phase retrieval models that can be cast as LSCC measurement schemes, and focus on understanding for which signals the "curse of dimensionality" can be avoided. The first model we discuss is a finite-dimensional model for locally supported measurements such as the windowed Fourier transform. For signals "without large holes", we show the stability constant exhibits only a mild polynomial growth in the dimension, in stark contrast with the exponential growth which uniform stability constants tend to suffer from; more precisely, in $R^d$ the constant grows proportionally to $d^{1/2}$, while in $C^d$ it grows proportionally to $d$. We also show the growth of the constant in the complex case cannot be reduced, suggesting that complex phase retrieval is substantially more difficult than real phase retrieval. The second model we consider is an infinite-dimensional phase retrieval problem in a principal shift invariant space. We show that despite the infinite dimensionality of this model, signals with monotone exponential decay will have a finite stability constant. In contrast, the stability bound provided by our results will be infinite if the signal's decay is polynomial.

cs.IT↗

arXiv2020

Tensor Ring Decomposition: Optimization Landscape and One-loop Convergence of Alternating Least Squares

In this work, we study the tensor ring decomposition and its associated numerical algorithms. We establish a sharp transition of algorithmic difficulty of the optimization problem as the bond dimension increases: On one hand, we show the existence of spurious local minima for the optimization landscape even when the tensor ring format is much over-parameterized, i.e., with bond dimension much larger than that of the true target tensor. On the other hand, when the bond dimension is further increased, we establish one-loop convergence for alternating least square algorithm for tensor ring decomposition. The theoretical results are complemented by numerical experiments for both local minimum and one-loop convergence for the alternating least square algorithm.

math.NA↗

arXiv2020

A Mean-field Analysis of Deep ResNet and Beyond: Towards Provable Optimization Via Overparameterization From Depth

Training deep neural networks with stochastic gradient descent (SGD) can often achieve zero training loss on real-world tasks although the optimization landscape is known to be highly non-convex. To understand the success of SGD for training deep neural networks, this work presents a mean-field analysis of deep residual networks, based on a line of works that interpret the continuum limit of the deep residual network as an ordinary differential equation when the network capacity tends to infinity. Specifically, we propose a new continuum limit of deep residual networks, which enjoys a good landscape in the sense that every local minimizer is global. This characterization enables us to derive the first global convergence result for multilayer neural networks in the mean-field regime. Furthermore, without assuming the convexity of the loss landscape, our proof relies on a zero-loss assumption at the global minimizer that can be achieved when the model shares a universal approximation property. Key to our result is the observation that a deep residual network resembles a shallow network ensemble, i.e. a two-layer network. We bound the difference between the shallow network and our ResNet model via the adjoint sensitivity method, which enables us to apply existing mean-field analyses of two-layer networks to deep networks. Furthermore, we propose several novel training schemes based on the new continuous model, including one training procedure that switches the order of the residual blocks and results in strong empirical performance on the benchmark datasets.

stat.ML↗

arXiv2020

End-to-end Learning for Inter-Vehicle Distance and Relative Velocity Estimation in ADAS with a Monocular Camera

Inter-vehicle distance and relative velocity estimations are two basic functions for any ADAS (Advanced driver-assistance systems). In this paper, we propose a monocular camera-based inter-vehicle distance and relative velocity estimation method based on end-to-end training of a deep neural network. The key novelty of our method is the integration of multiple visual clues provided by any two time-consecutive monocular frames, which include deep feature clue, scene geometry clue, as well as temporal optical flow clue. We also propose a vehicle-centric sampling mechanism to alleviate the effect of perspective distortion in the motion field (i.e. optical flow). We implement the method by a light-weight deep neural network. Extensive experiments are conducted which confirm the superior performance of our method over other state-of-the-art methods, in terms of estimation accuracy, computational speed, and memory footprint.

cs.CV↗

arXiv2020

Butterfly-Net: Optimal Function Representation Based on Convolutional Neural Networks

Deep networks, especially convolutional neural networks (CNNs), have been successfully applied in various areas of machine learning as well as to challenging problems in other scientific and engineering fields. This paper introduces Butterfly-Net, a low-complexity CNN with structured and sparse cross-channel connections, together with a Butterfly initialization strategy for a family of networks. Theoretical analysis of the approximation power of Butterfly-Net to the Fourier representation of input data shows that the error decays exponentially as the depth increases. Combining Butterfly-Net with a fully connected neural network, a large class of problems are proved to be well approximated with network complexity depending on the effective frequency bandwidth instead of the input dimension. Regular CNN is covered as a special case in our analysis. Numerical experiments validate the analytical results on the approximation of Fourier kernels and energy functionals of Poisson's equations. Moreover, all experiments support that training from Butterfly initialization outperforms training from random initialization. Also, adding the remaining cross-channel connections, although significantly increase the parameter number, does not much improve the post-training accuracy and is more sensitive to data distribution.

math.NA↗

arXiv2020

LightPAFF: A Two-Stage Distillation Framework for Pre-training and Fine-tuning

While pre-training and fine-tuning, e.g., BERT~\citep{devlin2018bert}, GPT-2~\citep{radford2019language}, have achieved great success in language understanding and generation tasks, the pre-trained models are usually too big for online deployment in terms of both memory cost and inference speed, which hinders them from practical online usage. In this paper, we propose LightPAFF, a Lightweight Pre-training And Fine-tuning Framework that leverages two-stage knowledge distillation to transfer knowledge from a big teacher model to a lightweight student model in both pre-training and fine-tuning stages. In this way the lightweight model can achieve similar accuracy as the big teacher model, but with much fewer parameters and thus faster online inference speed. LightPAFF can support different pre-training methods (such as BERT, GPT-2 and MASS~\citep{song2019mass}) and be applied to many downstream tasks. Experiments on three language understanding tasks, three language modeling tasks and three sequence to sequence generation tasks demonstrate that while achieving similar accuracy with the big BERT, GPT-2 and MASS models, LightPAFF reduces the model size by nearly 5x and improves online inference speed by 5x-7x.

cs.CL↗

arXiv2020

Synchronization of Kuramoto Oscillators in Dense Networks

We study synchronization properties of systems of Kuramoto oscillators. The problem can also be understood as a question about the properties of an energy landscape created by a graph. More formally, let $G=(V,E)$ be a connected graph and $(a_{ij})_{i,j=1}^{n}$ denotes its adjacency matrix. Let the function $f:\mathbb{T}^n \rightarrow \mathbb{R}$ be given by $$ f(θ_1, \dots, θ_n) = \sum_{i,j=1}^{n}{ a_{ij} \cos{(θ_i - θ_j)}}.$$ This function has a global maximum when $θ_i = θ$ for all $1\leq i \leq n$. It is known that if every vertex is connected to at least $μ(n-1)$ other vertices for $μ$ sufficiently large, then every local maximum is global. Taylor proved this for $μ\geq 0.9395$ and Ling, Xu \& Bandeira improved this to $μ\geq 0.7929$. We give a slight improvement to $μ\geq 0.7889$. Townsend, Stillman \& Strogatz suggested that the critical value might be $μ_c = 0.75$.

math.OC↗

arXiv2020

Bloch dynamics with second order Berry phase correction

We derive the semiclassical Bloch dynamics with the second-order Berry phase correction in the presence of the slow-varying scalar potential as perturbation. Our mathematical derivation is based on a two-scale WKB asymptotic analysis. For a uniform external electric field, the bi-characteristics system after a positional shift introduced by Berry connections agrees with the recent result in previous works. Moreover, for the case with a linear external electric field, we show that the extra terms arising in the bi-characteristics system after the positional shift are also gauge independent.

math.AP↗

arXiv2020

Convergence of Stochastic-extended Lagrangian molecular dynamics method for polarizable force field simulation

Extended Lagrangian molecular dynamics (XLMD) is a general method for performing molecular dynamics simulations using quantum and classical many-body potentials. Recently several new XLMD schemes have been proposed and tested on several classes of many-body polarization models such as induced dipoles or Drude charges, by creating an auxiliary set of these same degrees of freedom that are reversibly integrated through time. This gives rise to a singularly perturbed Hamiltonian system that provides a good approximation to the time evolution of the real mutual polarization field. To further improve upon the accuracy of the XLMD dynamics, and to potentially extend it to other many-body potentials, we introduce a stochastic modification which leads to a set of singularly perturbed Langevin equations with degenerate noise. We prove that the resulting Stochastic-XLMD converges to the accurate dynamics, and the convergence rate is both optimal and is independent of the accuracy of the initial polarization field. We carefully study the scaling of the damping factor and numerical noise for efficient numerical simulation for Stochastic-XLMD, and we demonstrate the effectiveness of the method for model polarizable force field systems.

math.NA↗

arXiv2020

Continuum limit and preconditioned Langevin sampling of the path integral molecular dynamics

We investigate the continuum limit that the number of beads goes to infinity in the ring polymer representation of thermal averages. Studying the continuum limit of the trajectory sampling equation sheds light on possible preconditioning techniques for sampling ring polymer configurations with large number of beads. We propose two preconditioned Langevin sampling dynamics, which are shown to have improved stability and sampling accuracy. We present a careful mode analysis of the preconditioned dynamics and show their connections to the normal mode, the staging coordinate and the Matsubara mode representation for ring polymers. In the case where the potential is quadratic, we show that the continuum limit of the preconditioned mass modified Langevin dynamics converges to its equilibrium exponentially fast, which suggests that the finite-dimensional counterpart has a dimension-independent convergence rate. In addition, the preconditioning techniques can be naturally applied to the multi-level quantum systems in the nonadiabatic regime, which are compatible with various numerical approaches.

physics.chem-ph↗