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Dong Li

Publications and source records attributed to Dong Li.

At least 433 records · Page 24Linked to original sources
arXiv2021

On a sinc-type MBE model

We introduce a new sinc-type molecular beam epitaxy model which is derived from a cosine-type energy functional. The landscape of the new functional is remarkably similar to the classical MBE model with double well potential but has the additional advantage that all its derivatives are uniformly bounded. We consider first order IMEX and second order BDF2 discretization schemes. For both cases we quantify explicit time step constraints for the energy dissipation which is in good accord with the practical numerical simulations. Furthermore we introduce a new theoretical framework and prove unconditional uniform energy boundedness with no size restrictions on the time step. This is the first unconditional (i.e. independent of the time step size) result for semi-implicit methods applied to the phase field models without introducing any artificial stabilization terms or fictitious variables.

math.NA↗
arXiv2021

On a Class of Nonlocal SIR Models

We revisit the classic Susceptible-Infected-Recovered (SIR) epidemic model and one of its nonlocal variations recently developed in \cite{Guan}. We introduce several new approaches to derive exact analytical solutions in the classical situation and analyze the corresponding effective approximations in the nonlocal setting. An interesting new feature of the nonlocal models, compared with the classic SIR model, is the appearance of multiple peak solutions for the infected population. We provide several rigorous results on the existence and non-existence of peak solutions with sharp asymptotics.

math.AP↗
arXiv2021

A regularity upgrade of pressure

For the incompressible Euler equations the pressure formally scales as a quadratic function of velocity. We provide several optimal regularity estimates on the pressure by using regularity of velocity in various Sobolev, Besov and Hardy spaces. Our proof exploits the incompressibility condition in an essential way and is deeply connected with the classic Div-Curl lemma which we also generalise as a fractional Leibniz rule in Hardy spaces. To showcase the sharpness of results, we construct a class of counterexamples at several end-points.

math.AP↗
arXiv2021

Energy-dissipation for time-fractional phase-field equations

We consider a class of time-fractional phase field models including the Allen-Cahn and Cahn-Hilliard equations. We establish several weighted positivity results for functionals driven by the Caputo time-fractional derivative. Several novel criterions are examined for showing the positive-definiteness of the associated kernel functions. We deduce strict energy-dissipation for a number of non-local energy functionals, thereby proving fractional energy dissipation laws.

math.AP↗
arXiv2021

On Sampling Top-K Recommendation Evaluation

Recently, Rendle has warned that the use of sampling-based top-$k$ metrics might not suffice. This throws a number of recent studies on deep learning-based recommendation algorithms, and classic non-deep-learning algorithms using such a metric, into jeopardy. In this work, we thoroughly investigate the relationship between the sampling and global top-$K$ Hit-Ratio (HR, or Recall), originally proposed by Koren[2] and extensively used by others. By formulating the problem of aligning sampling top-$k$ ($SHR@k$) and global top-$K$ ($HR@K$) Hit-Ratios through a mapping function $f$, so that $SHR@k\approx HR@f(k)$, we demonstrate both theoretically and experimentally that the sampling top-$k$ Hit-Ratio provides an accurate approximation of its global (exact) counterpart, and can consistently predict the correct winners (the same as indicate by their corresponding global Hit-Ratios).

cs.IR↗
arXiv2021

Stability of the semi-implicit method for the Cahn-Hilliard equation with logarithmic potentials

We consider the two-dimensional Cahn-Hilliard equation with logarithmic potentials and periodic boundary conditions. We employ the standard semi-implicit numerical scheme which treats the linear fourth-order dissipation term implicitly and the nonlinear term explicitly. Under natural constraints on the time step we prove strict phase separation and energy stability of the semi-implicit scheme. This appears to be the first rigorous result for the semi-implicit discretization of the Cahn-Hilliard equation with singular potentials.

math.NA↗
arXiv2021

Sharp convergence to steady states of Allen-Cahn

In our recent work we found a surprising breakdown of symmetry conservation: using standard numerical discretization with very high precision the computed numerical solutions corresponding to very nice initial data may converge to completely incorrect steady states due to the gradual accumulation of machine round-off error. We solved this issue by introducing a new Fourier filter technique for solutions with certain band gap properties. To further investigate the attracting basin of steady states we classify in this work all possible bounded nontrivial steady states for the Allen-Cahn equation. We characterize sharp dependence of nontrivial steady states on the diffusion coefficient and prove strict monotonicity of the associated energy. In particular, we establish a certain self-replicating property amongst the hierarchy of steady states and give a full classification of their energies and profiles. We develop a new modulation theory and prove sharp convergence to the steady state with explicit rates and profiles.

math.AP↗
arXiv2021

Towards a Better Understanding of Linear Models for Recommendation

Recently, linear regression models, such as EASE and SLIM, have shown to often produce rather competitive results against more sophisticated deep learning models. On the other side, the (weighted) matrix factorization approaches have been popular choices for recommendation in the past and widely adopted in the industry. In this work, we aim to theoretically understand the relationship between these two approaches, which are the cornerstones of model-based recommendations. Through the derivation and analysis of the closed-form solutions for two basic regression and matrix factorization approaches, we found these two approaches are indeed inherently related but also diverge in how they "scale-down" the singular values of the original user-item interaction matrix. This analysis also helps resolve the questions related to the regularization parameter range and model complexities. We further introduce a new learning algorithm in searching (hyper)parameters for the closed-form solution and utilize it to discover the nearby models of the existing solutions. The experimental results demonstrate that the basic models and their closed-form solutions are indeed quite competitive against the state-of-the-art models, thus, confirming the validity of studying the basic models. The effectiveness of exploring the nearby models are also experimentally validated.

cs.IR↗
arXiv2021

InfoBehavior: Self-supervised Representation Learning for Ultra-long Behavior Sequence via Hierarchical Grouping

E-commerce companies have to face abnormal sellers who sell potentially-risky products. Typically, the risk can be identified by jointly considering product content (e.g., title and image) and seller behavior. This work focuses on behavior feature extraction as behavior sequences can provide valuable clues for the risk discovery by reflecting the sellers' operation habits. Traditional feature extraction techniques heavily depend on domain experts and adapt poorly to new tasks. In this paper, we propose a self-supervised method InfoBehavior to automatically extract meaningful representations from ultra-long raw behavior sequences instead of the costly feature selection procedure. InfoBehavior utilizes Bidirectional Transformer as feature encoder due to its excellent capability in modeling long-term dependency. However, it is intractable for commodity GPUs because the time and memory required by Transformer grow quadratically with the increase of sequence length. Thus, we propose a hierarchical grouping strategy to aggregate ultra-long raw behavior sequences to length-processable high-level embedding sequences. Moreover, we introduce two types of pretext tasks. Sequence-related pretext task defines a contrastive-based training objective to correctly select the masked-out coarse-grained/fine-grained behavior sequences against other "distractor" behavior sequences; Domain-related pretext task designs a classification training objective to correctly predict the domain-specific statistical results of anomalous behavior. We show that behavior representations from the pre-trained InfoBehavior can be directly used or integrated with features from other side information to support a wide range of downstream tasks. Experimental results demonstrate that InfoBehavior significantly improves the performance of Product Risk Management and Intellectual Property Protection.

cs.CL↗
arXiv2021

On fractional smoothness of modulus of functions

We consider the Nemytskii operators $u\to |u|$ and $u\to u^{\pm}$ in a bounded domain $Ω$ with $C^2$ boundary. We give elementary proofs of the boundedness in $H^s(Ω)$ with $0\le s<3/2$.

math.AP↗
arXiv2021

On a parabolic sine-Gordon model

We consider a parabolic sine-Gordon model with periodic boundary conditions. We prove a fundamental maximum principle which gives a priori uniform control of the solution. In the one-dimensional case we classify all bounded steady states and exhibit some explicit solutions. For the numerical discretization we employ first order IMEX, and second order BDF2 discretization without any additional stabilization term. We rigorously prove the energy stability of the numerical schemes under nearly sharp and quite mild time step constraints. We demonstrate the striking similarity of the parabolic sine-Gordon model with the standard Allen-Cahn equations with double well potentials.

math.AP↗
arXiv2021

On symmetry breaking of Allen-Cahn

We consider numerical solutions for the Allen-Cahn equation with standard double well potential and periodic boundary conditions. Surprisingly it is found that using standard numerical discretizations with high precision computational solutions may converge to completely incorrect steady states. This happens for very smooth initial data and state-of-the-art algorithms. We analyze this phenomenon and showcase the resolution of this problem by a new symmetry-preserving filter technique. We develop a new theoretical framework and rigorously prove the convergence to steady states for the filtered solutions.

math.AP↗
arXiv2021

Cooperative Multi-Agent Transfer Learning with Level-Adaptive Credit Assignment

Extending transfer learning to cooperative multi-agent reinforcement learning (MARL) has recently received much attention. In contrast to the single-agent setting, the coordination indispensable in cooperative MARL constrains each agent's policy. However, existing transfer methods focus exclusively on agent policy and ignores coordination knowledge. We propose a new architecture that realizes robust coordination knowledge transfer through appropriate decomposition of the overall coordination into several coordination patterns. We use a novel mixing network named level-adaptive QTransformer (LA-QTransformer) to realize agent coordination that considers credit assignment, with appropriate coordination patterns for different agents realized by a novel level-adaptive Transformer (LA-Transformer) dedicated to the transfer of coordination knowledge. In addition, we use a novel agent network named Population Invariant agent with Transformer (PIT) to realize the coordination transfer in more varieties of scenarios. Extensive experiments in StarCraft II micro-management show that LA-QTransformer together with PIT achieves superior performance compared with state-of-the-art baselines.

cs.AI↗
arXiv2021

Uniform boundedness of highest norm for 2D quasilinear wave

We consider the two-dimensional quasilinear wave equations with quadratic nonlinearities. We introduce a new class of null forms and prove uniform boundedness of the highest order norm of the solution for all time. This class of null forms include several prototypical strong null conditions as special cases. To handle the critical decay near the light cone we inflate the nonlinearity through a new normal form type transformation which is based on a deep cancelation between the tangential and normal derivatives with respect to the light cone. Our proof does not employ the Lorentz boost and can be generalized to systems with multiple speeds.

math.AP↗
arXiv2021

Anisotropic superconducting properties of Kagome metal CsV3Sb5

We systematically measure the superconducting (SC) and mixed state properties of high-quality CsV3Sb5 single crystals with Tc ~ 3.5 K. We find that the upper critical field Hc2(T) exhibits a large anisotropic ratio of Hc2^(ab)/Hc2^(c) ~ 9 at zero temperature and fitting its temperature dependence requires a minimum two-band effective model. Moreover, the ratio of the lower critical field, Hc1^(ab)/Hc1^(c), is also found to be larger than 1, which indicates that the in-plane energy dispersion is strongly renormalized near Fermi energy. Both Hc1(T) and SC diamagnetic signal are found to change little initially below Tc ~ 3.5 K and then to increase abruptly upon cooling to a characteristic temperature of ~2.8 K. Furthermore, we identify a two-fold anisotropy of in-plane angular-dependent magnetoresistance in the mixed state. Interestingly, we find that, below the same characteristic T ~ 2.8 K, the orientation of this two-fold anisotropy displays a peculiar twist by an angle of 60o characteristic of the Kagome geometry. Our results suggest an intriguing superconducting state emerging in the complex environment of Kagome lattice, which, at least, is partially driven by electron-electron correlation.

cond-mat.supr-con↗