Sobolev inequalities and regularity of the linearized complex Monge-Ampere and Hessian equations

Let $u$ be a smooth, strictly $k$-plurisubharmonic function on a bounded domain $\Omega\in\mathbb C^n$ with $2\leq k\leq n$. The purpose of this paper is to study the regularity of solution to the linearized complex Monge-Amp\`ere and Hessian equations when the complex $k$-Hessian $H_k[u]$ of $u$ is bounded from above and below. We first establish some estimates of Green's functions associated to the linearized equations. Then we prove a class of new Sobolev inequalities. With these inequalities, we use Moser's iteration to investigate the a priori estimates of Hessian equations and their linearized equations, as well as the K\"ahler scalar curvature equation. In particular, we obtain the Harnack inequality for the linearized complex Monge-Amp\`ere and Hessian equations under an extra integrability condition on the coefficients. The approach works in both real and complex case.Comment: 34 pages. Comments welcome

DreamGaussian: Generative Gaussian Splatting for Efficient 3D Content Creation

Recent advances in 3D content creation mostly leverage optimization-based 3D generation via score distillation sampling (SDS). Though promising results have been exhibited, these methods often suffer from slow per-sample optimization, limiting their practical usage. In this paper, we propose DreamGaussian, a novel 3D content generation framework that achieves both efficiency and quality simultaneously. Our key insight is to design a generative 3D Gaussian Splatting model with companioned mesh extraction and texture refinement in UV space. In contrast to the occupancy pruning used in Neural Radiance Fields, we demonstrate that the progressive densification of 3D Gaussians converges significantly faster for 3D generative tasks. To further enhance the texture quality and facilitate downstream applications, we introduce an efficient algorithm to convert 3D Gaussians into textured meshes and apply a fine-tuning stage to refine the details. Extensive experiments demonstrate the superior efficiency and competitive generation quality of our proposed approach. Notably, DreamGaussian produces high-quality textured meshes in just 2 minutes from a single-view image, achieving approximately 10 times acceleration compared to existing methods.Comment: project page: https://dreamgaussian.github.io

Enhancing Large Language Models with Pseudo- and Multisource- Knowledge Graphs for Open-ended Question Answering

Mitigating the hallucinations of Large Language Models (LLMs) and enhancing them is a crucial task. Although some existing methods employ model self-enhancement techniques, they fall short of effectively addressing unknown factual hallucinations. Using Knowledge Graph (KG) enhancement approaches fails to address the generalization across different KG sources and the enhancement of open-ended answer questions simultaneously. To tackle these limitations, there is a framework that combines Pseudo-Graph Generation and Atomic Knowledge Verification proposed. The enhancement of LLM using KG in an open-ended question-answering setting is implemented by leveraging the Pseudo-Graph Generation. Atomic Knowledge Verification utilizes atomic-level knowledge querying and verification to achieve generalizability under different KG sources. Compared to the baseline, this approach yields a minimum improvement of 11.5 in the ROUGE-L score for open-ended questions. For precise questions, we observe a minimum accuracy improvement of 7.5. Moreover, there is also demonstration that this framework exhibits generalizability across different KG sources. In summary, our results pave the way for enhancing LLMs by incorporating Pseudo- and Multisource-KGs, particularly in the context of open-ended questions

Probing phase transition in neutron stars via the crust-core interfacial mode

Gravitational waves emitted from the binary neutron star (BNS) systems can carry information about the dense matter phase in these compact stars. The crust-core interfacial mode is an oscillation mode in a neutron star and it depends mostly on the equation of the state of the matter in the crust-core transition region. This mode can be resonantly excited by the tidal field of an inspiraling-in BNS system, thereby affecting the emitted gravitational waves, and hence could be used to probe the equation of state in the crust-core transition region. In this work, we investigate in detail how the first-order phase transition inside the neutron star affects the properties of the crust-core interfacial mode, using a Newtonian fluid perturbation theory on a general relativistic background solution of the stellar structure. Two possible types of phase transitions are considered: (1) the phase transitions happen in the fluid core but near the crust-core interface, which results in density discontinuities; and (2) the strong interaction phase transitions in the dense core (as in the conventional hybrid star case). These phase transitions' impacts on interfacial mode properties are discussed. In particular, the former phase transition has a minor effect on the M-R relation and the adiabatic tidal deformability, but can significantly affect the interfacial mode frequency and thereby could be probed using gravitational waves. For the BNS systems, we discuss the possible observational signatures of these phase transitions in the gravitational waveforms and their detectability. Our work enriches the exploration of the physical properties of the crust-core interfacial mode and provides a promising method for probing the phase transition using the seismology of a compact star.Comment: 18 pages, 14 figure

Accelerated Federated Learning with Decoupled Adaptive Optimization

The federated learning (FL) framework enables edge clients to collaboratively learn a shared inference model while keeping privacy of training data on clients. Recently, many heuristics efforts have been made to generalize centralized adaptive optimization methods, such as SGDM, Adam, AdaGrad, etc., to federated settings for improving convergence and accuracy. However, there is still a paucity of theoretical principles on where to and how to design and utilize adaptive optimization methods in federated settings. This work aims to develop novel adaptive optimization methods for FL from the perspective of dynamics of ordinary differential equations (ODEs). First, an analytic framework is established to build a connection between federated optimization methods and decompositions of ODEs of corresponding centralized optimizers. Second, based on this analytic framework, a momentum decoupling adaptive optimization method, FedDA, is developed to fully utilize the global momentum on each local iteration and accelerate the training convergence. Last but not least, full batch gradients are utilized to mimic centralized optimization in the end of the training process to ensure the convergence and overcome the possible inconsistency caused by adaptive optimization methods

Federated Learning of Large Language Models with Parameter-Efficient Prompt Tuning and Adaptive Optimization

Federated learning (FL) is a promising paradigm to enable collaborative model training with decentralized data. However, the training process of Large Language Models (LLMs) generally incurs the update of significant parameters, which limits the applicability of FL techniques to tackle the LLMs in real scenarios. Prompt tuning can significantly reduce the number of parameters to update, but it either incurs performance degradation or low training efficiency. The straightforward utilization of prompt tuning in the FL often raises non-trivial communication costs and dramatically degrades performance. In addition, the decentralized data is generally non-Independent and Identically Distributed (non-IID), which brings client drift problems and thus poor performance. This paper proposes a Parameter-efficient prompt Tuning approach with Adaptive Optimization, i.e., FedPepTAO, to enable efficient and effective FL of LLMs. First, an efficient partial prompt tuning approach is proposed to improve performance and efficiency simultaneously. Second, a novel adaptive optimization method is developed to address the client drift problems on both the device and server sides to enhance performance further. Extensive experiments based on 10 datasets demonstrate the superb performance (up to 60.8\% in terms of accuracy) and efficiency (up to 97.59\% in terms of training time) of FedPepTAO compared with 9 baseline approaches. Our code is available at https://github.com/llm-eff/FedPepTAO.Comment: 18 pages, accepted by EMNLP 202