A Unified Approach to Holomorphic Anomaly Equations and Quantum Spectral Curves

We present a unified approach to holomorphic anomaly equations and some well-known quantum spectral curves. We develop a formalism of abstract quantum field theory based on the diagrammatics of the Deligne-Mumford moduli spaces $\overline{{\mathcal M}}_{g,n}$ and derive a quadratic recursion relation for the abstract free energies in terms of the edge-cutting operators. This abstract quantum field theory can be realized by various choices of a sequence of holomorphic functions or formal power series and suitable propagators, and the realized quantum field theory can be represented by formal Gaussian integrals. Various applications are given.Comment: A section is adde

Efficient Algorithms for Sparse Moment Problems without Separation

We consider the sparse moment problem of learning a $k$-spike mixture in high-dimensional space from its noisy moment information in any dimension. We measure the accuracy of the learned mixtures using transportation distance. Previous algorithms either assume certain separation assumptions, use more recovery moments, or run in (super) exponential time. Our algorithm for the one-dimensional problem (also called the sparse Hausdorff moment problem) is a robust version of the classic Prony's method, and our contribution mainly lies in the analysis. We adopt a global and much tighter analysis than previous work (which analyzes the perturbation of the intermediate results of Prony's method). A useful technical ingredient is a connection between the linear system defined by the Vandermonde matrix and the Schur polynomial, which allows us to provide tight perturbation bound independent of the separation and may be useful in other contexts. To tackle the high-dimensional problem, we first solve the two-dimensional problem by extending the one-dimensional algorithm and analysis to complex numbers. Our algorithm for the high-dimensional case determines the coordinates of each spike by aligning a 1d projection of the mixture to a random vector and a set of 2d projections of the mixture. Our results have applications to learning topic models and Gaussian mixtures, implying improved sample complexity results or running time over prior work

A mixed precision Jacobi method for the symmetric eigenvalue problem

The eigenvalue problem is a fundamental problem in scientific computing. In this paper, we propose a mixed precision Jacobi method for the symmetric eigenvalue problem. We first compute the eigenvalue decomposition of a real symmetric matrix by an eigensolver at low precision and we obtain a low-precision matrix of eigenvectors. Then by using the modified Gram-Schmidt orthogonalization process to the low-precision eigenvector matrix in high precision, a high-precision orthogonal matrix is obtained, which is used as an initial guess for the Jacobi method. We give the rounding error analysis for the proposed method and the quadratic convergence of the proposed method is established under some sufficient conditions. We also present a mixed precision one-side Jacobi method for the singular value problem and the corresponding rounding error analysis and quadratic convergence are discussed. Numerical experiments on CPUs and GPUs are conducted to illustrate the efficiency of the proposed mixed precision Jacobi method over the original Jacobi method.Comment: 31 pages, 2 figure

Mantle upwelling beneath the South China Sea and links to surrounding subduction systems

© The Author(s), 2019. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in Lin, J., Xu, Y., Sun, Z., & Zhou, Z. Mantle upwelling beneath the South China Sea and links to surrounding subduction systems. National Science Review, 6(5), (2019): 877-881, doi:10.1093/nsr/nwz123.The evolution of the South China Sea (SCS) is directly linked to the complex subduction systems of the surrounding Pacific, Philippine Sea and Indo-Australian Plates (Fig. 1a). Major advances in the last several years are providing new insights into the SCS-mantle dynamics, through regional seismic imaging of the upper mantle [1,2], unprecedented IODP drilling expeditions (349/367/368/368X) [3–5] that obtained the oceanic basement basalt samples for the first time, geochemical analyses of the SCS-mantle source compositions [6–8] and geodynamic modeling [9,10]. Furthermore, new geological mapping, seismic imaging [11,12] and IODP drilling [13,14] have revealed evidence for significantly greater magma production at the northern SCS rifted margin, in comparison to the magma-poor end-member of the Atlantic rifted margins. This paper provides a new perspective of the SCS-mantle dynamics inspired by new observations and geodynamic modeling. We first highlight new geophysical evidence for a broad region of low-seismic-velocity anomalies in the upper mantle beneath the northern SCS, abundant magmatism during continental breakup and post-seafloor spreading, and geochemical evidence for recycled oceanic components beneath the SCS. We then present new models of layered flows in the mantle beneath the SCS, revealing two modes of plate- and subduction-driven mantle upwelling, including (i) narrow centers of mantle upwelling at shallow depths induced by divergent plate motion at seafloor-spreading centers and (ii) broad zones of mantle upwelling as a result of subduction-induced mantle-return flows at greater depths. These new observations and geodynamic studies suggest strong links between mantle upwelling beneath the SCS and surrounding subducting plates.This work was supported by the National Natural Science Foundation of China (41890813, 91628301, U1606401, 41976066, 91858207 and 41706056), the Chinese Academy of Sciences (Y4SL021001, QYZDY-SSW-DQC005 and 133244KYSB20180029), the Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou, GML2019ZD0205), the National Key R&D Program of China (2018YFC0309800 and 2018YFC0310100), the State Oceanic Administration (GASI-GEOGE-02) and China Ocean Mineral Resources R&D Association (DY135-S2–1-04)

A Hierarchical Framework of Cloud Resource Allocation and Power Management Using Deep Reinforcement Learning

Automatic decision-making approaches, such as reinforcement learning (RL), have been applied to (partially) solve the resource allocation problem adaptively in the cloud computing system. However, a complete cloud resource allocation framework exhibits high dimensions in state and action spaces, which prohibit the usefulness of traditional RL techniques. In addition, high power consumption has become one of the critical concerns in design and control of cloud computing systems, which degrades system reliability and increases cooling cost. An effective dynamic power management (DPM) policy should minimize power consumption while maintaining performance degradation within an acceptable level. Thus, a joint virtual machine (VM) resource allocation and power management framework is critical to the overall cloud computing system. Moreover, novel solution framework is necessary to address the even higher dimensions in state and action spaces. In this paper, we propose a novel hierarchical framework for solving the overall resource allocation and power management problem in cloud computing systems. The proposed hierarchical framework comprises a global tier for VM resource allocation to the servers and a local tier for distributed power management of local servers. The emerging deep reinforcement learning (DRL) technique, which can deal with complicated control problems with large state space, is adopted to solve the global tier problem. Furthermore, an autoencoder and a novel weight sharing structure are adopted to handle the high-dimensional state space and accelerate the convergence speed. On the other hand, the local tier of distributed server power managements comprises an LSTM based workload predictor and a model-free RL based power manager, operating in a distributed manner.Comment: accepted by 37th IEEE International Conference on Distributed Computing (ICDCS 2017