Quantum algorithm for neighborhood preserving embedding

Neighborhood preserving embedding (NPE) is an important linear dimensionality reduction technique that aims at preserving the local manifold structure. NPE contains three steps, i.e., finding the nearest neighbors of each data point, constructing the weight matrix, and obtaining the transformation matrix. Liang et al. proposed a variational quantum algorithm (VQA) for NPE [Phys. Rev. A 101 032323 (2020)]. The algorithm consists of three quantum sub-algorithms, corresponding to the three steps of NPE, and was expected to have an exponential speedup on the dimensionality n. However, the algorithm has two disadvantages: (i) It is not known how to efficiently obtain the input of the third sub-algorithm from the output of the second one. (ii) Its complexity cannot be rigorously analyzed because the third sub-algorithm in it is a VQA. In this paper, we propose a complete quantum algorithm for NPE, in which we redesign the three sub-algorithms and give a rigorous complexity analysis. It is shown that our algorithm can achieve a polynomial speedup on the number of data points m and an exponential speedup on the dimensionality n under certain conditions over the classical NPE algorithm, and achieve a significant speedup compared to Liang et al.'s algorithm even without considering the complexity of the VQA.     

硅和锗纳米线的原子排布和电荷分布的密度泛函紧束缚方法研究

低维硅锗材料是制备纳米电子器件的重要候选材料，是研发高效率、低能耗和超高速新一代纳米电子器件的基础材料之一，有着潜在的应用价值。采用密度泛函紧束缚方法分别对厚度相同、宽度在0.272 nm~0.554 nm之间的硅纳米线和宽度在0.283 nm~0.567 nm之间的锗纳米线的原子排布和电荷分布进行了计算研究。硅、锗纳米线宽度的改变使原子排布，纳米线的原子间键长和键角发生明显改变。纳米线表层结构的改变对各层内的电荷分布产生重要影响。纳米线中各原子的电荷转移量与该原子在表层内的位置相关。纳米线的尺寸和表层内原子排列结构对体系的稳定性产生重要影响。     

Semiconvergence analysis of the randomized row iterative method and its extended variants

The row iterative method is popular in solving the large‐scale ill‐posed problems due to its simplicity and efficiency. In this work we consider the randomized row iterative (RRI) method to tackle this issue. First, we present the semiconvergence analysis of RRI method for the overdetermined and inconsistent system, and derive upper bounds for the noise error propagation in the iteration vectors. To achieve a least squares solution, we then propose an extended version of the RRI (ERRI) method, which in fact can converge in expectation to the solution of the overdetermined or underdetermined, consistent or inconsistent systems. Finally, some numerical examples are given to demonstrate the convergence behaviors of the RRI and ERRI methods for these types of linear system.     

全局优化问题的一个新的无参数填充函数

求全局最优化问题的填充函数算法被提出以来，参数的选取和调整一直是制约算法有效性的因素。如何在实际的计算过程中选取合适的参数，直接影响和决定了运算速度和效率。因此，构造不含参数的填充函数就显得极为重要。提出一个新的无参数的填充函数，对其理论性质进行了分析，并给出相应的填充函数算法，数值计算验证了算法的有效性。     

Performance guarantees of forward and reverse greedy algorithms for minimizing nonsupermodular nonsubmodular functions on a matroid

This letter studies the problem of minimizing increasing set functions, equivalently, maximizing decreasing set functions, over the base matroid. This setting has received great interest, since it generalizes several applied problems including actuator and sensor placement problems in control, task allocation problems, video summarization, and many others. We study two greedy heuristics, namely, the forward and reverse greedy. We provide two novel performance guarantees for the approximate solutions obtained by them depending on both submodularity ratio and curvature.     

求解矩阵方程AXB+CXD=F参数迭代法的最优参数分析

本文针对求矩阵方程AXB+CXD=F唯一解的参数迭代法，分析当矩阵A，B，C，D均是Hermite正（负）定矩阵时，迭代矩阵的特征值表达式，给出了最优参数的确定方法，并提出了相应的加速算法.     

Linearly χ-bounding (P6, C4)-free graphs*

Given two graphs and , a graph is -free if it contains no induced subgraph isomorphic to or . Let and be the path on vertices and the cycle on vertices, respectively. In this paper we show that for any -free graph it holds that , where and are the chromatic number and clique number of , respectively. Our bound is attained by several graphs, for instance, the 5-cycle, the Petersen graph, the Petersen graph with an additional universal vertex, and all -critical -free graphs other than (see Hell and Huang [Discrete Appl. Math. 216 (2017), pp. 211–232]). The new result unifies previously known results on the existence of linear -binding functions for several graph classes. Our proof is based on a novel structure theorem on -free graphs that do not contain clique cutsets. Using this structure theorem we also design a polynomial time -approximation algorithm for coloring -free graphs. Our algorithm computes a coloring with colors for any -free graph in time.     

零误差计算

研究采用有误差的数值计算来获得无误差的准确值具有重要的理论价值和应用价值.这种通过近似的数值方法获得准确结果的计算被称为零误差计算.本文首先指出,只有一致离散集合中的数才能够开展零误差计算,即有非零隔离界的数集,这也是"数"可以进行零误差计算的一个充要条件.以此为基本出发点,本文分析代数数零误差计算的最低理论精度,该精度对应于恢复近似代数数的准确值时必要的误差控制条件,但由于所采用恢复算法的局限性,这一理论精度往往不能保证成功恢复出代数数的准确值.为此,本文给出采用PSLQ (partial-sum-LQ-decomposition)算法进行代数数零误差计算所需的精度控制条件,与基于LLL (Lenstra-Lenstra-Lovász)算法相比,该精度控制条件关于代数数次数的依赖程度由二次降为拟线性,从而可降低相应算法的复杂度.最后探讨零误差计算未来的发展趋势.     

硫化丁苯橡胶纳观界面耐磨机理及参数影响分析

硫化橡胶因其良好的力学和物理化学性能而被广泛作为摩擦副的基础材料. 本文提出了一种硫化交联算法, 实现了C—C键的硫化互交联和自交联, 构建了硫化丁苯橡胶的分子动力学磨损模型, 从微观摩擦学的角度阐明了硫化交联结构对改善丁苯橡胶磨损性能的机理, 研究了不同界面参数对硫化橡胶微观磨损性能的影响. 结果发现 硫化使丁苯橡胶分子链的界面黏附能力和活动能力更弱, 拉伸和解缠能力更低, 磨损过程中界面累积能量更低, 更不容易脱离橡胶基体, 因此可以表现出更好的摩擦学性能, 更强的抗磨损性能; 随着速度的增大, 硫化橡胶的磨损率降低, 与宏观实验结果一致, 原因是硫化橡胶的原子分布函数和相互作用能随着速度增大而降低, 说明橡胶分子链的黏附能力和活动能力随着速度增加趋弱, 温升更低, 导致较低的磨损率; 压入深度对磨损率的影响规律则呈现相反的结果和趋势.