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Fixed Boundary Flows [article]

Zhigang Yao, Yuqing Xia, Zengyan Fan
2019 arXiv   pre-print
We consider the fixed boundary flow with canonical interpretability as principal components extended on the non-linear Riemannian manifolds. We aim to find a flow with fixed starting and ending point for multivariate datasets lying on an embedded non-linear Riemannian manifold, differing from the principal flow that starts from the center of the data cloud. Both points are given in advance, using the intrinsic metric on the manifolds. From the perspective of geometry, the fixed boundary flow is
more » ... defined as an optimal curve that moves in the data cloud. At any point on the flow, it maximizes the inner product of the vector field, which is calculated locally, and the tangent vector of the flow. We call the new flow the fixed boundary flow. The rigorous definition is given by means of an Euler-Lagrange problem, and its solution is reduced to that of a Differential Algebraic Equation (DAE). A high level algorithm is created to numerically compute the fixed boundary. We show that the fixed boundary flow yields a concatenate of three segments, one of which coincides with the usual principal flow when the manifold is reduced to the Euclidean space. We illustrate how the fixed boundary flow can be used and interpreted, and its application in real data.
arXiv:1904.11332v1 fatcat:om2jcadymja65bho3qybyg6cle

Manifold Fitting under Unbounded Noise [article]

Zhigang Yao, Yuqing Xia
2019 arXiv   pre-print
There has been an emerging trend in non-Euclidean dimension reduction of aiming to recover a low dimensional structure, namely a manifold, underlying the high dimensional data. Recovering the manifold requires the noise to be of certain concentration. Existing methods address this problem by constructing an output manifold based on the tangent space estimation at each sample point. Although theoretical convergence for these methods is guaranteed, either the samples are noiseless or the noise is
more » ... bounded. However, if the noise is unbounded, which is a common scenario, the tangent space estimation of the noisy samples will be blurred, thereby breaking the manifold fitting. In this paper, we introduce a new manifold-fitting method, by which the output manifold is constructed by directly estimating the tangent spaces at the projected points on the underlying manifold, rather than at the sample points, to decrease the error caused by the noise. Our new method provides theoretical convergence, in terms of the upper bound on the Hausdorff distance between the output and underlying manifold and the lower bound on the reach of the output manifold, when the noise is unbounded. Numerical simulations are provided to validate our theoretical findings and demonstrate the advantages of our method over other relevant methods. Finally, our method is applied to real data examples.
arXiv:1909.10228v1 fatcat:qk7swndtrjcvrfcbof7ed35364

Principal Boundary on Riemannian Manifolds

Zhigang Yao, Zhenyue Zhang
2019 Figshare  
We consider the classification problem and focus on nonlinear methods for classification on manifolds. For multivariate datasets lying on an embedded nonlinear Riemannian manifold within the higher-dimensional ambient space, we aim to acquire a classification boundary for the classes with labels, using the intrinsic metric on the manifolds. Motivated by finding an optimal boundary between the two classes, we invent a novel approach – the principal boundary. From the perspective of
more » ... , the principal boundary is defined as an optimal curve that moves in between the principal flows traced out from two classes of data, and at any point on the boundary, it maximizes the margin between the two classes. We estimate the boundary in quality with its direction, supervised by the two principal flows. We show that the principal boundary yields the usual decision boundary found by the support vector machine in the sense that locally, the two boundaries coincide. Some optimality and convergence properties of the random principal boundary and its population counterpart are also shown. We illustrate how to find, use and interpret the principal boundary with an application in real data.
doi:10.6084/m9.figshare.8038301.v1 fatcat:vh5rnhpfonfmlf266whkqywutm

Principal Boundary on Riemannian Manifolds

Zhigang Yao, Zhenyue Zhang
2019 Figshare  
We consider the classification problem and focus on nonlinear methods for classification on manifolds. For multivariate datasets lying on an embedded nonlinear Riemannian manifold within the higher-dimensional ambient space, we aim to acquire a classification boundary for the classes with labels, using the intrinsic metric on the manifolds. Motivated by finding an optimal boundary between the two classes, we invent a novel approach—the principal boundary. From the perspective of classification,
more » ... the principal boundary is defined as an optimal curve that moves in between the principal flows traced out from two classes of data, and at any point on the boundary, it maximizes the margin between the two classes. We estimate the boundary in quality with its direction, supervised by the two principal flows. We show that the principal boundary yields the usual decision boundary found by the support vector machine in the sense that locally, the two boundaries coincide. Some optimality and convergence properties of the random principal boundary and its population counterpart are also shown. We illustrate how to find, use, and interpret the principal boundary with an application in real data. Supplementary materials for this article are available online.
doi:10.6084/m9.figshare.8038301 fatcat:h7kijelrufesfbxglcpsqallhq

Sufficient Dimension Reduction for Classification [article]

Xin Chen, Jingjing Wu, Zhigang Yao, Jia Zhang
2018 arXiv   pre-print
It has been verified that HCT performs quite well theoretically and practically when the signals are rare and weak; see Donoho and Jin (2008) and Fan, Jin and Yao (2013) for details.  ...  misclassification error; Donoho and Jin (2008) employed higher criticism thresholding for feature screening when the useful features are both rare and weak; and following their work, Fan, Jin and Yao  ... 
arXiv:1812.03775v1 fatcat:dcs57g527bdzza4gjo7vfnaqma

Principal Boundary on Riemannian Manifolds [article]

Zhigang Yao, Zhenyue Zhang
2019 arXiv   pre-print
We consider the classification problem and focus on nonlinear methods for classification on manifolds. For multivariate datasets lying on an embedded nonlinear Riemannian manifold within the higher-dimensional ambient space, we aim to acquire a classification boundary for the classes with labels, using the intrinsic metric on the manifolds. Motivated by finding an optimal boundary between the two classes, we invent a novel approach -- the principal boundary. From the perspective of
more » ... n, the principal boundary is defined as an optimal curve that moves in between the principal flows traced out from two classes of data, and at any point on the boundary, it maximizes the margin between the two classes. We estimate the boundary in quality with its direction, supervised by the two principal flows. We show that the principal boundary yields the usual decision boundary found by the support vector machine in the sense that locally, the two boundaries coincide. Some optimality and convergence properties of the random principal boundary and its population counterpart are also shown. We illustrate how to find, use and interpret the principal boundary with an application in real data.
arXiv:1711.06705v2 fatcat:ha6pat5dxfb5vmvkcedv7ut7ke

Principal Sub-manifolds [article]

Zhigang Yao, Benjamin Eltzner, Tung Pham
2021 arXiv   pre-print
We invent a novel method of finding principal components in multivariate data sets that lie on an embedded nonlinear Riemannian manifold within a higher-dimensional space. Our aim is to extend the geometric interpretation of PCA, while being able to capture non-geodesic modes of variation in the data. We introduce the concept of a principal sub-manifold, a manifold passing through the center of the data, and at any point on the manifold extending in the direction of highest variation in the
more » ... e spanned by the eigenvectors of the local tangent space PCA. Compared to recent work for the case where the sub-manifold is of dimension one –essentially a curve lying on the manifold attempting to capture one-dimensional variation–the current setting is much more general. The principal sub-manifold is therefore an extension of the principal flow, accommodating to capture higher dimensional variation in the data. We show the principal sub-manifold yields the ball spanned by the usual principal components in Euclidean space. By means of examples, we illustrate how to find, use and interpret a principal sub-manifold and we present an application in shape analysis.
arXiv:1604.04318v3 fatcat:a6xve7svbvfbvpb3hmksiuh5d4

Manifold Fitting in Ambient Space [article]

Zhigang Yao, Bingjie Li, Wee Chin Tan
2022 arXiv   pre-print
Modern sample points in many applications no longer comprise real vectors in a real vector space but sample points of much more complex structures, which may be represented as points in a space with a certain underlying geometric structure, namely a manifold. Manifold learning is an emerging field for learning the underlying structure. The study of manifold learning can be split into two main branches: dimension reduction and manifold fitting. With the aim of combining statistics and geometry,
more » ... e address the problem of manifold fitting in the ambient space. Inspired by the relation between the eigenvalues of the Laplace-Beltrami operator and the geometry of a manifold, we aim to find a small set of points that preserve the geometry of the underlying manifold. From this relationship, we extend the idea of subsampling to sample points in high-dimensional space and employ the Moving Least Squares (MLS) approach to approximate the underlying manifold. We analyze the two core steps in our proposed method theoretically and also provide the bounds for the MLS approach. Our simulation results and theoretical analysis demonstrate the superiority of our method in estimating the underlying manifold.
arXiv:1909.13492v2 fatcat:3x26mdvw7zdwpi3h2fpoguyxim

I-III-VI chalcogenide semiconductor nanocrystals: Synthesis, properties, and applications

Shiqi Li, Xiaosheng Tang, Zhigang Zang, Yao Yao, Zhiqiang Yao, Haizheng Zhong, Bingkun Chen
2018 Chinese Journal of Catalysis  
Colloidal semiconductor nanocrystals have been proven to be promising candidates for applications in low-cost and high-performance photovoltaics, bioimaging, and photocatalysis due to their novel size-and shape-dependent properties. Among the colloidal systems, I-III-VI semiconductor nanocrystals (NCs) have drawn much attention in the past few decades. Compared to binary NCs, ternary I-III-VI NCs not only exhibit low toxicity, but also a high performance similar to that of binary NCs. In this
more » ... view, we mainly focus on the synthesis, properties, and applications of I-III-VI NCs. We summarize the major synthesis methods, analyze their photophysical and electronic properties, and highlight some of the latest applications of I-III-VI NCs in solar cells, light-emitting diodes, bioimaging, and photocatalysis. Finally, based on the information reviewed, we highlight the existing problems and challenges.
doi:10.1016/s1872-2067(18)63052-9 fatcat:lxkelgcdfje4hm4kakxwn5lxea

Advances in nutrient retention of dams on river

RAN Xiangbin, YU Zhigang, YAO Qingzheng, CHEN Hongtao, MI Tiezhu, YAO Peng
2009 Journal of Lake Sciences  
The progress of reservoir nutrient retention was reviewed. Challenges had been risen about whether the huge reservoirs altered river-borne nutrient transport significantly since controversial results for nutrient retention of reservoirs were found. Most documents supported the conclusion that the reservoirs changed the riverine biogeochemistry process, which resulted in a serial of impacts on downstream and estuary. In this paper, we classified nutrient retention by "sediment filter" and
more » ... ical and biochemical filter", and listed some estimating methods of nutrient retention and efficiencies.
doi:10.18307/2009.0502 fatcat:igzqu5x6zzhljfap6enj2vvtsq

Extraterrestrial artificial photosynthetic materials for in-situ resource utilization

Liuqing Yang, Ce Zhang, Xiwen Yu, Yingfang Yao, Zhaosheng Li, Congping Wu, Wei Yao, Zhigang Zou
2021 National Science Review  
Zhigang Zou et al. synthesized CeO 2 CO 2 RR.  ...  In 2001, Zhigang Zou proposed a new theory and method to regulate the band structure of photocatalytic materials and broadened the response range of photocatalytic materials.  ... 
doi:10.1093/nsr/nwab104 pmid:34691720 pmcid:PMC8363334 fatcat:jdp67hmyhfbtvcjbcvdg6vg25q

Quantifying Time-Varying Sources in Magnetoencephalography – A Discrete Approach [article]

Zhigang Yao, Zengyan Fan, Masahito Hayashi, William F. Eddy
2019 arXiv   pre-print
We study the distribution of brain source from the most advanced brain imaging technique, Magnetoencephalography (MEG), which measures the magnetic fields outside the human head produced by the electrical activity inside the brain. Common time-varying source localization methods assume the source current with a time-varying structure and solve the MEG inverse problem by mainly estimating the source moment parameters. These methods use the fact that the magnetic fields linearly depend on the
more » ... nt parameters of the source, and work well under the linear dynamic system. However, magnetic fields are known to be non-linearly related to the location parameters of the source. The existing work on estimating the time-varying unknown location parameters is limited. We are motivated to investigate the source distribution for the location parameters based on a dynamic framework, where the posterior distribution of the source is computed in a closed form discretely. The new framework allows us not only to directly approximate the posterior distribution of the source current, where sequential sampling methods may suffer from slow convergence due to the large volume of measurement, but also to quantify the source distribution at any time point from the entire set of measurements reflecting the distribution of the source, rather than using only the measurements up to the time point of interest. Both a dynamic procedure and a switch procedure are proposed for the new discrete approach, balancing estimation accuracy and computational efficiency when multiple sources are present. In both simulation and real data, we illustrate that the new method is able to provide comprehensive insight into the time evolution of the sources at different stages of the MEG and EEG experiment.
arXiv:1908.03926v1 fatcat:fogr5zsjiffqnhhwcl77eyzs7m

Probing nanoscale functionalities of metal–organic framework nanocrystals

Yao Sun, Zhigang Hu, Dan Zhao, Kaiyang Zeng
2017 Nanoscale  
We report the nanoscale piezo- and ferro-electric properties and elasticity as a function of temperature for NUS-6-based MOF nanocrystals.
doi:10.1039/c7nr04245k pmid:28805847 fatcat:azywozhehvbzjgwsdimbgixqqy

Estimating the Number of Sources in Magnetoencephalography Using Spiked Population Eigenvalues [article]

Zhigang Yao, Ye Zhang, Zhidong Bai, William F. Eddy
2017 arXiv   pre-print
Most recently, several works based on time-varying dipoles (Yao and Eddy 2014) have been proposed, where it is suggested that the number of varying dipoles is estimated in a dynamic fashion.  ... 
arXiv:1707.01225v1 fatcat:dy6iuwwevfdvnekp5wtsmymkru

Emergency Scheduling Optimization Simulation of Cloud Computing Platform Network Public Resources

Dingrong Liu, Zhigang Yao, Liukui Chen
2021 Complexity  
Emergency scheduling of public resources on the cloud computing platform network can effectively improve the network emergency rescue capability of the cloud computing platform. To schedule the network common resources, it is necessary to generate the initial population through the Hamming distance constraint and improve the objective function as the fitness function to complete the emergency scheduling of the network common resources. The traditional method, from the perspective of public
more » ... rce fairness and priority mapping, uses incremental optimization algorithm to realize emergency scheduling of public resources, neglecting the improvement process of the objective function, which leads to unsatisfactory scheduling effect. An emergency scheduling method of cloud computing platform network public resources based on genetic algorithm is proposed. With emergency public resource scheduling time cost and transportation cost minimizing target, initial population by Hamming distance constraints, emergency scheduling model, and the corresponding objective function improvement as the fitness function, the genetic algorithm to individual selection and crossover and mutation probability were optimized and complete the public emergency resources scheduling. Experimental results show that the proposed method can effectively improve the efficiency of emergency resource scheduling, and the reliability of emergency scheduling is better.
doi:10.1155/2021/9950198 doaj:701c4f463ecf42ab9ce0547410701803 fatcat:teuz7rn47vdv3drhw7xvcjiuuy
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