arXiv subjects

Zhigang Yao

Publications and source records attributed to Zhigang Yao.

27 records · Page 2Linked to original sources

arXiv2022

Manifold Fitting in Ambient Space

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, we 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.

stat.ML↗

arXiv2019

Quantifying Time-Varying Sources in Magnetoencephalography -- A Discrete Approach

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 moment 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.

stat.AP↗

arXiv2019

Community Detection Based on the $L_\infty$ convergence of eigenvectors in DCBM

Spectral clustering is one of the most popular algorithms for community detection in network analysis. Based on this rationale, in this paper we give the convergence rate of eigenvectors for the adjacency matrix in the $l_\infty$ norm, under the stochastic block model (BM) and degree corrected stochastic block model (DCBM), adding some mild and rational conditions. We also extend this result to a more general model, presented based on the DCBM such that the value of random variables in the adjacency matrix is not 0 or 1, but an arbitrary real number. During the process of proving the above conclusion, we obtain the relationship of the eigenvalues in the adjacency matrix and the corresponding `population' matrix, which vary in dimension from the community-wise edge probability matrix. Using that result, we can give an estimate of the number of the communities in a known set of network data. Meanwhile we proved the consistency of the estimator. Furthermore, according to the derivation of proof for the convergence of eigenvectors, we propose a new approach to community detection -- Spectral Clustering based on Difference of Ratios of Eigenvectors (SCDRE). Our simulation experiments demonstrate the superiority of our method in community detection.

math.ST↗

arXiv2019

Principal Boundary on Riemannian Manifolds

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, 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.

stat.ML↗

arXiv2018

Sufficient Dimension Reduction for Classification

We propose a new sufficient dimension reduction approach designed deliberately for high-dimensional classification. This novel method is named maximal mean variance (MMV), inspired by the mean variance index first proposed by Cui, Li and Zhong (2015), which measures the dependence between a categorical random variable with multiple classes and a continuous random variable. Our method requires reasonably mild restrictions on the predicting variables and keeps the model-free advantage without the need to estimate the link function. The consistency of the MMV estimator is established under regularity conditions for both fixed and diverging dimension (p) cases and the number of the response classes can also be allowed to diverge with the sample size n. We also construct the asymptotic normality for the estimator when the dimension of the predicting vector is fixed. Furthermore, our method works pretty well when n < p. The surprising classification efficiency gain of the proposed method is demonstrated by simulation studies and real data analysis.

stat.ME↗

arXiv2017

Estimating the Number of Sources in Magnetoencephalography Using Spiked Population Eigenvalues

Magnetoencephalography (MEG) is an advanced imaging technique used to measure the magnetic fields outside the human head produced by the electrical activity inside the brain. Various source localization methods in MEG require the knowledge of the underlying active sources, which are identified by a priori. Common methods used to estimate the number of sources include principal component analysis or information criterion methods, both of which make use of the eigenvalue distribution of the data, thus avoiding solving the time-consuming inverse problem. Unfortunately, all these methods are very sensitive to the signal-to-noise ratio (SNR), as examining the sample extreme eigenvalues does not necessarily reflect the perturbation of the population ones. To uncover the unknown sources from the very noisy MEG data, we introduce a framework, referred to as the intrinsic dimensionality (ID) of the optimal transformation for the SNR rescaling functional. It is defined as the number of the spiked population eigenvalues of the associated transformed data matrix. It is shown that the ID yields a more reasonable estimate for the number of sources than its sample counterparts, especially when the SNR is small. By means of examples, we illustrate that the new method is able to capture the number of signal sources in MEG that can escape PCA or other information criterion based methods.

stat.ME↗

arXiv2014

Partial Correlation Screening for Estimating Large Precision Matrices, with Applications to Classification

We propose Partial Correlation Screening (PCS) as a new row-by-row approach to estimating a large precision matrix $Ω$. To estimate the $i$-th row of $Ω$, $1 \leq i \leq p$, PCS uses a Screen step and a Clean step. In the Screen step, PCS recruits a (small) subset of indices using a stage-wise algorithm, where in each stage, the algorithm updates the set of recruited indices by adding the index $j$ that has the largest (in magnitude) empirical partial correlation with $i$. In the Clean step, PCS re-investigates all recruited indices and use them to reconstruct the $i$-th row of $Ω$. PCS is computationally efficient and modest in memory use: to estimate a row of $Ω$, it only needs a few rows (determined sequentially) of the empirical covariance matrix. This enables PCS to execute the estimation of a large precision matrix (e.g., $p=10K$) in a few minutes, and open doors to estimating much larger precision matrices. We use PCS for classification. Higher Criticism Thresholding (HCT) is a recent classifier that enjoys optimality, but to exploit its full potential in practice, one needs a good estimate of the precision matrix $Ω$. Combining HCT with any approach to estimating $Ω$ gives a new classifier: examples include HCT-PCS and HCT-glasso. We have applied HCT-PCS to two large microarray data sets ($p = 8K$ and $10K$) for classification, where it not only significantly outperforms HCT-glasso, but also is competitive to the Support Vector Machine (SVM) and Random Forest (RF). The results suggest that PCS gives more useful estimates of $Ω$ than the glasso. We set up a general theoretical framework and show that in a broad context, PCS fully recovers the support of $Ω$ and HCT-PCS yields optimal classification behavior. Our proofs shed interesting light on the behavior of stage-wise procedures.

stat.ME↗

arXiv2014

A statistical approach to the inverse problem in magnetoencephalography

Magnetoencephalography (MEG) is an imaging technique used to measure the magnetic field outside the human head produced by the electrical activity inside the brain. The MEG inverse problem, identifying the location of the electrical sources from the magnetic signal measurements, is ill-posed, that is, there are an infinite number of mathematically correct solutions. Common source localization methods assume the source does not vary with time and do not provide estimates of the variability of the fitted model. Here, we reformulate the MEG inverse problem by considering time-varying locations for the sources and their electrical moments and we model their time evolution using a state space model. Based on our predictive model, we investigate the inverse problem by finding the posterior source distribution given the multiple channels of observations at each time rather than fitting fixed source parameters. Our new model is more realistic than common models and allows us to estimate the variation of the strength, orientation and position. We propose two new Monte Carlo methods based on sequential importance sampling. Unlike the usual MCMC sampling scheme, our new methods work in this situation without needing to tune a high-dimensional transition kernel which has a very high cost. The dimensionality of the unknown parameters is extremely large and the size of the data is even larger. We use Parallel Virtual Machine (PVM) to speed up the computation.

stat.CO↗

arXiv2013

Optimal classification in sparse Gaussian graphic model

Consider a two-class classification problem where the number of features is much larger than the sample size. The features are masked by Gaussian noise with mean zero and covariance matrix $Σ$, where the precision matrix $Ω=Σ^{-1}$ is unknown but is presumably sparse. The useful features, also unknown, are sparse and each contributes weakly (i.e., rare and weak) to the classification decision. By obtaining a reasonably good estimate of $Ω$, we formulate the setting as a linear regression model. We propose a two-stage classification method where we first select features by the method of Innovated Thresholding (IT), and then use the retained features and Fisher's LDA for classification. In this approach, a crucial problem is how to set the threshold of IT. We approach this problem by adapting the recent innovation of Higher Criticism Thresholding (HCT). We find that when useful features are rare and weak, the limiting behavior of HCT is essentially just as good as the limiting behavior of ideal threshold, the threshold one would choose if the underlying distribution of the signals is known (if only). Somewhat surprisingly, when $Ω$ is sufficiently sparse, its off-diagonal coordinates usually do not have a major influence over the classification decision. Compared to recent work in the case where $Ω$ is the identity matrix [Proc. Natl. Acad. Sci. USA 105 (2008) 14790-14795; Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367 (2009) 4449-4470], the current setting is much more general, which needs a new approach and much more sophisticated analysis. One key component of the analysis is the intimate relationship between HCT and Fisher's separation. Another key component is the tight large-deviation bounds for empirical processes for data with unconventional correlation structures, where graph theory on vertex coloring plays an important role.

stat.ML↗