arXiv subjects

Zhihua Zhang

Publications and source records attributed to Zhihua Zhang.

142 records · Page 8Linked to original sources

arXiv2014

Group Orbit Optimization: A Unified Approach to Data Normalization

In this paper we propose and study an optimization problem over a matrix group orbit that we call \emph{Group Orbit Optimization} (GOO). We prove that GOO can be used to induce matrix decomposition techniques such as singular value decomposition (SVD), LU decomposition, QR decomposition, Schur decomposition and Cholesky decomposition, etc. This gives rise to a unified framework for matrix decomposition and allows us to bridge these matrix decomposition methods. Moreover, we generalize GOO for tensor decomposition. As a concrete application of GOO, we devise a new data decomposition method over a special linear group to normalize point cloud data. Experiment results show that our normalization method is able to obtain recovery well from distortions like shearing, rotation and squeezing.

math.NA↗

arXiv2014

Kinetic Energy Plus Penalty Functions for Sparse Estimation

In this paper we propose and study a family of sparsity-inducing penalty functions. Since the penalty functions are related to the kinetic energy in special relativity, we call them \emph{kinetic energy plus} (KEP) functions. We construct the KEP function by using the concave conjugate of a $χ^2$-distance function and present several novel insights into the KEP function with $q=1$. In particular, we derive a thresholding operator based on the KEP function, and prove its mathematical properties and asymptotic properties in sparsity modeling. Moreover, we show that a coordinate descent algorithm is especially appropriate for the KEP function. Additionally, we discuss the relationship of KEP with the penalty functions $\ell_{1/2}$ and MCP. The theoretical and empirical analysis validates that the KEP function is effective and efficient in high-dimensional data modeling.

stat.ML↗

arXiv2014

Self-organized vanadium and nitrogen co-doped titania nanotube arrays with enhanced photocatalytic reduction of CO2 into CH4

Self-organized V-N co-doped TiO2 nanotube arrays (TNAs) with various doping amount were synthesized by anodizing in association with hydrothermal treatment. Impacts of V-N co-doping on the morphologies, phase structures, and photoelectrochemical properties of the TNAs films were thoroughly investigated. The co-doped TiO2 photocatalysts show remarkably enhanced photocatalytic activity for the CO2 photoreduction to methane under ultraviolet illumination. The mechanism of the enhanced photocatalytic activity is discussed in detail.

cond-mat.mes-hall↗

arXiv2014

Some operator convex functions of several variables

We obtain operator concavity (convexity) of some functions of two or three variables by using perspectives of regular operator mappings of one or several variables. As an application, we obtain, for $ 0<p < 1,$ concavity, respectively convexity, of the Frechét differential mapping associated with the functions $t \to t^{1+p} $ and $t \to t^{1-p}.$

math.FA↗

arXiv2014

Enhancement of Visible-Light-Induced Photocurrent and Photocatalytic Activity of V and N Codoped TiO2 Nanotube Array Films

Highly ordered TiO2 nanotube arrays (TNAs) codoped with V and N were synthesized by electrochemical anodization in association with hydrothermal treatment. The samples were characterized by field emission scanning electron microscopy, X-ray diffraction and X-ray photoelectron spectroscopy. The photocurrent and photocatalytic activity of codoped TiO2 nanotube arrays were investigated under visible light irradiation. Moreover, the production of hydroxyl radicals on the surface of visible light-irradiated samples is detected by a photoluminescence technique using terephthalic acid (TA) as a probe molecule. It was found that the V+N co-doped TiO2 nanotube arrays showed remarkably enhanced photocurrent and photocatalytic activity than undoped sample due to the V and N codoping.

cond-mat.mtrl-sci↗

arXiv2014

Efficient Algorithms and Error Analysis for the Modified Nystrom Method

Many kernel methods suffer from high time and space complexities and are thus prohibitive in big-data applications. To tackle the computational challenge, the Nyström method has been extensively used to reduce time and space complexities by sacrificing some accuracy. The Nyström method speedups computation by constructing an approximation of the kernel matrix using only a few columns of the matrix. Recently, a variant of the Nyström method called the modified Nyström method has demonstrated significant improvement over the standard Nyström method in approximation accuracy, both theoretically and empirically. In this paper, we propose two algorithms that make the modified Nyström method practical. First, we devise a simple column selection algorithm with a provable error bound. Our algorithm is more efficient and easier to implement than and nearly as accurate as the state-of-the-art algorithm. Second, with the selected columns at hand, we propose an algorithm that computes the approximation in lower time complexity than the approach in the previous work. Furthermore, we prove that the modified Nyström method is exact under certain conditions, and we establish a lower error bound for the modified Nyström method.

cs.LG↗

arXiv2013

The Matrix Ridge Approximation: Algorithms and Applications

We are concerned with an approximation problem for a symmetric positive semidefinite matrix due to motivation from a class of nonlinear machine learning methods. We discuss an approximation approach that we call {matrix ridge approximation}. In particular, we define the matrix ridge approximation as an incomplete matrix factorization plus a ridge term. Moreover, we present probabilistic interpretations using a normal latent variable model and a Wishart model for this approximation approach. The idea behind the latent variable model in turn leads us to an efficient EM iterative method for handling the matrix ridge approximation problem. Finally, we illustrate the applications of the approximation approach in multivariate data analysis. Empirical studies in spectral clustering and Gaussian process regression show that the matrix ridge approximation with the EM iteration is potentially useful.

stat.ML↗

arXiv2013

The Bernstein Function: A Unifying Framework of Nonconvex Penalization in Sparse Estimation

In this paper we study nonconvex penalization using Bernstein functions. Since the Bernstein function is concave and nonsmooth at the origin, it can induce a class of nonconvex functions for high-dimensional sparse estimation problems. We derive a threshold function based on the Bernstein penalty and give its mathematical properties in sparsity modeling. We show that a coordinate descent algorithm is especially appropriate for penalized regression problems with the Bernstein penalty. Additionally, we prove that the Bernstein function can be defined as the concave conjugate of a $φ$-divergence and develop a conjugate maximization algorithm for finding the sparse solution. Finally, we particularly exemplify a family of Bernstein nonconvex penalties based on a generalized Gamma measure and conduct empirical analysis for this family.

stat.ML↗

arXiv2013

Improving CUR Matrix Decomposition and the Nyström Approximation via Adaptive Sampling

The CUR matrix decomposition and the Nyström approximation are two important low-rank matrix approximation techniques. The Nyström method approximates a symmetric positive semidefinite matrix in terms of a small number of its columns, while CUR approximates an arbitrary data matrix by a small number of its columns and rows. Thus, CUR decomposition can be regarded as an extension of the Nyström approximation. In this paper we establish a more general error bound for the adaptive column/row sampling algorithm, based on which we propose more accurate CUR and Nyström algorithms with expected relative-error bounds. The proposed CUR and Nyström algorithms also have low time complexity and can avoid maintaining the whole data matrix in RAM. In addition, we give theoretical analysis for the lower error bounds of the standard Nyström method and the ensemble Nyström method. The main theoretical results established in this paper are novel, and our analysis makes no special assumption on the data matrices.

cs.LG↗

arXiv2012

A Scalable CUR Matrix Decomposition Algorithm: Lower Time Complexity and Tighter Bound

The CUR matrix decomposition is an important extension of Nyström approximation to a general matrix. It approximates any data matrix in terms of a small number of its columns and rows. In this paper we propose a novel randomized CUR algorithm with an expected relative-error bound. The proposed algorithm has the advantages over the existing relative-error CUR algorithms that it possesses tighter theoretical bound and lower time complexity, and that it can avoid maintaining the whole data matrix in main memory. Finally, experiments on several real-world datasets demonstrate significant improvement over the existing relative-error algorithms.

cs.LG↗

arXiv2012

Bayesian Multicategory Support Vector Machines

We show that the multi-class support vector machine (MSVM) proposed by Lee et. al. (2004), can be viewed as a MAP estimation procedure under an appropriate probabilistic interpretation of the classifier. We also show that this interpretation can be extended to a hierarchical Bayesian architecture and to a fully-Bayesian inference procedure for multi-class classification based on data augmentation. We present empirical results that show that the advantages of the Bayesian formalism are obtained without a loss in classification accuracy.

cs.LG↗

arXiv2012

EP-GIG Priors and Applications in Bayesian Sparse Learning

In this paper we propose a novel framework for the construction of sparsity-inducing priors. In particular, we define such priors as a mixture of exponential power distributions with a generalized inverse Gaussian density (EP-GIG). EP-GIG is a variant of generalized hyperbolic distributions, and the special cases include Gaussian scale mixtures and Laplace scale mixtures. Furthermore, Laplace scale mixtures can subserve a Bayesian framework for sparse learning with nonconvex penalization. The densities of EP-GIG can be explicitly expressed. Moreover, the corresponding posterior distribution also follows a generalized inverse Gaussian distribution. These properties lead us to EM algorithms for Bayesian sparse learning. We show that these algorithms bear an interesting resemblance to iteratively re-weighted $\ell_2$ or $\ell_1$ methods. In addition, we present two extensions for grouped variable selection and logistic regression.

stat.ML↗

arXiv2012

Coherence Functions with Applications in Large-Margin Classification Methods

Support vector machines (SVMs) naturally embody sparseness due to their use of hinge loss functions. However, SVMs can not directly estimate conditional class probabilities. In this paper we propose and study a family of coherence functions, which are convex and differentiable, as surrogates of the hinge function. The coherence function is derived by using the maximum-entropy principle and is characterized by a temperature parameter. It bridges the hinge function and the logit function in logistic regression. The limit of the coherence function at zero temperature corresponds to the hinge function, and the limit of the minimizer of its expected error is the minimizer of the expected error of the hinge loss. We refer to the use of the coherence function in large-margin classification as C-learning, and we present efficient coordinate descent algorithms for the training of regularized ${\cal C}$-learning models.

stat.ML↗

arXiv2011

Multiway Spectral Clustering: A Margin-Based Perspective

Spectral clustering is a broad class of clustering procedures in which an intractable combinatorial optimization formulation of clustering is "relaxed" into a tractable eigenvector problem, and in which the relaxed solution is subsequently "rounded" into an approximate discrete solution to the original problem. In this paper we present a novel margin-based perspective on multiway spectral clustering. We show that the margin-based perspective illuminates both the relaxation and rounding aspects of spectral clustering, providing a unified analysis of existing algorithms and guiding the design of new algorithms. We also present connections between spectral clustering and several other topics in statistics, specifically minimum-variance clustering, Procrustes analysis and Gaussian intrinsic autoregression.

stat.ME↗

arXiv2010

On Edwards-Child's inequality

In this paper, by making use of one of Chen's theorems and the method of mathematical analysis, we refine Edwards-Child's inequality and solve a conjecture posed by Liu.

math.MG↗

arXiv2010

On the circumradius of a special class of n-simplices

An n-simplex is called circumscriptible (or edge-incentric) if there is a sphere tangent to all its n(n + 1)/2 edges. We obtain a closed formula for the radius of the circumscribed sphere of the circumscriptible n-simplex, and also prove a double inequality involving the circumradius and the edge-inradius of such simplices. Among this inequality settles affirmatively a part of a problem posed by the authors.

math.MG↗