arXiv subjects

Zongben Xu

Publications and source records attributed to Zongben Xu.

At least 109 records · Page 6Linked to original sources

arXiv2015

Sparse Regularization: Convergence Of Iterative Jumping Thresholding Algorithm

In recent studies on sparse modeling, non-convex penalties have received considerable attentions due to their superiorities on sparsity-inducing over the convex counterparts. Compared with the convex optimization approaches, however, the non-convex approaches have more challenging convergence analysis. In this paper, we study the convergence of a non-convex iterative thresholding algorithm for solving sparse recovery problems with a certain class of non-convex penalties, whose corresponding thresholding functions are discontinuous with jump discontinuities. Therefore, we call the algorithm the iterative jumping thresholding (IJT) algorithm. The finite support and sign convergence of IJT algorithm is firstly verified via taking advantage of such jump discontinuity. Together with the assumption of the introduced restricted Kurdyka-Łojasiewicz (rKL) property, then the strong convergence of IJT algorithm can be proved.Furthermore, we can show that IJT algorithm converges to a local minimizer at an asymptotically linear rate under some additional conditions. Moreover, we derive a posteriori computable error estimate, which can be used to design practical terminal rules for the algorithm. It should be pointed out that the $l_q$ quasi-norm ($0<q<1$) is an important subclass of the class of non-convex penalties studied in this paper. In particular, when applied to the $l_q$ regularization, IJT algorithm can converge to a local minimizer with an asymptotically linear rate under certain concentration conditions. We provide also a set of simulations to support the correctness of theoretical assertions and compare the time efficiency of IJT algorithm for the $l_{q}$ regularization ($q=1/2, 2/3$) with other known typical algorithms like the iterative reweighted least squares (IRLS) algorithm and the iterative reweighted $l_{1}$ minimization (IRL1) algorithm.

math.OC↗

arXiv2015

An Efficient Optimization Approach for a Cardinality-Constrained Index Tracking Problem

In the practical business environment, portfolio managers often face business-driven requirements that limit the number of constituents in their tracking portfolio. A natural index tracking model is thus to minimize a tracking error measure while enforcing an upper bound on the number of assets in the portfolio. In this paper we consider such a cardinality-constrained index tracking model. In particular, we propose an efficient nonmonotone projected gradient (NPG) method for solving this problem. At each iteration, this method usually solves several projected gradient subproblems. We show that each subproblem has a closed-form solution, which can be computed in linear time. Under some suitable assumptions, we establish that any accumulation point of the sequence generated by the NPG method is a local minimizer of the cardinality-constrained index tracking problem. We also conduct empirical tests to compare our method with the hybrid evolutionary algorithm and the hybrid half thresholding algorithm \cite{L1/2} for index tracking. The computational results demonstrate that our approach generally produces sparse portfolios with smaller out-of-sample tracking error and higher consistency between in-sample and out-of-sample tracking errors. Moreover, our method outperforms the other two approaches in terms of speed.

math.OC↗

arXiv2015

Sparse Index Tracking Based On $L_{1/2}$ Model And Algorithm

Recently, $L_1$ regularization have been attracted extensive attention and successfully applied in mean-variance portfolio selection for promoting out-of-sample properties and decreasing transaction costs. However, $L_1$ regularization approach is ineffective in promoting sparsity and selecting regularization parameter on index tracking with the budget and no-short selling constraints, since the 1-norm of the asset weights will have a constant value of one. Our recent research on $L_{1/2}$ regularization has found that the half thresholding algorithm with optimal regularization parameter setting strategy is the fast solver of $L_{1/2}$ regularization, which can provide the more sparse solution. In this paper we apply $L_{1/2}$ regularization method to stock index tracking and establish a new sparse index tracking model. A hybrid half thresholding algorithm is proposed for solving the model. Empirical tests of model and algorithm are carried out on the eight data sets from OR-library. The optimal tracking portfolio obtained from the new model and algorithm has lower out-of-sample prediction error and consistency both in-sample and out-of-sample. Moreover, since the automatic regularization parameters are selected for the fixed number of optimal portfolio, our algorithm is a fast solver, especially for the large scale problem.

math.OC↗

arXiv2015

Shrinkage degree in $L_2$-re-scale boosting for regression

Re-scale boosting (RBoosting) is a variant of boosting which can essentially improve the generalization performance of boosting learning. The key feature of RBoosting lies in introducing a shrinkage degree to re-scale the ensemble estimate in each gradient-descent step. Thus, the shrinkage degree determines the performance of RBoosting. The aim of this paper is to develop a concrete analysis concerning how to determine the shrinkage degree in $L_2$-RBoosting. We propose two feasible ways to select the shrinkage degree. The first one is to parameterize the shrinkage degree and the other one is to develope a data-driven approach of it. After rigorously analyzing the importance of the shrinkage degree in $L_2$-RBoosting learning, we compare the pros and cons of the proposed methods. We find that although these approaches can reach the same learning rates, the structure of the final estimate of the parameterized approach is better, which sometimes yields a better generalization capability when the number of sample is finite. With this, we recommend to parameterize the shrinkage degree of $L_2$-RBoosting. To this end, we present an adaptive parameter-selection strategy for shrinkage degree and verify its feasibility through both theoretical analysis and numerical verification. The obtained results enhance the understanding of RBoosting and further give guidance on how to use $L_2$-RBoosting for regression tasks.

cs.LG↗

arXiv2015

Convergence of multi-block Bregman ADMM for nonconvex composite problems

The alternating direction method with multipliers (ADMM) has been one of most powerful and successful methods for solving various composite problems. The convergence of the conventional ADMM (i.e., 2-block) for convex objective functions has been justified for a long time, and its convergence for nonconvex objective functions has, however, been established very recently. The multi-block ADMM, a natural extension of ADMM, is a widely used scheme and has also been found very useful in solving various nonconvex optimization problems. It is thus expected to establish convergence theory of the multi-block ADMM under nonconvex frameworks. In this paper we present a Bregman modification of 3-block ADMM and establish its convergence for a large family of nonconvex functions. We further extend the convergence results to the $N$-block case ($N \geq 3$), which underlines the feasibility of multi-block ADMM applications in nonconvex settings. Finally, we present a simulation study and a real-world application to support the correctness of the obtained theoretical assertions.

math.OC↗

arXiv2015

Learning a Convolutional Neural Network for Non-uniform Motion Blur Removal

In this paper, we address the problem of estimating and removing non-uniform motion blur from a single blurry image. We propose a deep learning approach to predicting the probabilistic distribution of motion blur at the patch level using a convolutional neural network (CNN). We further extend the candidate set of motion kernels predicted by the CNN using carefully designed image rotations. A Markov random field model is then used to infer a dense non-uniform motion blur field enforcing motion smoothness. Finally, motion blur is removed by a non-uniform deblurring model using patch-level image prior. Experimental evaluations show that our approach can effectively estimate and remove complex non-uniform motion blur that is not handled well by previous approaches.

cs.CV↗

arXiv2015

Video Primal Sketch: A Unified Middle-Level Representation for Video

This paper presents a middle-level video representation named Video Primal Sketch (VPS), which integrates two regimes of models: i) sparse coding model using static or moving primitives to explicitly represent moving corners, lines, feature points, etc., ii) FRAME /MRF model reproducing feature statistics extracted from input video to implicitly represent textured motion, such as water and fire. The feature statistics include histograms of spatio-temporal filters and velocity distributions. This paper makes three contributions to the literature: i) Learning a dictionary of video primitives using parametric generative models; ii) Proposing the Spatio-Temporal FRAME (ST-FRAME) and Motion-Appearance FRAME (MA-FRAME) models for modeling and synthesizing textured motion; and iii) Developing a parsimonious hybrid model for generic video representation. Given an input video, VPS selects the proper models automatically for different motion patterns and is compatible with high-level action representations. In the experiments, we synthesize a number of textured motion; reconstruct real videos using the VPS; report a series of human perception experiments to verify the quality of reconstructed videos; demonstrate how the VPS changes over the scale transition in videos; and present the close connection between VPS and high-level action models.

cs.CV↗

arXiv2014

Convergence of Bregman alternating direction method with multipliers for nonconvex composite problems

The alternating direction method with multipliers (ADMM) has been one of most powerful and successful methods for solving various convex or nonconvex composite problems that arise in the fields of image & signal processing and machine learning. In convex settings, numerous convergence results have been established for ADMM as well as its varieties. However, due to the absence of convexity, the convergence analysis of nonconvex ADMM is generally very difficult. In this paper we study the Bregman modification of ADMM (BADMM), which includes the conventional ADMM as a special case and often leads to an improvement of the performance of the algorithm. Under certain assumptions, we prove that the iterative sequence generated by BADMM converges to a stationary point of the associated augmented Lagrangian function. The obtained results underline the feasibility of ADMM in applications under nonconvex settings.

math.OC↗

arXiv2014

Greedy metrics in orthogonal greedy learning

Orthogonal greedy learning (OGL) is a stepwise learning scheme that adds a new atom from a dictionary via the steepest gradient descent and build the estimator via orthogonal projecting the target function to the space spanned by the selected atoms in each greedy step. Here, "greed" means choosing a new atom according to the steepest gradient descent principle. OGL then avoids the overfitting/underfitting by selecting an appropriate iteration number. In this paper, we point out that the overfitting/underfitting can also be avoided via redefining "greed" in OGL. To this end, we introduce a new greedy metric, called $δ$-greedy thresholds, to refine "greed" and theoretically verifies its feasibility. Furthermore, we reveals that such a greedy metric can bring an adaptive termination rule on the premise of maintaining the prominent learning performance of OGL. Our results show that the steepest gradient descent is not the unique greedy metric of OGL and some other more suitable metric may lessen the hassle of model-selection of OGL.

cs.LG↗

arXiv2014

On parallel multisplitting block iterative methods for linear systems arising in the numerical solution of Euler equations

The paper studies the convergence of some parallel multisplitting block iterative methods for the solution of linear systems arising in the numerical solution of Euler equations. Some sufficient conditions for convergence are proposed. As special cases the convergence of the parallel block generalized AOR (BGAOR), the parallel block AOR (BAOR), the parallel block generalized SOR (BGSOR), the parallel block SOR (BSOR), the extrapolated parallel BAOR and the extrapolated parallel BSOR methods are presented. Furthermore, the convergence of the parallel block iterative methods for linear systems with special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper.

math.NA↗

arXiv2014

Learning rates of $l^q$ coefficient regularization learning with Gaussian kernel

Regularization is a well recognized powerful strategy to improve the performance of a learning machine and $l^q$ regularization schemes with $0<q<\infty$ are central in use. It is known that different $q$ leads to different properties of the deduced estimators, say, $l^2$ regularization leads to smooth estimators while $l^1$ regularization leads to sparse estimators. Then, how does the generalization capabilities of $l^q$ regularization learning vary with $q$? In this paper, we study this problem in the framework of statistical learning theory and show that implementing $l^q$ coefficient regularization schemes in the sample dependent hypothesis space associated with Gaussian kernel can attain the same almost optimal learning rates for all $0<q<\infty$. That is, the upper and lower bounds of learning rates for $l^q$ regularization learning are asymptotically identical for all $0<q<\infty$. Our finding tentatively reveals that, in some modeling contexts, the choice of $q$ might not have a strong impact with respect to the generalization capability. From this perspective, $q$ can be arbitrarily specified, or specified merely by other no generalization criteria like smoothness, computational complexity, sparsity, etc..

cs.LG↗

arXiv2014

Learning and approximation capability of orthogonal super greedy algorithm

We consider the approximation capability of orthogonal super greedy algorithms (OSGA) and its applications in supervised learning. OSGA is concerned with selecting more than one atoms in each iteration step, which, of course, greatly reduces the computational burden when compared with the conventional orthogonal greedy algorithm (OGA). We prove that even for function classes that are not the convex hull of the dictionary, OSGA does not degrade the approximation capability of OGA provided the dictionary is incoherent. Based on this, we deduce a tight generalization error bound for OSGA learning. Our results show that in the realm of supervised learning, OSGA provides a possibility to further reduce the computational burden of OGA in the premise of maintaining its prominent generalization capability.

cs.LG↗

arXiv2014

A Cyclic Coordinate Descent Algorithm for lq Regularization

In recent studies on sparse modeling, $l_q$ ($0<q<1$) regularization has received considerable attention due to its superiorities on sparsity-inducing and bias reduction over the $l_1$ regularization.In this paper, we propose a cyclic coordinate descent (CCD) algorithm for $l_q$ regularization. Our main result states that the CCD algorithm converges globally to a stationary point as long as the stepsize is less than a positive constant. Furthermore, we demonstrate that the CCD algorithm converges to a local minimizer under certain additional conditions. Our numerical experiments demonstrate the efficiency of the CCD algorithm.

math.OC↗

arXiv2014

On the Optimal Solution of Weighted Nuclear Norm Minimization

In recent years, the nuclear norm minimization (NNM) problem has been attracting much attention in computer vision and machine learning. The NNM problem is capitalized on its convexity and it can be solved efficiently. The standard nuclear norm regularizes all singular values equally, which is however not flexible enough to fit real scenarios. Weighted nuclear norm minimization (WNNM) is a natural extension and generalization of NNM. By assigning properly different weights to different singular values, WNNM can lead to state-of-the-art results in applications such as image denoising. Nevertheless, so far the global optimal solution of WNNM problem is not completely solved yet due to its non-convexity in general cases. In this article, we study the theoretical properties of WNNM and prove that WNNM can be equivalently transformed into a quadratic programming problem with linear constraints. This implies that WNNM is equivalent to a convex problem and its global optimum can be readily achieved by off-the-shelf convex optimization solvers. We further show that when the weights are non-descending, the globally optimal solution of WNNM can be obtained in closed-form.

cs.CV↗

arXiv2014

$L_{1/2}$ Regularization: Convergence of Iterative Half Thresholding Algorithm

In recent studies on sparse modeling, the nonconvex regularization approaches (particularly, $L_{q}$ regularization with $q\in(0,1)$) have been demonstrated to possess capability of gaining much benefit in sparsity-inducing and efficiency. As compared with the convex regularization approaches (say, $L_{1}$ regularization), however, the convergence issue of the corresponding algorithms are more difficult to tackle. In this paper, we deal with this difficult issue for a specific but typical nonconvex regularization scheme, the $L_{1/2}$ regularization, which has been successfully used to many applications. More specifically, we study the convergence of the iterative \textit{half} thresholding algorithm (the \textit{half} algorithm for short), one of the most efficient and important algorithms for solution to the $L_{1/2}$ regularization. As the main result, we show that under certain conditions, the \textit{half} algorithm converges to a local minimizer of the $L_{1/2}$ regularization, with an eventually linear convergence rate. The established result provides a theoretical guarantee for a wide range of applications of the \textit{half} algorithm. We provide also a set of simulations to support the correctness of theoretical assertions and compare the time efficiency of the \textit{half} algorithm with other known typical algorithms for $L_{1/2}$ regularization like the iteratively reweighted least squares (IRLS) algorithm and the iteratively reweighted $l_{1}$ minimization (IRL1) algorithm.

math.NA↗

arXiv2014

Sparse K-Means with $\ell_{\infty}/\ell_0$ Penalty for High-Dimensional Data Clustering

Sparse clustering, which aims to find a proper partition of an extremely high-dimensional data set with redundant noise features, has been attracted more and more interests in recent years. The existing studies commonly solve the problem in a framework of maximizing the weighted feature contributions subject to a $\ell_2/\ell_1$ penalty. Nevertheless, this framework has two serious drawbacks: One is that the solution of the framework unavoidably involves a considerable portion of redundant noise features in many situations, and the other is that the framework neither offers intuitive explanations on why this framework can select relevant features nor leads to any theoretical guarantee for feature selection consistency. In this article, we attempt to overcome those drawbacks through developing a new sparse clustering framework which uses a $\ell_{\infty}/\ell_0$ penalty. First, we introduce new concepts on optimal partitions and noise features for the high-dimensional data clustering problems, based on which the previously known framework can be intuitively explained in principle. Then, we apply the suggested $\ell_{\infty}/\ell_0$ framework to formulate a new sparse k-means model with the $\ell_{\infty}/\ell_0$ penalty ($\ell_0$-k-means for short). We propose an efficient iterative algorithm for solving the $\ell_0$-k-means. To deeply understand the behavior of $\ell_0$-k-means, we prove that the solution yielded by the $\ell_0$-k-means algorithm has feature selection consistency whenever the data matrix is generated from a high-dimensional Gaussian mixture model. Finally, we provide experiments with both synthetic data and the Allen Developing Mouse Brain Atlas data to support that the proposed $\ell_0$-k-means exhibits better noise feature detection capacity over the previously known sparse k-means with the $\ell_2/\ell_1$ penalty ($\ell_1$-k-means for short).

stat.ML↗

arXiv2014

Is Extreme Learning Machine Feasible? A Theoretical Assessment (Part II)

An extreme learning machine (ELM) can be regarded as a two stage feed-forward neural network (FNN) learning system which randomly assigns the connections with and within hidden neurons in the first stage and tunes the connections with output neurons in the second stage. Therefore, ELM training is essentially a linear learning problem, which significantly reduces the computational burden. Numerous applications show that such a computation burden reduction does not degrade the generalization capability. It has, however, been open that whether this is true in theory. The aim of our work is to study the theoretical feasibility of ELM by analyzing the pros and cons of ELM. In the previous part on this topic, we pointed out that via appropriate selection of the activation function, ELM does not degrade the generalization capability in the expectation sense. In this paper, we launch the study in a different direction and show that the randomness of ELM also leads to certain negative consequences. On one hand, we find that the randomness causes an additional uncertainty problem of ELM, both in approximation and learning. On the other hand, we theoretically justify that there also exists an activation function such that the corresponding ELM degrades the generalization capability. In particular, we prove that the generalization capability of ELM with Gaussian kernel is essentially worse than that of FNN with Gaussian kernel. To facilitate the use of ELM, we also provide a remedy to such a degradation. We find that the well-developed coefficient regularization technique can essentially improve the generalization capability. The obtained results reveal the essential characteristic of ELM and give theoretical guidance concerning how to use ELM.

cs.LG↗

arXiv2013

Compressed Sensing SAR Imaging with Multilook Processing

Multilook processing is a widely used speckle reduction approach in synthetic aperture radar (SAR) imaging. Conventionally, it is achieved by incoherently summing of some independent low-resolution images formulated from overlapping subbands of the SAR signal. However, in the context of compressive sensing (CS) SAR imaging, where the samples are collected at sub-Nyquist rate, the data spectrum is highly aliased that hinders the direct application of the existing multilook techniques. In this letter, we propose a new CS-SAR imaging method that can realize multilook processing simultaneously during image reconstruction. The main idea is to replace the SAR observation matrix by the inverse of multilook procedures, which is then combined with random sampling matrix to yield a multilook CS-SAR observation model. Then a joint sparse regularization model, considering pixel dependency of subimages, is derived to form multilook images. The suggested SAR imaging method can not only reconstruct sparse scene efficiently below Nyquist rate, but is also able to achieve a comparable reduction of speckles during reconstruction. Simulation results are finally provided to demonstrate the effectiveness of the proposed method.

cs.IT↗