The divide-and-conquer strategy is a very effective method of dealing with big data. Noisy samples in big data usually have a great impact on algorithmic performance. In this article, we introduce Markov sampling and different weights for distributed learning with the classical support vector machine (cSVM). We first estimate the generalization error of weighted distributed cSVM algorithm with uniformly ergodic Markov chain (u.e.M.c.) samples and obtain its optimal convergence rate. As applications, we obtain the generalization bounds of weighted distributed cSVM with strong mixing observations and independent and identically distributed (i.i.d.) samples, respectively. We also propose a novel weighted distributed cSVM based on Markov sampling (DM-cSVM). The numerical studies of benchmark datasets show that the DM-cSVM algorithm not only has better performance but also has less total time of sampling and training compared to other distributed algorithms.

Facing with rapidly increasing demands for analyzing high-order data or multiway data, feature-extracting methods become imperative for analysis and processing. The traditional feature-extracting methods, however, either need to overly vectorize the data and smash the original structure hidden in data, such as PCA and PCA-like methods, which is unfavorable to the data recovery, or cannot eliminate the redundant information very well, such as tucker decomposition (TD) and TD-like methods. To overcome these limitations, we propose a more flexible and more powerful tool, called the multiview principal components analysis (Multiview-PCA) in this article. By segmenting a random tensor into equal-sized subarrays called sections and maximizing variations caused by orthogonal projections of these sections, the Multiview-PCA finds principal components in a parsimonious and flexible way. In so doing, two new operations on tensors, the S -direction inner/outer product, are introduced to formulate tensor projection and recovery. With different segmentation ways characterized by section depth and direction, the Multiview-PCA can be implemented many times in different ways, which defines the sequential and global Multiview-PCA, respectively. These multiple Multiview-PCA take the PCA and PCA-like, and TD and TD-like as the special cases, which correspond to the deepest section depth and the shallowest section depth, respectively. We propose an adaptive depth and direction selection algorithm for the implementation of Multiview-PCA. The Multiview-PCA is then tested in terms of subspace recovery ability, compression ability, and feature extraction performance when applied to a set of artificial data, surveillance videos, and hyperspectral imaging data. All numerical results support the flexibility, effectiveness, and usefulness of Multiview-PCA.

Although deep neural networks (DNNs) have been widely applied to hyperspectral image (HSI) denoising, most DNN-based HSI denoising methods are designed by stacking convolution layer, which can only model and reason local relations, and thus ignore the global contextual information. To address this issue, we propose a deep spatial-spectral global reasoning network to consider both the local and global information for HSI noise removal. Specifically, two novel modules are proposed to model and reason global relational information. The first one aims to model global spatial relations between pixels in feature maps, and the second one models the global relations across the channels. Compared to traditional convolution operations, the two proposed modules enable the network to extract representations from new dimensions. For the HSI denoising task, the two modules, as well as the densely connected structures, are embedded into the U-Net architecture. Thus, the new-designed global reasoning network can help tackle complex noise by exploiting multiple representations, e.g., hierarchical local feature, global spatial coherence, cross-channel correlation, and multi-scale abstract representation. Experiments on both synthetic and real HSI data demonstrate that our proposed network can obtain comparable or even better denoising results than other state-of-the-art methods.

Tensor principal component pursuit (TPCP) is a powerful approach in the tensor robust principal component analysis (TRPCA), where the goal is to decompose a data tensor to a low-tubal-rank part plus a sparse residual. TPCP is shown to be effective under certain tensor incoherence conditions, which can be restrictive in practice. In this paper, we propose a Modified-TPCP, which incorporates the prior subspace information in the analysis. With the aid of prior info, the proposed method is able to recover the low-tubal-rank and the sparse components under a significantly weaker incoherence assumption. We further design an efficient algorithm to implement Modified-TPCP based upon the alternating direction method of multipliers (ADMM). The promising performance of the proposed method is supported by simulations and real data applications.

Subsampling is a widely used and effective method to deal with the challenges brought by big data. Most subsampling procedures are designed based on the importance sampling framework, where samples with high importance measures are given corresponding sampling probabilities. However, in the highly noisy case, these samples may cause an unstable estimator which could lead to a misleading result. To tackle this issue, we propose a gradient-based Markov subsampling (GMS) algorithm to achieve robust estimation. The core idea is to construct a subset which allows us to conservatively correct a crude initial estimate towards the true signal. Specifically, GMS selects samples with small gradients via a probabilistic procedure, constructing a subset that is likely to exclude noisy samples and provide a safe improvement over the initial estimate. We show that the GMS estimator is statistically consistent at a rate which matches the optimal in the minimax sense. The promising performance of GMS is supported by simulation studies and real data examples.

Additive models have attracted much attention for high-dimensional regression estimation and variable selection. However, the existing models are usually limited to the single-task learning framework under the mean squared error (MSE) criterion, where the utilization of variable structure depends heavily on priori knowledge among variables. For high-dimensional observations in real environment, e.g., Coronal Mass Ejections (CMEs) data, the learning performance of previous methods may be degraded seriously due to the complex non-Gaussian noise and the insufficiency of prior knowledge on variable structure. To tackle this problem, we propose a new class of additive models, called Multi-task Additive Models (MAM), by integrating the mode-induced metric, the structure-based regularizer, and additive hypothesis spaces into a bilevel optimization framework. Our approach does not require any priori knowledge of variable structure and suits for high-dimensional data with complex noise, e.g., skewed noise, heavy-tailed noise, and outliers. A smooth iterative optimization algorithm with convergence guarantees is provided to implement MAM efficiently. Experiments on simulations and the CMEs analysis demonstrate the competitive performance of our approach for robust estimation and automatic structure discovery.

Feature screening is a powerful tool in processing high-dimensional data. When the sample size N and the number of features p are both large, the implementation of classic screening methods can be numerically challenging. In this paper, we propose a distributed screening framework for big data setup. In the spirit of 'divide-and-conquer', the proposed framework expresses a correlation measure as a function of several component parameters, each of which can be distributively estimated using a natural U-statistic from data segments. With the component estimates aggregated, we obtain a final correlation estimate that can be readily used for screening features. This framework enables distributed storage and parallel computing and thus is computationally attractive. Due to the unbiased distributive estimation of the component parameters, the final aggregated estimate achieves a high accuracy that is insensitive to the number of data segments m. Under mild conditions, we show that the aggregated correlation estimator is as efficient as the centralized estimator in terms of the probability convergence bound and the mean squared error rate; the corresponding screening procedure enjoys sure screening property for a wide range of correlation measures. The promising performances of the new method are supported by extensive numerical examples.

Incremental learning is one of the most effective methods of learning accumulated data and large-scale data. The newly increased samples of the previously known works on incremental learning are usually independent and identically distributed. To study how dependent sampling methods influence the learning ability of incremental support vector machines (ISVM) algorithm, in this paper we introduce an ISVM based on Markov resampling (MR-ISVM), and give the experimental research on the learning ability of the MR-ISVM algorithm. The experimental results indicate that the MR-ISVM algorithm has not only smaller misclassification rates and sparser of the obtained classifiers, but also less total time of sampling and training compared to ISVM based on randomly independent sampling. We also compare it with other ISVM algorithms.

In this paper, we propose four new variants of the backpropagation algorithm to improve the generalization ability for feedforward neural networks. The basic idea of these methods stems from the Group Lasso concept which deals with the variable selection problem at the group level. There are two main drawbacks when the Group Lasso penalty has been directly employed during network training. They are numerical oscillations and theoretical challenges in computing the gradients at the origin. To overcome these obstacles, smoothing functions have then been introduced by approximating the Group Lasso penalty. Numerical experiments for classification and regression problems demonstrate that the proposed algorithms perform better than the other three classical penalization methods, Weight Decay, Weight Elimination, and Approximate Smoother, on both generalization and pruning efficiency. In addition, detailed simulations based on a specific data set have been performed to compare with some other common pruning strategies, which verify the advantages of the proposed algorithm. The pruning abilities of the proposed strategy have been investigated in detail for a relatively large data set, MNIST, in terms of various smoothing approximation cases.

Support vector machine (SVM) is one of the most widely used learning algorithms for classification problems. Although SVM has good performance in practical applications, it has high algorithmic complexity as the size of training samples is large. In this paper, we introduce SVM classification (SVMC) algorithm based on -times Markov sampling and present the numerical studies on the learning performance of SVMC with -times Markov sampling for benchmark data sets. The experimental results show that the SVMC algorithm with -times Markov sampling not only have smaller misclassification rates, less time of sampling and training, but also the obtained classifier is more sparse compared with the classical SVMC and the previously known SVMC algorithm based on Markov sampling. We also give some discussions on the performance of SVMC with -times Markov sampling for the case of unbalanced training samples and large-scale training samples.

In Big Data applications, massive datasets with huge numbers of observations are frequently encountered. To deal with such massive datasets, a divide-and-conquer scheme (e.g., MapReduce) is often used for the analysis of Big Data. With such a strategy, a large dataset (e.g., a centralized real database or a virtual database with distributed data sources) is first divided into smaller manageable segments; the final output is then aggregated from the individual outputs of the segments. Despite its popularity in practice, it remains largely unknown whether such a distributive strategy provides valid theoretical inferences to the original data. In this paper, we address this fundamental issue for the distributed kernel regression (DKR) problem, where the algorithmic feasibility is measured by the generalization performance of the resulting estimator. To justify DKR, a uniform convergence rate is needed for bounding the generalization error over the individual outputs, which brings new and challenging issues in the Big Data setup. Using a sample dependent kernel dictionary, we show that, with proper data segmentation, DKR leads to an estimator that is generalization consistent to the unknown regression function. This result theoretically justifies DKR and sheds light on more advanced distributive algorithms for processing Big Data. The promising performance of the method is supported by both simulation and real data examples.