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Sparse unmixing (SU) algorithms use the existing spectral library as prior knowledge to analyze the endmembers and estimate abundance maps. The majority of SU algorithms use loss functions based on the $L_{2,1}$ -norm or $F$ -norm to minimize reconstruction error. They have different advantages and shortcomings. In short, $F$ -norm has a differentiable characteristic, and it is easy to minimize as a loss function. However, it is very sensitive to heavy noise and outliers. While the $L_{2,1}$ -norm emphasizes the reconstruction error on each band and is robust to noise with different intensities in different bands. But the $L_{2,1}$ -norm is nondifferentiable at zero-point. This article introduces an adaptive loss function based on the $\sigma $ -norm for SU, which combines the advantages of the $L_{2,1}$ -norm and $F$ -norm. The adaptive loss function is related to a nonnegative parameter $\sigma $. By adjusting the parameter $\sigma $ , the adaptive loss function can approach the $F$ -norm or $L_{2,1}$ -norm. To the best of our knowledge, it is the first time to apply an adaptive loss function to SU. Moreover, the adaptive loss function is globally differentiable, and we propose an optimization algorithm for the adaptive loss function and verify its convergence. Experiments on real-world and synthetic HSIs show that the adaptive loss function effectively enhances the performance of the SU algorithms. [ABSTRACT FROM AUTHOR]

Semisupervised band selection (BS) technique plays an important role in processing hyperspectral images (HSIs) because of its superiority of using the limited labeled data and plentiful unlabeled data to select the discriminative and informative feature, which copes with the problem of high-dimensional and scare labeled samples of HSIs. Among semisupervised BS models, graph-based methods are superior to others in many situations and have received more and more attention. However, traditional graph-based models construct a similarity matrix and select valuable bands independently. The similarity matrix remains constant, which will damage the local manifold structure and lead to a suboptimal result. To solve this problem, we propose a semisupervised band selection with an optimal graph (BSOG) approach, which performs BS and local structure learning simultaneously. Instead of fixing the input similarity matrix, the similarity matrix is updated constantly to learn a better local structure. Besides, the learned similarity matrix is adaptive. Then, the optimal band subset can be selected by analyzing the obtained projection matrix $\boldsymbol W$. An efficient and simple optimization algorithm is proposed to solve this model. Experiments on four real HSIs data validate the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

Linear Discriminant Analysis (LDA) is one of the most successful supervised dimensionality reduction methods and has been widely used in many real-world applications. However, ℓ2-norm is employed as the distance metric in the objective of LDA, which is sensitive to outliers. Many previous works improve the robustness of LDA by using ℓ1-norm distance. However, the robustness against outliers is limited and the solver of ℓ1-norm is mostly based on the greedy search strategy, which is time-consuming and easy to get stuck in a local optimum. In this paper, we propose a novel robust LDA measured by ℓ2,1-norm to learn robust discriminative projections. The proposed model is challenging to solve since it needs to minimize and maximize (minmax) ℓ2,1-norm terms simultaneously. As a result, we first systematically derive an efficient iterative optimization algorithm to solve a general ratio minimization problem, and then rigorously prove its convergence. More importantly, an alternately non-greedy iterative re-weighted optimization algorithm is developed based on the preceding approach for solving proposed ℓ2,1-norm minmax problem. Besides, an optimal weighted mean mechanism is driven according to the designed objective and solver, which can be applied to other approaches for robustness improvement. Experimental results on several real-world datasets show the effectiveness of proposed method. [ABSTRACT FROM AUTHOR]

Low Rank Regularization (LRR), in essence, involves introducing a low rank or approximately low rank assumption to target we aim to learn, which has achieved great success in many data analysis tasks. Over the last decade, much progress has been made in theories and applications. Nevertheless, the intersection between these two lines is rare. In order to construct a bridge between practical applications and theoretical studies, in this paper we provide a comprehensive survey for LRR. Specifically, we first review the recent advances in two issues that all LRR models are faced with: (1) rank-norm relaxation, which seeks to find a relaxation to replace the rank minimization problem; (2) model optimization, which seeks to use an efficient optimization algorithm to solve the relaxed LRR models. For the first issue, we provide a detailed summarization for various relaxation functions and conclude that the non-convex relaxations can alleviate the punishment bias problem compared with the convex relaxations. For the second issue, we summarize the representative optimization algorithms used in previous studies, and analyze their advantages and disadvantages. As the main goal of this paper is to promote the application of non-convex relaxations, we conduct extensive experiments to compare different relaxation functions. The experimental results demonstrate that the non-convex relaxations generally provide a large advantage over the convex relaxations. Such a result is inspiring for further improving the performance of existing LRR models. [ABSTRACT FROM AUTHOR]

Spectral clustering has been playing a vital role in various research areas. Most traditional spectral clustering algorithms comprise two independent stages (e.g., first learning continuous labels and then rounding the learned labels into discrete ones), which may cause unpredictable deviation of resultant cluster labels from genuine ones, thereby leading to severe information loss and performance degradation. In this work, we study how to achieve discrete clustering as well as reliably generalize to unseen data. We propose a novel spectral clustering scheme which deeply explores cluster label properties, including discreteness, nonnegativity, and discrimination, as well as learns robust out-of-sample prediction functions. Specifically, we explicitly enforce a discrete transformation on the intermediate continuous labels, which leads to a tractable optimization problem with a discrete solution. Besides, we preserve the natural nonnegative characteristic of the clustering labels to enhance the interpretability of the results. Moreover, to further compensate the unreliability of the learned clustering labels, we integrate an adaptive robust module with \ell 2,p loss to learn prediction function for grouping unseen data. We also show that the out-of-sample component can inject discriminative knowledge into the learning of cluster labels under certain conditions. Extensive experiments conducted on various data sets have demonstrated the superiority of our proposal as compared to several existing clustering approaches. [ABSTRACT FROM PUBLISHER]

In this paper, we study Stochastic Composite Optimization (SCO) for sparse learning that aims to learn a sparse solution from a composite function. Most of the recent SCO algorithms have already reached the optimal expected convergence rate \mathcal O(1/\lambda T) , but they often fail to deliver sparse solutions at the end either due to the limited sparsity regularization during stochastic optimization (SO) or due to the limitation in online-to-batch conversion. Even when the objective function is strongly convex, their high probability bounds can only attain \mathcal O(\sqrt\log (1/\delta)/T) with $\delta$ is the failure probability, which is much worse than the expected convergence rate. To address these limitations, we propose a simple yet effective two-phase Stochastic Composite Optimization scheme by adding a novel powerful sparse online-to-batch conversion to the general Stochastic Optimization algorithms. We further develop three concrete algorithms, OptimalSL, LastSL and AverageSL, directly under our scheme to prove the effectiveness of the proposed scheme. Both the theoretical analysis and the experiment results show that our methods can really outperform the existing methods at the ability of sparse learning and at the meantime we can improve the high probability bound to approximately \mathcal {O}(\log (\log (T)/\delta)/\lambda T) . [ABSTRACT FROM AUTHOR]

Device-to-device (D2D) communications have recently attracted much attention for the potential capability to improve spectral efficiency (SE) underlaying the existing heterogeneous networks (HetNets). Due to no sophisticated control, D2D-worked user equipments (DUEs) themselves cannot resist eavesdropping or security attacks. It is urgent to maximize the secure capacity for both cellular users and DUEs. This paper formulates the radio resource-allocation problem to maximize the secure capacity of DUEs for D2D communication underlaying HetNets, which consist of high-power nodes (HPNs) and low-power nodes (LPNs). The optimization objective function with transmit bit rate and power constraints, which is nonconvex and hard to directly derive, is first transformed into a matrix form. Then, the equivalent convex form of the optimization problem is derived according to Perron–Frobenius theory. A heuristic iterative algorithm based on the proximal theory is proposed to solve this equivalent convex problem through evaluating the proximal operator of the Lagrange function. Numerical results show that the proposed radio resource-allocation solution significantly improves the secure capacity with a fast convergence speed. [ABSTRACT FROM PUBLISHER]

We consider the problem of image representation from the perspective of statistical design. Recent studies have shown that images are possibly sampled from a low dimensional manifold despite of the fact that the ambient space is usually very high dimensional. Learning low dimensional image representations is crucial for many image processing tasks such as recognition and retrieval. Most of the existing approaches for learning low dimensional representations, such as principal component analysis (PCA) and locality preserving projections (LPP), aim at discovering the geometrical or discriminant structures in the data. In this paper, we take a different perspective from statistical experimental design, and propose a novel dimensionality reduction algorithm called A-Optimal Projection (AOP). AOP is based on a linear regression model. Specifically, AOP finds the optimal basis functions so that the expected prediction error of the regression model can be minimized if the new representations are used for training the model. Experimental results suggest that the proposed approach provides a better representation and achieves higher accuracy in image retrieval. [ABSTRACT FROM PUBLISHER]

Recovering a large matrix from a small subset of its entries is a challenging problem arising in many real applications, such as image inpainting and recommender systems. Many existing approaches formulate this problem as a general low-rank matrix approximation problem. Since the rank operator is nonconvex and discontinuous, most of the recent theoretical studies use the nuclear norm as a convex relaxation. One major limitation of the existing approaches based on nuclear norm minimization is that all the singular values are simultaneously minimized, and thus the rank may not be well approximated in practice. In this paper, we propose to achieve a better approximation to the rank of matrix by truncated nuclear norm, which is given by the nuclear norm subtracted by the sum of the largest few singular values. In addition, we develop a novel matrix completion algorithm by minimizing the Truncated Nuclear Norm. We further develop three efficient iterative procedures, TNNR-ADMM, TNNR-APGL, and TNNR-ADMMAP, to solve the optimization problem. TNNR-ADMM utilizes the alternating direction method of multipliers (ADMM), while TNNR-AGPL applies the accelerated proximal gradient line search method (APGL) for the final optimization. For TNNR-ADMMAP, we make use of an adaptive penalty according to a novel update rule for ADMM to achieve a faster convergence rate. Our empirical study shows encouraging results of the proposed algorithms in comparison to the state-of-the-art matrix completion algorithms on both synthetic and real visual datasets. [ABSTRACT FROM AUTHOR]