The Canonical Polyadic Decomposition (CPD) is the tensor analog of the Singular Value Decomposition (SVD) for a matrix and has many data science applications including signal processing and machine learning. For the CPD, the Alternating Least Squares (ALS) algorithm has been used extensively. Although the ALS algorithm is simple, it is sensitive to a noise of a data tensor in the applications. In this paper, we propose a novel strategy to realize the noise suppression for the CPD. The proposed strategy is decomposed into two steps: (Step 1) denoising the given tensor and (Step 2) solving the exact CPD of the denoised tensor. Step 1 can be realized by solving a structured low-rank approximation with the Douglas-Rachford splitting algorithm and then Step 2 can be realized by solving the simultaneous diagonalization of a matrix tuple constructed by the denoised tensor with the DODO method. Numerical experiments show that the proposed algorithm works well even in typical cases where the ALS algorithm suffers from the so-called bottleneck/swamp effect.

Approximate Simultaneous Diagonalization (ASD) is a problem to find a common similarity transformation which approximately diagonalizes a given square-matrix tuple. Many data science problems have been reduced into ASD through ingenious modelling. For ASD, the so-called Jacobi-like methods have been extensively used. However, the methods have no guarantee to suppress the magnitude of off-diagonal entries of the transformed tuple even if the given tuple has an exact common diagonalizer, i.e., the given tuple is simultaneously diagonalizable. In this paper, to establish an alternative powerful strategy for ASD, we present a novel two-step strategy, called Approximate-Then-Diagonalize-Simultaneously (ATDS) algorithm. The ATDS algorithm decomposes ASD into (Step 1) finding a simultaneously diagonalizable tuple near the given one; and (Step 2) finding a common similarity transformation which diagonalizes exactly the tuple obtained in Step 1. The proposed approach to Step 1 is realized by solving a Structured Low-Rank Approximation (SLRA) with Cadzow's algorithm. In Step 2, by exploiting the idea in the constructive proof regarding the conditions for the exact simultaneous diagonalizability, we obtain an exact common diagonalizer of the obtained tuple in Step 1 as a solution for the original ASD. Unlike the Jacobi-like methods, the ATDS algorithm has a guarantee to find an exact common diagonalizer if the given tuple happens to be simultaneously diagonalizable. Numerical experiments show that the ATDS algorithm achieves better performance than the Jacobi-like methods.

In the age of big data, recommendation systems provide users with fast access to interesting information, resulting to a significant commercial value. However, the extreme sparseness of user assessment data is one of the key factors that lead to the poor performance of recommendation algorithms. To address this problem, we propose a spectral clustering recommendation scheme with low-rank matrix completion and spectral clustering. Our scheme exploits spectral clustering to achieve the division of a similar user group. Meanwhile, the low-rank matrix completion is used to effectively predict un-rated items in the sub-matrix of the spectral clustering. With the real dataset experiment, the results show that our proposed scheme can effectively improve the prediction accuracy of un-rated items.

A monaural speech enhancement method combining deep neural network (DNN) with low rank analysis and speech present probability is proposed in this letter. Low rank and sparse analysis is first applied on the noisy speech spectrogram to get the approximate low rank representation of noise. Then a joint feature training strategy for DNN based speech enhancement is presented, which helps the DNN better predict the target speech. To reduce the residual noise in highly overlapping regions and high frequency domain, speech present probability (SPP) weighted post-processing is employed to further improve the quality of the speech enhanced by trained DNN model. Compared with the supervised non-negative matrix factorization (NMF) and the conventional DNN method, the proposed method obtains improved speech enhancement performance under stationary and non-stationary conditions.

Human action recognition in videos draws huge research interests in computer vision. The Bag-of-Word model is quite commonly used to obtain the video level representations, however, BoW model roughly assigns each feature vector to its nearest visual word and the collection of unordered words ignores the interest points' spatial information, inevitably causing nontrivial quantization errors and impairing improvements on classification rates. To address these drawbacks, we propose an approach for action recognition by encoding spatio-temporal log Euclidean covariance matrix (ST-LECM) features within the low-rank and sparse representation framework. Motivated by low rank matrix recovery, local descriptors in a spatial temporal neighborhood have similar representation and should be approximately low rank. The learned coefficients can not only capture the global data structures, but also preserve consistent. Experimental results showed that the proposed approach yields excellent recognition performance on synthetic video datasets and are robust to action variability, view variations and partial occlusion.

Compressive spectral imaging (CSI) systems capture the 3D spatiospectral data by measuring the 2D compressed focal plane array (FPA) coded projection with the help of reconstruction algorithms exploiting the sparsity of signals. However, the contradiction between the multi-dimension of the scenes and the limited dimension of the sensors has limited improvement of recovery performance. In order to solve the problem, a novel CSI system based on a coded aperture snapshot spectral imager, RGB-CASSI, is proposed, which has two branches, one for CASSI, another for RGB images. In addition, considering that conventional reconstruction algorithms lead to oversmoothing, a RGB-guided low-rank (RGBLR) method for compressive hyperspectral image reconstruction based on compressed sensing and coded aperture spectral imaging system is presented, in which the available additional RGB information is used to guide the reconstruction and a low-rank regularization for compressive sensing and a non-convex surrogate of the rank is also used instead of nuclear norm for seeking a preferable solution. Experiments show that the proposed algorithm performs better in both PSNR and subjective effects compared with other state-of-art methods.

Subspace representation model is an important subset of visual tracking algorithms. Compared with models performed on the original data space, subspace representation model can effectively reduce the computational complexity, and filter out high dimensional noises. However, for some complicated situations, e.g., dramatic illumination changing, large area of occlusion and abrupt object drifting, traditional subspace representation models may fail to handle the visual tracking task. In this paper, we propose a novel subspace representation algorithm for robust visual tracking by using low-rank representation with graph constraints (LRGC). Low-rank representation has been well-known for its superiority of handling corrupted samples, and graph constraint is flexible to characterize sample relationship. In this paper, we aim to exploit benefits from both low-rank representation and graph constraint, and deploy it to handle challenging visual tracking problems. Specifically, we first propose a novel graph structure to characterize the relationship of target object in different observation states. Then we learn a subspace by jointly optimizing low-rank representation and graph embedding in a unified framework. Finally, the learned subspace is embedded into a Bayesian inference framework by using the dynamical model and the observation model. Experiments on several video benchmarks demonstrate that our algorithm performs better than traditional ones, especially in dynamically changing and drifting situations.

Recovery of low-rank matrices has seen significant activity in many areas of science and engineering, motivated by theoretical results for exact reconstruction guarantees and interesting practical applications. Recently, numerous methods incorporated the nuclear norm to pursue the convexity of the optimization. However, this greatly restricts its capability and flexibility in dealing with many practical problems, where the singular values have clear physical meanings. This paper studies a generalized non-convex low-rank approximation, where the singular values are in lp-heuristic. Then specific results are derived for image restoration, including denoising and deblurring. Extensive experimental results on natural images demonstrate the improvement of the proposed method over the recent image restoration methods.

The alternating direction implicit (ADI) method is proposed for low-rank solution of projected generalized continuous-time algebraic Lyapunov equations. The low-rank solution is expressed by Cholesky factor that is similar to that of Cholesky factorization for linear system of equations. The Cholesky factor is represented in a real form so that it is useful for balanced truncation of sparsely connected RLC networks. Moreover, we show how to determine the shift parameters which are required for the ADI iterations, where Krylov subspace method is used for finding the shift parameters that reduce the residual error quickly. In the illustrative examples, we confirm that the real Cholesky factor certainly provides low-rank solution of projected generalized continuous-time algebraic Lyapunov equations. Effectiveness of the shift parameters determined by Krylov subspace method is also demonstrated.

Detecting small infrared targets is a difficult but important task in highly cluttered coastal surveillance. The paper proposed a method called low-rank and sparse decomposition based frame difference to improve the detection performance of a surveillance system. First, the frame difference is used in adjacent frames to detect the candidate object regions which we are most interested in. Then we further exclude clutters by low-rank and sparse matrix recovery. Finally, the targets are extracted from the recovered target component by a local self-adaptive threshold. The experiment results show that, the method could effectively enhance the system's signal-to-clutter ratio gain and background suppression factor, and precisely extract target in highly cluttered coastal scene.

Traditional low-rank feature lose the temporal information among action sequence. To obtain the temporal information, we split an action video into multiple action subsequences and concatenate all the low-rank features of subsequences according to their time order. Then we recognize actions by learning a novel dictionary model from concatenated low-rank features. However, traditional dictionary learning models usually neglect the similarity among the coding coefficients and have bad performance in dealing with non-linearly separable data. To overcome these shortcomings, we present a novel similarity constrained discriminative kernel dictionary learning for action recognition. The effectiveness of the proposed method is verified on three benchmarks, and the experimental results show the promising results of our method for action recognition.

Robust crater recognition is a research focus on deep space exploration mission, and sparse representation methods can achieve desirable robustness and accuracy. Due to destruction and noise incurred by complex topography and varied illumination in planetary images, a robust crater recognition approach is proposed based on dictionary learning with a low-rank error correction model in a sparse representation framework. In this approach, all the training images are learned as a compact and discriminative dictionary. A low-rank error correction term is introduced into the dictionary learning to deal with gross error and corruption. Experimental results on crater images show that the proposed method achieves competitive performance in both recognition accuracy and efficiency.

Chronic liver disease is a major worldwide health problem. Diagnosis and staging of chronic liver diseases is an important issue. In this paper, we propose a quantitative method of analyzing local morphological changes for accurate and practical computer-aided diagnosis of cirrhosis. Our method is based on sparse and low-rank matrix decomposition, since the matrix of the liver shapes can be decomposed into two parts: a low-rank matrix, which can be considered similar to that of a normal liver, and a sparse error term that represents the local deformation. Compared with the previous global morphological analysis strategy based on the statistical shape model (SSM), our proposed method improves the accuracy of both normal and abnormal classifications. We also propose using the norm of the sparse error term as a simple measure for classification as normal or abnormal. The experimental results of the proposed method are better than those of the state-of-the-art SSM-based methods.

Kernel methods such as the support vector machine are one of the most successful algorithms in modern machine learning. Their advantage is that linear algorithms are extended to non-linear scenarios in a straightforward way by the use of the kernel trick. However, naive use of kernel methods is computationally expensive since the computational complexity typically scales cubically with respect to the number of training samples. In this article, we review recent advances in the kernel methods, with emphasis on scalability for massive problems.