GoDec+: Fast and Robust Low-Rank Matrix Decomposition Based on Maximum Correntropy

Compact Model Training by Low-Rank Projection With Energy Transfer

Low-rankness plays an important role in traditional machine learning but is not so popular in deep learning. Most previous low-rank network compression methods compress networks by approximating pretrained models and retraining. However, the optimal solution in the Euclidean space may be quite different from the one with low-rank constraint. A well-pretrained model is not a good initialization for the model with low-rank constraints. Thus, the performance of a low-rank compressed network degrades significantly. Compared with other network compression methods such as pruning, low-rank methods attract less attention in recent years. In this article, we devise a new training method, low-rank projection with energy transfer (LRPET), that trains low-rank compressed networks from scratch and achieves competitive performance. We propose to alternately perform stochastic gradient descent training and projection of each weight matrix onto the corresponding low-rank manifold. Compared to retraining on the compact model, this enables full utilization of model capacity since solution space is relaxed back to Euclidean space after projection. The matrix energy (the sum of squares of singular values) reduction caused by projection is compensated by energy transfer. We uniformly transfer the energy of the pruned singular values to the remaining ones. We theoretically show that energy transfer eases the trend of gradient vanishing caused by projection. In modern networks, a batch normalization (BN) layer can be merged into the previous convolution layer for inference, thereby influencing the optimal low-rank approximation (LRA) of the previous layer. We propose BN rectification to cut off its effect on the optimal LRA, which further improves the performance. Comprehensive experiments on CIFAR-10 and ImageNet have justified that our method is superior to other low-rank compression methods and also outperforms recent state-of-the-art pruning methods. For object detection and semantic segmentation, our method still achieves good compression results. In addition, we combine LRPET with quantization and hashing methods and achieve even better compression than the original single method. We further apply it in Transformer-based models to demonstrate its transferability. Our code is available at https://github.com/BZQLin/LRPET.

Noise Robust Face Hallucination Based on Smooth Correntropy Representation

Face hallucination technologies have been widely developed during the past decades, among which the sparse manifold learning (SML)-based approaches have become the popular ones and achieved promising performance. However, these SML methods always failed in handling noisy images due to the least-square regression (LSR) they used for error approximation. To this end, we propose, in this article, a smooth correntropy representation (SCR) model for noisy face hallucination. In SCR, the correntropy regularization and smooth constraint are combined into one unified framework to improve the resolution of noisy face images. Specifically, we introduce the correntropy induced metric (CIM) rather than the LSR to regularize the encoding errors, which admits the proposed method robust to noise with uncertain distributions. Besides, the fused LASSO penalty is added into the feature space to ensure similar training samples holding similar representation coefficients. This encourages the SCR not only robust to noise but also can well exploit the inherent typological structure of patch manifold, resulting in more accurate representations in noise environment. Comparison experiments against several state-of-the-art methods demonstrate the superiority of SCR in super-resolving noisy low-resolution (LR) face images.

Weighted Error Entropy-Based Information Theoretic Learning for Robust Subspace Representation

In most of the existing representation learning frameworks, the noise contaminating the data points is often assumed to be independent and identically distributed (i.i.d.), where the Gaussian distribution is often imposed. This assumption, though greatly simplifies the resulting representation problems, may not hold in many practical scenarios. For example, the noise in face representation is usually attributable to local variation, random occlusion, and unconstrained illumination, which is essentially structural, and hence, does not satisfy the i.i.d. property or the Gaussianity. In this article, we devise a generic noise model, referred to as independent and piecewise identically distributed (i.p.i.d.) model for robust presentation learning, where the statistical behavior of the underlying noise is characterized using a union of distributions. We demonstrate that our proposed i.p.i.d. model can better describe the complex noise encountered in practical scenarios and accommodate the traditional i.i.d. one as a special case. Assisted by the proposed noise model, we then develop a new information-theoretic learning framework for robust subspace representation through a novel minimum weighted error entropy criterion. Thanks to the superior modeling capability of the i.p.i.d. model, our proposed learning method achieves superior robustness against various types of noise. When applying our scheme to the subspace clustering and image recognition problems, we observe significant performance gains over the existing approaches.

Robust Flexible Preserving Embedding

Neighborhood preserving embedding (NPE) has been proposed to encode overall geometry manifold embedding information. However, the class-special structure of the data is destroyed by noise or outliers existing in the data. To address this problem, in this article, we propose a novel embedding approach called robust flexible preserving embedding (RFPE). First, RFPE recovers the noisy data by low-rank learning and obtains clean data. Then, the clean data are used to learn the projection matrix. In this way, the projective learning is totally unaffected by noise or outliers. By encoding a flexible regularization term, RFPE can keep the property of the data points with a nonlinear manifold and be more flexible. RFPE searches the optimal projective subspace for feature extraction. In addition, we also extend the proposed RFPE to a kernel case and propose kernel RFPE (KRFPE). Extensive experiments on six public image databases show the superiority of the proposed methods over other state-of-the-art methods.

Laplacian-Uniform Mixture-Driven Iterative Robust Coding With Applications to Face Recognition Against Dense Errors

Outliers due to occlusion, pixel corruption, and so on pose serious challenges to face recognition despite the recent progress brought by sparse representation. In this article, we show that robust statistics implemented by the state-of-the-art methods are insufficient for robustness against dense gross errors. By modeling the distribution of coding residuals with a Laplacian-uniform mixture, we obtain a sparse representation that is significantly more robust than the previous methods. The nonconvex error term of the implemented objective function is nondifferentiable at zero and cannot be properly addressed by the usual iteratively reweighted least-squares formulation. We show that an iterative robust coding algorithm can be derived by local linear approximation of the nonconvex error term, which is both effective and efficient. With iteratively reweighted l₁ minimization of the error term, the proposed algorithm is capable of handling the sparsity assumption of the coding errors more appropriately than the previous methods. Notably, it has the distinct property of addressing error detection and error correction cooperatively in the robust coding process. The proposed method demonstrates significantly improved robustness for face recognition against dense gross errors, either contiguous or discontiguous, as verified by extensive experiments.

Bayesian deep matrix factorization network for multiple images denoising

This paper aims at proposing a robust and fast low rank matrix factorization model for multiple images denoising. To this end, a novel model, Bayesian deep matrix factorization network (BDMF), is presented, where a deep neural network (DNN) is designed to model the low rank components and the model is optimized via stochastic gradient variational Bayes. By the virtue of deep learning and Bayesian modeling, BDMF makes significant improvement on synthetic experiments and real-world tasks (including shadow removal and hyperspectral image denoising), compared with existing state-of-the-art models.

PCA via joint graph Laplacian and sparse constraint: Identification of differentially expressed genes and sample clustering on gene expression data

Background

In recent years, identification of differentially expressed genes and sample clustering have become hot topics in bioinformatics. Principal Component Analysis (PCA) is a widely used method in gene expression data. However, it has two limitations: first, the geometric structure hidden in data, e.g., pair-wise distance between data points, have not been explored. This information can facilitate sample clustering; second, the Principal Components (PCs) determined by PCA are dense, leading to hard interpretation. However, only a few of genes are related to the cancer. It is of great significance for the early diagnosis and treatment of cancer to identify a handful of the differentially expressed genes and find new cancer biomarkers.

Results

In this study, a new method gLSPCA is proposed to integrate both graph Laplacian and sparse constraint into PCA. gLSPCA on the one hand improves the clustering accuracy by exploring the internal geometric structure of the data, on the other hand identifies differentially expressed genes by imposing a sparsity constraint on the PCs.

Conclusions

Experiments of gLSPCA and its comparison with existing methods, including Z-SPCA, GPower, PathSPCA, SPCArt, gLPCA, are performed on real datasets of both pancreatic cancer (PAAD) and head & neck squamous carcinoma (HNSC). The results demonstrate that gLSPCA is effective in identifying differentially expressed genes and sample clustering. In addition, the applications of gLSPCA on these datasets provide several new clues for the exploration of causative factors of PAAD and HNSC.

Accurate vessel extraction via tensor completion of background layer in X-ray coronary angiograms

This paper proposes an effective method for accurately recovering vessel structures and intensity information from the X-ray coronary angiography (XCA) images of moving organs or tissues. Specifically, a global logarithm transformation of XCA images is implemented to fit the X-ray attenuation sum model of vessel/background layers into a low-rank, sparse decomposition model for vessel/background separation. The contrast-filled vessel structures are extracted by distinguishing the vessels from the low-rank backgrounds by using a robust principal component analysis and by constructing a vessel mask via Radon-like feature filtering plus spatially adaptive thresholding. Subsequently, the low-rankness and inter-frame spatio-temporal connectivity in the complex and noisy backgrounds are used to recover the vessel-masked background regions using tensor completion of all other background regions, while the twist tensor nuclear norm is minimized to complete the background layers. Finally, the method is able to accurately extract vessels' intensities from the noisy XCA data by subtracting the completed background layers from the overall XCA images. We evaluated the vessel visibility of resulting images on real X-ray angiography data and evaluated the accuracy of vessel intensity recovery on synthetic data. Experiment results show the superiority of the proposed method over the state-of-the-art methods.