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Random matrices have played an important role in many fields including machine learning, quantum information theory and optimization. One of the main research focuses is on the deviation inequalities for eigenvalues of random matrices. Although there are intensive studies on the large-deviation inequalities for random matrices, only a few of works discuss the small-deviation behavior of random matrices. In this paper, we present the small-deviation inequalities for the largest eigenvalues of sums of random matrices. Since the resulting inequalities are independent of the matrix dimension, they are applicable to the high-dimensional and even the infinite-dimensional cases.

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Low-light images are a common phenomenon when taking photos in low-light environments with inappropriate camera equipment, leading to shortcomings such as low contrast, color distortion, uneven brightness, and high loss of detail. These shortcomings are not only subjectively annoying but also affect the performance of many computer vision systems. Enhanced low-light images can be better applied to image recognition, object detection and image segmentation. This paper proposes a novel RetinexDIP method to enhance images. Noise is considered as a factor in image decomposition using deep learning generative strategies. The involvement of noise makes the image more real, weakens the coupling relationship between the three components, avoids overfitting, and improves generalization. Extensive experiments demonstrate that our method outperforms existing methods qualitatively and quantitatively.

As major components of the random matrix theory, Gaussian random matrices have been playing an important role in many fields, because they are both unitary invariant and have independent entries and can be used as models for multivariate data or multivariate phenomena. Tail bounds for eigenvalues of Gaussian random matrices are one of the hot study problems. In this paper, we present tail and expectation bounds for the ℓ 1 norm of Gaussian random matrices, respectively. Moreover, the tail and expectation bounds for the ℓ 1 norm of the Gaussian Wigner matrix are calculated based on the resulting bounds. Compared with existing results, our results are more suitable for the high-dimensional matrix case. Finally, we study the tail bounds for the parameter vector of some existing regularization algorithms.