tight frame
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2021 ◽  
Vol 118 (43) ◽  
pp. e2103091118
Author(s):  
Cong Fang ◽  
Hangfeng He ◽  
Qi Long ◽  
Weijie J. Su

In this paper, we introduce the Layer-Peeled Model, a nonconvex, yet analytically tractable, optimization program, in a quest to better understand deep neural networks that are trained for a sufficiently long time. As the name suggests, this model is derived by isolating the topmost layer from the remainder of the neural network, followed by imposing certain constraints separately on the two parts of the network. We demonstrate that the Layer-Peeled Model, albeit simple, inherits many characteristics of well-trained neural networks, thereby offering an effective tool for explaining and predicting common empirical patterns of deep-learning training. First, when working on class-balanced datasets, we prove that any solution to this model forms a simplex equiangular tight frame, which, in part, explains the recently discovered phenomenon of neural collapse [V. Papyan, X. Y. Han, D. L. Donoho, Proc. Natl. Acad. Sci. U.S.A. 117, 24652–24663 (2020)]. More importantly, when moving to the imbalanced case, our analysis of the Layer-Peeled Model reveals a hitherto-unknown phenomenon that we term Minority Collapse, which fundamentally limits the performance of deep-learning models on the minority classes. In addition, we use the Layer-Peeled Model to gain insights into how to mitigate Minority Collapse. Interestingly, this phenomenon is first predicted by the Layer-Peeled Model before being confirmed by our computational experiments.


Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1873
Author(s):  
Yanfeng Shen ◽  
Shuli Sun ◽  
Fengsheng Xu ◽  
Yanqin Liu ◽  
Xiuling Yin ◽  
...  

X-ray computed tomography (CT) is widely used in medical applications, where many efforts have been made for decades to eliminate artifacts caused by incomplete projection. In this paper, we propose a new CT image reconstruction model based on nonlocal low-rank regularity and data-driven tight frame (NLR-DDTF). Unlike the Spatial-Radon domain data-driven tight frame regularization, the proposed NLR-DDTF model uses an asymmetric treatment for image reconstruction and Radon domain inpainting, which combines the nonlocal low-rank approximation method for spatial domain CT image reconstruction and data-driven tight frame-based regularization for Radon domain image inpainting. An alternative direction minimization algorithm is designed to solve the proposed model. Several numerical experiments and comparisons are provided to illustrate the superior performance of the NLR-DDTF method.


Author(s):  
Zhiwei Zhang ◽  
Pengfei Chai ◽  
Yong Chen ◽  
Jie Tian ◽  
Hua Ouyang

Abstract Blade tip timing (BTT) data are usually an under-sampled signal and are vulnerable to noise and sensor failures. In this paper , based on an arbitrary-angle compressed-sensing method and equiangular tight frame theory, combined with a niching micro-genetic algorithm, a method for placing BTT sensors is proposed to ensure higher reconstruction accuracy and reliability. If the dimensions of the sensing matrix are moderate, the index range of arrangements with excellent performance in multi-frequency signal reconstruction is determined by enumerating all the uniform-distribution extraction placements. A two-parameter search method is then proposed. Reconstruction of a mixed signal is carried out to verify the asynchronous signal-reconstruction performance. Thus, to achieve a larger frequency multiplication recognition range and probe-installation flexibility, a method for optimizing the BTT sensor placement is proposed. Finally, a finite-element simulation of the signal from an aero-engine fan blade is used to verify the reconstruction ability of the proposed method. The results show that the placement determined by the optimization algorithm can achieve similar or even better performance than the optimal placement under uniform-distribution extraction. The proposed sensor-placement optimization method has a high reconstruction success rate and the BTT system is robust. This approach has significant value for engineering applications.


2021 ◽  
pp. 1-23
Author(s):  
Xiao-Juan Yang ◽  
Jin Jing

Abstract In this paper, we propose a variation model which takes advantage of the wavelet tight frame and nonconvex shrinkage penalties for compressed sensing recovery. We address the proposed optimization problem by introducing a adjustable parameter and a firm thresholding operations. Numerical experiment results show that the proposed method outperforms some existing methods in terms of the convergence speed and reconstruction errors. JEL classification numbers: 68U10, 65K10, 90C25, 62H35. Keywords: Compressed Sensing, Nonconvex, Firm thresholding, Wavelet tight frame.


2021 ◽  
Author(s):  
Zhiwei Zhang ◽  
Pengfei Chai ◽  
Yong Chen ◽  
Jie Tian ◽  
Hua Ouyang

Abstract Blade tip timing (BTT) data are usually an under-sampled signal and are vulnerable to noise and sensor failures. In this paper, based on an arbitrary-angle compressed-sensing method and equiangular tight frame theory, combined with a niching micro-genetic algorithm, a method for placing BTT sensors is proposed to ensure higher reconstruction accuracy and reliability. If the dimensions of the sensing matrix are moderate, the index range of arrangements with excellent performance in multi-frequency signal reconstruction is determined by enumerating all the uniform-distribution extraction placements. A two-parameter search method is then proposed. Reconstruction of a mixed signal is carried out to verify the asynchronous signal-reconstruction performance. Thus, to achieve a larger frequency multiplication recognition range and probe-installation flexibility, a method for optimizing the BTT sensor placement is proposed. Finally, a finite-element simulation of the signal from an aero-engine fan blade is used to verify the reconstruction ability of the proposed method. The results show that the placement determined by the optimization algorithm can achieve similar or even better performance than the optimal placement under uniform-distribution extraction. The proposed sensor-placement optimization method has a high reconstruction success rate and the BTT system is robust. This approach has significant value for engineering applications.


Sensors ◽  
2021 ◽  
Vol 21 (4) ◽  
pp. 1229
Author(s):  
Qiangrong Xu ◽  
Zhichao Sheng ◽  
Yong Fang ◽  
Liming Zhang

Compressed sensing (CS) has been proposed to improve the efficiency of signal processing by simultaneously sampling and compressing the signal of interest under the assumption that the signal is sparse in a certain domain. This paper aims to improve the CS system performance by constructing a novel sparsifying dictionary and optimizing the measurement matrix. Owing to the adaptability and robustness of the Takenaka–Malmquist (TM) functions in system identification, the use of it as the basis function of a sparsifying dictionary makes the represented signal exhibit a sparser structure than the existing sparsifying dictionaries. To reduce the mutual coherence between the dictionary and the measurement matrix, an equiangular tight frame (ETF) based iterative minimization algorithm is proposed. In our approach, we modify the singular values without changing the properties of the corresponding Gram matrix of the sensing matrix to enhance the independence between the column vectors of the Gram matrix. Simulation results demonstrate the promising performance of the proposed algorithm as well as the superiority of the CS system, designed with the constructed sparsifying dictionary and the optimized measurement matrix, over existing ones in terms of signal recovery accuracy.


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