For the real planar autonomous differential system, the questionsof detection between center and focus, successor function, formal series, central integration, integration factor, focal values, values of singular poin...For the real planar autonomous differential system, the questionsof detection between center and focus, successor function, formal series, central integration, integration factor, focal values, values of singular point and bifurcation of limit cycles for a class of higher-degree critical points and infinite points are expounded.展开更多
The convolution operation possesses the characteristic of translation group equivariance. To achieve more group equivariances, rotation group equivariant convolutions(RGEC) are proposed to acquire both translation and...The convolution operation possesses the characteristic of translation group equivariance. To achieve more group equivariances, rotation group equivariant convolutions(RGEC) are proposed to acquire both translation and rotation group equivariances.However, previous work paid more attention to the number of parameters and usually ignored other resource costs. In this paper, we construct our networks without introducing extra resource costs. Specifically, a convolution kernel is rotated to different orientations for feature extractions of multiple channels. Meanwhile, much fewer kernels than previous works are used to ensure that the output channel does not increase. To further enhance the orthogonality of kernels in different orientations, we construct the non-maximum-suppression loss on the rotation dimension to suppress the other directions except the most activated one. Considering that the low-level-features benefit more from the rotational symmetry, we only share weights in the shallow layers(SWSL) via RGEC. Extensive experiments on multiple datasets(i.e., Image Net, CIFAR, and MNIST) demonstrate that SWSL can effectively benefit from the higher-degree weight sharing and improve the performances of various networks, including plain and Res Net architectures. Meanwhile, the convolutional kernels and parameters are much fewer(e.g., 75%, 87.5% fewer) in the shallow layers, and no extra computation costs are introduced.展开更多
基金This work was supported by the Natural Science Foundation of Hunan Province (Grant No. 10071016) .
文摘For the real planar autonomous differential system, the questionsof detection between center and focus, successor function, formal series, central integration, integration factor, focal values, values of singular point and bifurcation of limit cycles for a class of higher-degree critical points and infinite points are expounded.
基金supported by National Natural Science Foundation of China(Nos.61976209 and 62020106015)CAS International Collaboration Key Project(No.173211KYSB20190024)Strategic Priority Research Program of CAS(No.XDB32040000)。
文摘The convolution operation possesses the characteristic of translation group equivariance. To achieve more group equivariances, rotation group equivariant convolutions(RGEC) are proposed to acquire both translation and rotation group equivariances.However, previous work paid more attention to the number of parameters and usually ignored other resource costs. In this paper, we construct our networks without introducing extra resource costs. Specifically, a convolution kernel is rotated to different orientations for feature extractions of multiple channels. Meanwhile, much fewer kernels than previous works are used to ensure that the output channel does not increase. To further enhance the orthogonality of kernels in different orientations, we construct the non-maximum-suppression loss on the rotation dimension to suppress the other directions except the most activated one. Considering that the low-level-features benefit more from the rotational symmetry, we only share weights in the shallow layers(SWSL) via RGEC. Extensive experiments on multiple datasets(i.e., Image Net, CIFAR, and MNIST) demonstrate that SWSL can effectively benefit from the higher-degree weight sharing and improve the performances of various networks, including plain and Res Net architectures. Meanwhile, the convolutional kernels and parameters are much fewer(e.g., 75%, 87.5% fewer) in the shallow layers, and no extra computation costs are introduced.
