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We characterize conformally flat spaces as the only compact self-dual manifolds which are U(1)-equivariantly and conformally decomposable into two complete self-dual Einstein manifolds with common conformal infinity. A geometric characterization of such conformally flat spaces is also given.

Let <i>H</i> denote a spherical subgroup within a semisimple algebraic group <i>G</i>. In this paper we study the closures of the finitely many <i>H</i>-orbits in the flag variety of <i>G</i>. Using the language of Frobenius splitting we provide a criterion for these closures to have nice geometric and cohomological properties. We then show how the criterion applies to the spherical subgroups of minimal rank studied by N. Ressayre. Finally, we also provide applications of the criterion to orbit closures which are not multiplicity-free in the sense defined by M. Brion.

Remarkable achievements have been obtained by deep neural networks in the last several years. However, the breakthrough in neural networks accuracy is always accompanied by explosive growth of computation and parameters, which leads to a severe limitation of model deployment. In this paper, we propose a novel knowledge distillation technique named self-distillation to address this problem. Self-distillation attaches several attention modules and shallow classifiers at different depths of neural networks and distills knowledge from the deepest classifier to the shallower classifiers. Different from the conventional knowledge distillation methods where the knowledge of the teacher model is transferred to another student model, self-distillation can be considered as knowledge transfer in the same model - from the deeper layers to the shallow layers. Moreover, the additional classifiers in self-distillation allow the neural network to work in a dynamic manner, which leads to a much higher acceleration. Experiments demonstrate that self-distillation has consistent and significant effectiveness on various neural networks and datasets. On average, 3.49% and 2.32% accuracy boost are observed on CIFAR100 and ImageNet. Besides, experiments show that self-distillation can be combined with other model compression methods, including knowledge distillation, pruning and lightweight model design.

Harmonic functions are the critical points of a Dirichlet energy functional, the linear projections of conformal maps. They play an important role in computer graphics, particularly for gradient-domain image processing and shape-preserving geometric computation. We propose Poisson coordinates, a novel transfinite interpolation scheme based on the Poisson integral formula, as a rapid way to estimate a harmonic function on a certain domain with desired boundary values. Poisson coordinates are an extension of the Mean Value coordinates (MVCs) which inherit their linear precision, smoothness, and kernel positivity. We give explicit formulae for Poisson coordinates in both continuous and 2D discrete forms. Superior to MVCs, Poisson coordinates are proved to be pseudoharmonic (i.e., they reproduce harmonic functions on n-dimensional balls). Our experimental results show that Poisson coordinates have lower Dirichlet energies than MVCs on a number of typical 2D domains (particularly convex domains). As well as presenting a formula, our approach provides useful insights for further studies on coordinates-based interpolation and fast estimation of harmonic functions.