365 research outputs found
Spectral multipliers for maximally subelliptic operators
Consider a non-negative, self-adjoint, maximally subelliptic operator on a
compact manifold. We show that the spectral multiplier is a singular integral
operator under an appropriate Mihlin-H\"ormander type condition. We also give a
Mihlin-H\"ormander type condition for the boundedness of the spectral
multiplier on non-isotropic Sobolev spaces. We establish the equivalence
between non-isotropic Besov and Triebel-Lizorkin spaces adapted to the operator
and those adapted to a Carnot-Carath\'eodory geometry on the manifold.Comment: 21 page
The Euclidean Space is Evil: Hyperbolic Attribute Editing for Few-shot Image Generation
Few-shot image generation is a challenging task since it aims to generate
diverse new images for an unseen category with only a few images. Existing
methods suffer from the trade-off between the quality and diversity of
generated images. To tackle this problem, we propose Hyperbolic Attribute
Editing (HAE), a simple yet effective method. Unlike other methods that work in
Euclidean space, HAE captures the hierarchy among images using data from seen
categories in hyperbolic space. Given a well-trained HAE, images of unseen
categories can be generated by moving the latent code of a given image toward
any meaningful directions in the Poincar\'e disk with a fixing radius. Most
importantly, the hyperbolic space allows us to control the semantic diversity
of the generated images by setting different radii in the disk. Extensive
experiments and visualizations demonstrate that HAE is capable of not only
generating images with promising quality and diversity using limited data but
achieving a highly controllable and interpretable editing process
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