Deformations of higher-page analogues of $$\partial {{\bar{\partial }}}$$-manifolds

We extend the notion of small essential deformations of Calabi–Yau complex structures from the case of the Iwasawa manifold, for which they were introduced recently by the first-named author, to the general case of page-1-$$\partial {{\bar{\partial }}}$$ ∂ ∂ ¯ -manifolds that were jointly introduced very recently by all three authors. We go on to obtain an analogue of the unobstructedness theorem of Bogomolov, Tian and Todorov for Calabi–Yau page-1-$$\partial {{\bar{\partial }}}$$ ∂ ∂ ¯ -manifolds. As applications of this discussion, we study the small deformations of certain Nakamura solvmanifolds and reinterpret the cases of the Iwasawa manifold and its 5-dimensional analogue from this standpoint.

Scene Image Retrieval Based on Manifold Structures of Canonical Images

Image retrieval methods have been dramatically developed in the last decade. In this paper, we propose a novel method for image retrieval based on manifold structures of canonical images. Firstly, we present the image normalization process to find a set of canonical images that anchors the probabilistic distributions around the real data manifolds to learn the representations that better encode the manifold structures in general high-dimensional image space. In addition, we employ the canonical images as the centers of the conditional multivariate Gaussian distributions. This approach allows to learn more detailed structures of the partial manifolds resulting in improved representation of the high level properties of scene images. Furthermore, we use the probabilistic framework of the extended model to retrieve images based on the similarity measure of reciprocal likelihood of pairs of images and the sum of likelihood of one of two images based on the other’s best distributions. We estimate our method using SUN database. In the experiments on scene image retrieval, the proposed method is efficient, and exhibits superior capabilities compared to other methods, such as GIST.