arXiv subjects
Publications and source records attributed to Jiachen Kang.
We define a class of links in handlebodies called ``charm bracelets," which are a subset of staked links. We provide tools to construct infinitely many such hyperbolic links and bound the corresponding volumes from below in terms of volumes corresponding to the individual charms.
Let $M_0$ be a compact and orientable 3-manifold. After capping off spherical boundaries with balls and removing any torus boundaries, we prove that the resulting manifold $M$ contains handlebodies of arbitrary genus such that the closure of their complement is hyperbolic. We then extend the octahedral decomposition to obtain bounds on volume for some of these handlebody complements.
We consider a simple but infinite class of staked links known as bongles. We provide necessary and sufficient conditions for these bongles to be hyperbolic. Then, we prove that all balanced hyperbolic $n$-bongles have the same volume and the corresponding volume is an upper bound on the volume of any hyperbolic $n$-bongle for $n$ even. Moreover, all hyperbolic $n$-bongles have volume strictly less than $5n(1.01494\dots)$. We also include explicit volume calculations for all hyperbolic 3-bongles through 6-bongles.
Self-supervised representation learning (SSRL) has gained increasing attention in point cloud understanding, in addressing the challenges posed by 3D data scarcity and high annotation costs. This paper presents PCExpert, a novel SSRL approach that reinterprets point clouds as "specialized images". This conceptual shift allows PCExpert to leverage knowledge derived from large-scale image modality in a more direct and deeper manner, via extensively sharing the parameters with a pre-trained image encoder in a multi-way Transformer architecture. The parameter sharing strategy, combined with a novel pretext task for pre-training, i.e., transformation estimation, empowers PCExpert to outperform the state of the arts in a variety of tasks, with a remarkable reduction in the number of trainable parameters. Notably, PCExpert's performance under LINEAR fine-tuning (e.g., yielding a 90.02% overall accuracy on ScanObjectNN) has already approached the results obtained with FULL model fine-tuning (92.66%), demonstrating its effective and robust representation capability.
The existing deep learning models suffer from out-of-distribution (o.o.d.) performance drop in computer vision tasks. In comparison, humans have a remarkable ability to interpret images, even if the scenes in the images are rare, thanks to the systematicity of acquired knowledge. This work focuses on 1) the acquisition of systematic knowledge of 2D transformations, and 2) architectural components that can leverage the learned knowledge in image classification tasks in an o.o.d. setting. With a new training methodology based on synthetic datasets that are constructed under the causal framework, the deep neural networks acquire knowledge from semantically different domains (e.g. even from noise), and exhibit certain level of systematicity in parameter estimation experiments. Based on this, a novel architecture is devised consisting of a classifier, an estimator and an identifier (abbreviated as "CED"). By emulating the "hypothesis-verification" process in human visual perception, CED improves the classification accuracy significantly on test sets under covariate shift.