We present a rotated hyperbolic wrapped normal distribution (RoWN), a simple yet effective alteration of a hyperbolic wrapped normal distribution (HWN). The HWN expands the domain of probabilistic modeling from Euclidean to hyperbolic space, where a tree can be embedded with arbitrary low distortion in theory. In this work, we analyze the geometric properties of the diagonal HWN, a standard choice of distribution in probabilistic modeling. The analysis shows that the distribution is inappropriate to represent the data points at the same hierarchy level through their angular distance with the same norm in the Poincar\'e disk model. We then empirically verify the presence of limitations of HWN, and show how RoWN, the proposed distribution, can alleviate the limitations on various hierarchical datasets, including noisy synthetic binary tree, WordNet, and Atari 2600 Breakout.

Hyperbolic space has served as an effective medium to learn parsimonious representations of hierarchical data, including vocabulary with relationships (Nickel and Kiela, 2017, 2018; Tifrea et al.,

We present a rotated hyperbolic wrapped normal distribution (RoWN), a simple yet effective alteration of a hyperbolic wrapped normal distribution (HWN). The HWN expands the domain of probabilistic modeling from Euclidean to hyperbolic space, where a tree can be embedded with arbitrary low distortion in theory. In this work, we analyze the geometric properties of the diagonal HWN, a standard choice of distribution in probabilistic modeling. The analysis shows that the distribution is inappropriate to represent the data points at the same hierarchy level through their angular distance with the same norm in the Poincar\'e disk model. We then empirically verify the presence of limitations of HWN, and show how RoWN, the proposed distribution, can alleviate the limitations on various hierarchical datasets, including noisy synthetic binary tree, WordNet, and Atari 2600 Breakout.

Clustered Federated Multitask Learning (CFL) has gained considerable attention as an effective strategy for overcoming statistical challenges, particularly when dealing with non independent and identically distributed (non IID) data across multiple users. However, much of the existing research on CFL operates under the unrealistic premise that devices have access to accurate ground truth labels. This assumption becomes especially problematic in hierarchical wireless networks (HWNs), where edge networks contain a large amount of unlabeled data, resulting in slower convergence rates and increased processing times, particularly when dealing with two layers of model aggregation. To address these issues, we introduce a novel framework, Clustered Federated Semi-Supervised Learning (CFSL), designed for more realistic HWN scenarios. Our approach leverages a best-performing specialized model algorithm, wherein each device is assigned a specialized model that is highly adept at generating accurate pseudo-labels for unlabeled data, even when the data stems from diverse environments. We validate the efficacy of CFSL through extensive experiments, comparing it with existing methods highlighted in recent literature. Our numerical results demonstrate that CFSL significantly improves upon key metrics such as testing accuracy, labeling accuracy, and labeling latency under varying proportions of labeled and unlabeled data while also accommodating the non-IID nature of the data and the unique characteristics of wireless edge networks.

We present a rotated hyperbolic wrapped normal distribution (RoWN), a simple yet effective alteration of a hyperbolic wrapped normal distribution (HWN). The HWN expands the domain of probabilistic modeling from Euclidean to hyperbolic space, where a tree can be embedded with arbitrary low distortion in theory. In this work, we analyze the geometric properties of the diagonal HWN, a standard choice of distribution in probabilistic modeling. The analysis shows that the distribution is inappropriate to represent the data points at the same hierarchy level through their angular distance with the same norm in the Poincar\'e disk model. We then empirically verify the presence of limitations of HWN, and show how RoWN, the proposed distribution, can alleviate the limitations on various hierarchical datasets, including noisy synthetic binary tree, WordNet, and Atari 2600 Breakout. The code is available at https://github.com/ml-postech/RoWN.