Related papers: Poincar\'e Wasserstein Autoencoder
Data living on manifolds commonly appear in many applications. Often this results from an inherently latent low-dimensional system being observed through higher dimensional measurements. We show that under certain conditions, it is possible…
A standard Variational Autoencoder, with a Euclidean latent space, is structurally incapable of capturing topological properties of certain datasets. To remove topological obstructions, we introduce Diffusion Variational Autoencoders with…
This paper aims at building the theoretical foundations for manifold learning algorithms in the space of absolutely continuous probability measures $\mathcal{P}_{\mathrm{a.c.}}(\Omega)$ with $\Omega$ a compact and convex subset of…
We develop Riemannian approaches to variational autoencoders (VAEs) for PDE-type ambient data with regularizing geometric latent dynamics, which we refer to as VAE-DLM, or VAEs with dynamical latent manifolds. We redevelop the VAE framework…
Autoencoders receive latent models of input data. It was shown in recent works that they also estimate probability density functions of the input. This fact makes using the Bayesian decision theory possible. If we obtain latent models of…
Hyperbolic neural networks have shown great potential for modeling complex data. However, existing hyperbolic networks are not completely hyperbolic, as they encode features in a hyperbolic space yet formalize most of their operations in…
Network data is ubiquitous in various scientific disciplines, including sociology, economics, and neuroscience. Latent space models are often employed in network data analysis, but the geometric effect of latent space curvature remains a…
Autoencoder-based learning has emerged as a staple for disciplining representations in unsupervised and semi-supervised settings. This paper analyzes a framework for improving generalization in a purely supervised setting, where the target…
Taxonomies are valuable resources for many applications, but the limited coverage due to the expensive manual curation process hinders their general applicability. Prior works attempt to automatically expand existing taxonomies to improve…
The extreme anisotropy of hyperbolic materials enables extreme wave confinement, but it is also associated with an inherent misalignment between phase and energy flow, which complicates device modeling and design. Here we introduce a…
Learning hyperbolic embeddings for knowledge graph (KG) has gained increasing attention due to its superiority in capturing hierarchies. However, some important operations in hyperbolic space still lack good definitions, making existing…
Visualization is a crucial step in exploratory data analysis. One possible approach is to train an autoencoder with low-dimensional latent space. Large network depth and width can help unfolding the data. However, such expressive networks…
Learning in the latent variable model is challenging in the presence of the complex data structure or the intractable latent variable. Previous variational autoencoders can be low effective due to the straightforward encoder-decoder…
Variational Autoencoders for multimodal data hold promise for many tasks in data analysis, such as representation learning, conditional generation, and imputation. Current architectures either share the encoder output, decoder input, or…
To gain insight into the mechanisms behind machine learning methods, it is crucial to establish connections among the features describing data points. However, these correlations often exhibit a high-dimensional and strongly nonlinear…
We establish a general scale-dependent Poincar\'{e}-Hardy type identity involving a vector field on the hyperbolic space. By choosing suitable parameter, potential and vector field in this identity, we can recover, as well as derive new…
Theoretical studies and experiments in the last six years have revealed the potential for novel behaviours and functionalities in device physics through the synthetic engineering of negatively-curved spaces. For instance, recent…
The hyperbolic manifold is a smooth manifold of negative constant curvature. While the hyperbolic manifold is well-studied in the literature, it has gained interest in the machine learning and natural language processing communities lately…
In the era of foundation models and Large Language Models (LLMs), Euclidean space is the de facto geometric setting of our machine learning architectures. However, recent literature has demonstrated that this choice comes with fundamental…
Hyperbolic representations are effective in modeling knowledge graph data which is prevalently used to facilitate multi-hop reasoning. However, a rigorous and detailed comparison of the two spaces for this task is lacking. In this paper,…