Related papers: Topological Constraints on Homeomorphic Auto-Encod…
The manifold hypothesis states that many kinds of high-dimensional data are concentrated near a low-dimensional manifold. If the topology of this data manifold is non-trivial, a continuous encoder network cannot embed it in a one-to-one…
Topological classification of the 4-manifolds bridges computation theory and physics. A proof of the undecidability of the homeomorphy problem for 4-manifolds is outlined here in a clarifying way. It is shown that an arbitrary Turing…
Given a "data manifold" $M\subset \mathbb{R}^n$ and "latent space" $\mathbb{R}^\ell$, an autoencoder is a pair of continuous maps consisting of an "encoder" $E\colon \mathbb{R}^n\to \mathbb{R}^\ell$ and "decoder" $D\colon \mathbb{R}^\ell\to…
Incorporating geometric inductive biases into models can aid interpretability and generalization, but encoding to a specific geometric structure can be challenging due to the imposed topological constraints. In this paper, we theoretically…
We propose a novel approach for preserving topological structures of the input space in latent representations of autoencoders. Using persistent homology, a technique from topological data analysis, we calculate topological signatures of…
Autoencoding is a popular method in representation learning. Conventional autoencoders employ symmetric encoding-decoding procedures and a simple Euclidean latent space to detect hidden low-dimensional structures in an unsupervised way.…
Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…
We prove that under some assumptions on how points escape to infinity in the universal cover, homeomorphisms of hyperbolic 3-manifolds are forced to have several invariant sets (in particular, they cannot be minimal). For this, we use some…
A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded C1-submanifold without boundary in…
Choosing a suitable filtering function for the Mapper algorithm can be difficult due to its arbitrariness and domain-specific requirements. Finding a general filtering function that can be applied across domains is therefore of interest,…
This paper presents the geometric aspect of the autoencoder framework, which, despite its importance, has been relatively less recognized. Given a set of high-dimensional data points that approximately lie on some lower-dimensional…
Autoencoders have been proposed as a powerful tool for model-independent anomaly detection in high-energy physics. The operating principle is that events which do not belong to the space of training data will be reconstructed poorly, thus…
Representing a manifold of very high-dimensional data with generative models has been shown to be computationally efficient in practice. However, this requires that the data manifold admits a global parameterization. In order to represent…
Autoencoders are commonly used in representation learning. They consist of an encoder and a decoder, which provide a straightforward way to map n-dimensional data in input space to a lower m-dimensional representation space and back. The…
The encoder network of an autoencoder is an approximation of the nearest point projection onto the manifold spanned by the decoder. A concern with this approximation is that, while the output of the encoder is always unique, the projection…
Omnidirectional images and spherical representations of $3D$ shapes cannot be processed with conventional 2D convolutional neural networks (CNNs) as the unwrapping leads to large distortion. Using fast implementations of spherical and…
We present a topological proof of the existence of invariant manifolds for maps with normally hyperbolic-like properties. The proof is conducted in the phase space of the system. In our approach we do not require that the map is a…
Manifold learning aims to discover and represent low-dimensional structures underlying high-dimensional data while preserving critical topological and geometric properties. Existing methods often fail to capture local details with global…
We consider closed and orientable immersed hypersurfaces of translational manifolds. Given a vector field on such a hypersurface, we define a perturbation of its Gauss map, which allows us to obtain topological invariants for the immersion…
Among plenty of applications, low-dimensional homogeneous spaces appear in cosmological models as both, classical factor spaces of multidimensional geometry and minisuperspaces in canonical quantization. Here a new tool to restrict their…