Related papers: Non-Euclidean Self-Organizing Maps
In this paper, we introduce and study the concepts of semi open SOM) and semi closed (SCM) M-sets in multiset topological spaces.With this generalization of the notions of open and closed sets in M-topology, we generalize the concept of…
I describe the manifestation of the non-Euclidean geometry in the behavior of collective observables of some complex physical systems. Specifically, I consider the formation of equilibrium shapes of plants and statistics of sparse random…
We introduce aweSOM, an open-source Python package for machine learning (ML) clustering and classification, using a Self-organizing Maps (SOM) algorithm that incorporates CPU/GPU acceleration to accommodate large ($N > 10^6$, where $N$ is…
A smooth map between smooth manifolds is called a special generic map if it has only definite fold points as its singularities. In this paper, we give conditions for a special generic map into the 3-dimensional Euclidean space to be…
We present simplicial neural networks (SNNs), a generalization of graph neural networks to data that live on a class of topological spaces called simplicial complexes. These are natural multi-dimensional extensions of graphs that encode not…
We show how complexity theory can be introduced in machine learning to help bring together apparently disparate areas of current research. We show that this new approach requires less training data and is more generalizable as it shows…
This work describes the implementation and application of a correlation determination method based on Self Organizing Maps and Bayesian Inference (SOMBI). SOMBI aims to automatically identify relations between different observed parameters…
Unsupervised clustering, also known as natural clustering, stands for the classification of data according to their similarities. Here we study this problem from the perspective of complex networks. Mapping the description of data…
Piecewise Euclidean structures (identified solid Euclidean polyhedra) on topological 3-dimensional manifolds and pseudo-manifolds are constructed so that they admit pseudo-foliations, a generalized type of foliation. The construction of…
Graph neural networks have pushed state-of-the-arts in graph classifications recently. Typically, these methods are studied within the context of supervised end-to-end training, which necessities copious task-specific labels. However, in…
A map from a manifold to a Euclidean space is said to be k-regular if the image of any distinct k points are linearly independent. In this paper, we give some lower bounds of the dimension of the ambient Euclidean space for complex…
Modern information processing relies on the axiom that high-dimensional data lie near low-dimensional geometric structures. This paper revisits the problem of data-driven learning of these geometric structures and puts forth two new…
Embedding graphs in continous spaces is a key factor in designing and developing algorithms for automatic information extraction to be applied in diverse tasks (e.g., learning, inferring, predicting). The reliability of graph embeddings…
In machine learning and computer graphics, a fundamental task is the approximation of a probability density function through a well-dispersed collection of samples. Providing a formal metric for measuring the distance between probability…
We consider different ways to control the magnification in self-organizing maps (SOM) and neural gas (NG). Starting from early approaches of magnification control in vector quantization, we then concentrate on different approaches for SOM…
We introduce a new notion, called quasi-holomorphic maps. These are real smooth maps equipped with a structure that imitates the singularities and singularity stratifications of holomorphic maps on the source and target manifolds, although…
The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators,…
Topographic feature maps are low dimensional representations of data, that preserve spatial dependencies. Current methods of training such maps (e.g. self organizing maps - SOM, generative topographic maps) require centralized control and…
We present Self-Organizing Visual Prototypes (SOP), a new training technique for unsupervised visual feature learning. Unlike existing prototypical self-supervised learning (SSL) methods that rely on a single prototype to encode all…
We introduce and study a class of Thurston maps from the 2-sphere to itself which we call nearly Euclidean Thurston (NET) maps. These are simple generalizations of Euclidean Thurston maps.