International Scientific Journals
of Scientific Technical Union of Mechanical Engineering "Industry 4.0"

This paper examines the geometric structure of sentence embeddings through the lens of persistent homology. The goal is to determine whether semantic similarity produces distinctive topological patterns in a controlled embedding environment. To isolate semantic effects, a single sentence template was combined with different target words, forming two point clouds in a transformer embedding space: one derived from semantically similar words and one from dissimilar words. A Vietoris–Rips filtration was applied to both clouds, and the resulting persistence diagrams were summarized by average lifetime, entropy of birth–death intervals, and the area under the Betti curve. The results show a coherent difference across topological dimensions: similar words generate stable connected components with lower variability, while dissimilar words produce a richer set of cycle features that persist across a broader range of scales. These findings indicate that persistent homology can capture multi-scale structural differences in embedding spaces that are not visible through standard distance-based comparisons. Although the experiment is intentionally simple, it highlights the potential of topological methods for studying how semantic structure is distributed across levels of a neural embedding space.

Topological Data Analysis (TDA) is relatively new filed of Applied Mathematics that emerged rapidly last years. The main tool of Topological Data Analysis is Persistent Homology. Persistent Homology provides some topological characteristics of the datasets. In this paper we will discuss classification of digital images using their topological signatures computed with Persistent Homology. We will experiment on the Fashion-MNIST dataset. Using Topological Data Analysis, the classification was improved.

In this paper we introduce, analyze and apply persistent homology, one of the main algorithms of TDA, on some real data sets from the bio-medical field. Topological data analysis (TDA) is a field which is a synergy between mathematics, data science and computer science. The main goal of TDA is studying the shape of data using topological techniques. TDA proposes new algorithms that deal with these problems based on tools or concepts from algebraic topology and pure mathematics. We analyze the results and give a topological characterization of the dataset and propose to use them in future work.

In this paper we analyze and apply one of the main algorithms of TDA (Topological Data Analysis), Mapper, on some real data sets. We use Mapper for visualization of a data sets, and we tend to get some insights if some key characteristics of the data are captured by the visualization and how they are connected with human perception of the data. Also, we will discuss if the visualization can make progress in further work.