🍩 Database of Original & Non-Theoretical Uses of Topology

1. The Shape of Word Embeddings: Quantifying Non-Isometry With Topological Data Analysis (2024)

   Ondřej Draganov, Steven Skiena

   Abstract

   Word embeddings represent language vocabularies as clouds of d-dimensional points. We investigate how information is conveyed by the general shape of these clouds, instead of representing the semantic meaning of each token. Specifically, we use the notion of persistent homology from topological data analysis (TDA) to measure the distances between language pairs from the shape of their unlabeled embeddings. These distances quantify the degree of non-isometry of the embeddings. To distinguish whether these differences are random training errors or capture real information about the languages, we use the computed distance matrices to construct language phylogenetic trees over 81 Indo-European languages. Careful evaluation shows that our reconstructed trees exhibit strong and statistically-significant similarities to the reference.

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   • Language
   • linguistics
   • Textual document
   • word embedding
   • Barcode statistics
   • Bottleneck distance
   • Clustering
   • Wasserstein distance
   • persistent homology
   • persistent homology:persistence diagrams
   • persistent homology:persistence images
   • vietoris-rips

2. Knowledge Gaps in the Early Growth of Semantic Feature Networks (2018)

   Ann E. Sizemore, Elisabeth A. Karuza, Chad Giusti, Danielle S. Bassett

   Abstract

   Understanding language learning and more general knowledge acquisition requires the characterization of inherently qualitative structures. Recent work has applied network science to this task by creating semantic feature networks, in which words correspond to nodes and connections correspond to shared features, and then by characterizing the structure of strongly interrelated groups of words. However, the importance of sparse portions of the semantic network—knowledge gaps—remains unexplored. Using applied topology, we query the prevalence of knowledge gaps, which we propose manifest as cavities in the growing semantic feature network of toddlers. We detect topological cavities of multiple dimensions and find that, despite word order variation, the global organization remains similar. We also show that nodal network measures correlate with filling cavities better than basic lexical properties. Finally, we discuss the importance of semantic feature network topology in language learning and speculate that the progression through knowledge gaps may be a robust feature of knowledge acquisition.

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   • Education
   • Knowledge acquisition
   • Language learning
   • Networks
   • semantic feature networks
   • Innovate
   • Node orders
   • Persistent homology