Totsvectors

Totsvectors are a type of data object used in theoretical and applied contexts to encode both quantitative measurements and topological information within a single representation. The core idea is to pair a standard numerical feature vector with a topological descriptor that captures structure or shape in the data, such as connectivity or persistence features.

Formally, a totsvector can be described as a pair (v, t), where v is a real-valued vector in R^n and t is a topological descriptor. The descriptor t can take several forms, such as a simplicial complex, a summarized persistence diagram, or another concise representation of topology on the same index set as v. This construction allows one to carry both magnitude information and shape information through subsequent analysis. Variants exist that combine multiple topological descriptors into a single object, sometimes denoted as k-totsvectors, which pair a numeric vector with several topological components.

Operations on totsvectors are defined to respect both parts. Addition and scalar multiplication primarily act on the numerical vector component, while the topological descriptor may be updated through a predefined merge or union operation. A topology-aware distance metric combines a standard Euclidean distance on the numeric part with a topology-distance measure on the descriptor, such as a bottleneck or Wasserstein distance when the descriptor is a persistence diagram.

Applications of totsvectors appear in machine learning and data analysis where it is useful to integrate quantitative features with topological summaries. They are used to enhance classifiers or clustering tasks by embedding geometric or connectivity information alongside feature values. See also topological data analysis, persistent homology, and feature vectors.