Designing for Interpretation Uncertainty: Architecture and Principles for Topological Learning Analytics Dashboards

Topological data analysis (TDA), particularly persistent homology (PH), captures geometric structural properties in medical images (e.g., connected components, loops, shape characteristics), which conventional pixel-level deep learning approaches often neglect. While many topological descriptors are known for converting persistence diagrams (PDs) or raw images into topological feature vectors, existing methods mostly default to a single fixed descriptor (e.g., persistence images), leaving the diversity of topological representations largely unexplored. To the best of our knowledge, there is no known large language model (LLM)-based agentic framework that can automatically determine the most suitable topological descriptors for a given image dataset and produce the corresponding topological feature vectors for downstream tasks. To fill this gap, we propose \textbf{TopoAgent}, an LLM-based agentic framework that automates topology learning for medical image analysis.TopoAgent operates through a Perception--Reasoning--Action--Reflection loop supported by 21 domain-specific tools and dual memory that accumulates experience across runs. Its skill set is distilled from systematic evaluation of 15 topological descriptors across 26 datasets with six classifiers. TopoAgent analyzes input images and their topological characteristics, reasons about which topological descriptors best suit the input, and determines the optimal descriptor and its configuration, all without task-specific training.

Self-Organizing Maps (SOMs) have long been used as exploratory tools for high-dimensional data: they organize objects into a two-dimensional topology that reveals clusters, gradients, sparse regions, dense regions, and boundaries. Yet, in modern data systems, SOMs are typically trained and visualized outside the DBMS, disconnected from the relational data they summarize. We introduce the abstraction of a queryable data map: a learned topological artifact consisting of representatives, neighborhood relations, object assignments, and derived summaries. We instantiate this idea with MapDB, a lightweight prototype that makes SOM artifacts queryable so users can explore data topology without leaving the database. Experimental study shows that SOM training is feasible at moderate analytical scale, that map queries are interactive after materialization, and that SOM regions provide meaningful targets for exploratory SQL.

Real-time process monitoring requires methods that extract actionable information from high-dimensional time-series data. In this work, we present a new approach for process monitoring that combines tools of topological data analysis (TDA) and machine learning. In the proposed approach, we represent multivariate time-series data as manifolds and use topological descriptors to summarize the structure of such data; we then use a neural ordinary differential equation to learn the dynamic evolution of the topological structure of the system. Using real data from an industrial process, we show that this trajectory-based event detection approach is effective at detecting diverse types of events. We contrast this approach against reconstruction-based approaches such as principal component analysis and autoencoders and against a trajectory-based approach that uses Koopman autoencoders.