Exploring ESC Winners with Nested Diagrams

We present ConceptFlow, a scikit-learn-compatible Python library for Formal Concept Analysis that constructs and renders nested line diagrams from many-valued formal contexts. Given a many-valued context and a partition of its attributes into conceptual scales, ConceptFlow performs conceptual scaling, computes the factor lattices, identifies filled nodes of the corresponding subdirect product, and produces an interactive visualization. We apply ConceptFlow to the winners of the Eurovision Song Contest from 1975 to 2025, exploring relationships between voting patterns and musical characteristics. Voting support is captured by an outer scale spanning regional, cultural, historical, and political dimensions, while an inner scale captures musical characteristics via tempo and key. The resulting nested line diagram reveals implications across both scales, exposing dependencies between how winning entries were voted for and the musical properties they share.

Verifiable Knowledge Expansion through Retrieval-Grounded Formal Concept Analysis

Ontology construction requires deciding which objects, attributes, and structural relations should be accepted as valid knowledge. Language models can propose such structures from text, but their outputs can still be unsupported or inconsistent. This paper proposes a retrieval-augmented small language model (SLM) framework that uses formal concept analysis (FCA) as a symbolic verification loop for knowledge expansion. Starting from seed attributes, FCA proposes implications over a growing formal context. A retrieval-grounded SLM oracle then validates each implication or returns a counterexample. The oracle also supports incidence judgments, consistency checks, and attribute proposals, making accepted implications, counterexamples, contradictions, and corrections inspectable. In a rare ataxia setting constructed from Orphadata resources, retrieval-grounded 10-seed runs obtain relation F1 of 0.29-0.52 and closure-based implication F1 of 0.22-0.30. Larger seed sets increase the number of evaluated implications and often improve implication F1. The lower implication scores reflect a stricter evaluation of derived implications, where one missed or extra relation can affect several implication judgments. Ablations show that incidence judgments in a fixed object-attribute setting can improve closure-based implication scores. However, identifying positive object-attribute pairs remains difficult even when the candidate objects and attributes are fixed.

A Variability-Based Framework for Interpretable Naming in Formal and Relational Concept Analysis

Knowledge extraction from symbolic data often produces abstractions that are formally defined but not immediately interpretable by users. Formal Concept Analysis (FCA) and Relational Concept Analysis (RCA) provide representative settings for this issue: they generate explicit conceptual structures, implications, and relational dependencies from object descriptions and relations. Although these structures are explainable by design, their concepts are often identified by technical labels, which limits their use as human-interpretable knowledge units. Assigning meaningful names to such concepts is therefore a key issue for interpretation, navigation, validation, and reuse by domain experts. This paper investigates concept naming in FCA and RCA from a symbolic knowledge representation perspective. We first characterize the linguistic and terminological challenges involved in naming generated symbolic abstractions, including ambiguity, discrimination, concision, and consistency across related concepts. We then propose a configurable framework for LLM-assisted concept naming. The framework relies on a variability model that controls which sources of information are exposed during naming, such as intent, extent, inherited information, neighboring concepts, implications, and relational attributes. It thereby makes explicit the semantic choices involved in moving from formal concept descriptions to human-readable names. The approach is illustrated as a proof of concept on a small relational dataset in the pizzeria domain. This illustration shows how different configurations influence the names suggested by an LLM, and how naming variability can reveal interpretation choices, relational dependencies, and possible modeling issues in the underlying symbolic data.

A Geometric View for Understanding Concept Learning and Neuron Interpretation in Sparse Autoencoders

We propose a unified mathematical framework for a geometric understanding of concept learning and neuron interpretation in sparse autoencoders (SAEs). While SAEs improve interpretability of neural networks by learning sparse feature representations, a principled definition of ''concept'' and ''learning'' remains unclear. We formalize concepts as sets of data points and cast concept learning as a set-alignment problem between human-defined and model-induced concepts. This formulation distinguishes three increasingly strong notions of learning -- detection, separation, and approximation -- and yields geometric conditions, error bounds, and capacity constraints for when concepts can be represented by individual neurons or multi-neuron units. It also provides a set-theoretic account for common SAE phenomena, including feature splitting, feature absorption, feature families, and hierarchical concepts. Finally, we connect concept learning and neuron interpretation through formal concept analysis, showing that the two directions need not agree and that their many-to-many structure can be organized by concept lattices. Experiments on synthetic data with ReLU and Top-$K$ SAEs illustrate the theory and reveal the effects of SAE size and sparsity on concept learning.

Formal Concept Lattices are Good Semantic Scaffolds for Concept-Based Learning

Learning semantics is essential for deep learning models to be interpretable and better aligned with human reasoning. Concept-based models approach this by representing classes through meaningful semantic abstractions, but typically treat all concepts as a flat, unstructured set learned at a single neural network layer. This overlooks a fundamental property of human semantic understanding: concepts being organized hierarchically, from general to specific. While deep networks do learn a hierarchy of visual features, this structure is rarely aligned with explicit semantic hierarchies. Drawing on Formal Concept Analysis, we demonstrate that formal concept lattices provide principled semantic scaffolds to guide neural network learning. These lattices naturally identify where in the network concepts should be learned based on their level of generality. This allows the model to develop staged, semantically grounded representations throughout its depth. Empirical results on real-world datasets show that our models produce more interpretable embeddings, support more effective interventions, and learn concept representations that are both meaningful and hierarchically structured.