Robust inter-contextual linking: A theory of approximate bonds in FCA

The standard definition of a bond in FCA is too strict for noisy data. This paper introduces ‘approximate bonds,’ which use similarity operators and tolerance thresholds to allow for a controlled degree of imperfection. This robust framework is better suited for practical applications.

Keywords

Approximate bonds, Inter-contextual linking, Similarity operators, Tolerance thresholds, Robustness

The standard definition of a bond in Formal Concept Analysis (FCA) imposes strict conditions on the relationship between two contexts, making it overly restrictive for noisy or incomplete real-world data. This paper introduces the concept of “approximate bonds,” a generalization that allows for a controlled degree of imperfection in the inter-contextual dependencies. We relax the bond definition by incorporating similarity operators and tolerance thresholds, permitting a certain level of “disagreement” or “nearness” in the mappings. We formalize the notion of an $(\epsilon, \delta)$-approximate bond and study its properties. This framework leads to the construction of bonds that are more robust to data imperfections while retaining significant inferential power, making them better suited for practical applications like recommender systems in noisy environments.

Introduction

Bonds provide the theoretical foundation for linking information between disparate datasets. However, their definition is crisp: the extents of a bond $\mathbb{B}$ between $\mathbb{K}_1=(G_1, M_1, I_1)$ and $\mathbb{K}_2=(G_2, M_2, I_2)$ must be a subset of the extents of $\mathbb{K}_1$, and its intents must correspond exactly to the extents of $\mathbb{K}_2$. This perfect correspondence is rarely found in real data. A single noisy data point can break an otherwise strong bond.

To overcome this brittleness, we propose a framework for approximate bonds. Our approach is to flexibilize the rigid conditions of the bond definition.

Our contributions are:

1. A formal definition of an $(\epsilon, \delta)$-approximate bond, parameterized by similarity operators and tolerance thresholds.
2. A theoretical analysis of the properties of these approximate bonds and the structure of the set of all such bonds.
3. A demonstration of how approximate bonds can successfully link contexts where no exact bond exists, enhancing the robustness of bond-based applications.

Methodology and expected theoretical results

Formalizing approximation

Let $sim_{ext}: 2^{G_1} \times 2^{G_1} \to [0,1]$ be a similarity operator on sets of objects (e.g., Jaccard index) and $sim_{int}: 2^{G_2} \times 2^{G_2} \to [0,1]$ be a similarity on sets of attributes (or objects from the second context).

A context $\mathbb{B}=(G_1, G_2, I)$ is an $(\epsilon, \delta)$-approximate bond between $\mathbb{K}_1$ and $\mathbb{K}_2$ if: 1. For every extent $E \in Ext(\mathbb{B})$, there exists an extent $E' \in Ext(\mathbb{K}_1)$ such that $sim_{ext}(E, E') \ge 1-\epsilon$. 2. For every intent $I \in Int(\mathbb{B})$, there exists an extent $E'' \in Ext(\mathbb{K}_2)$ such that $sim_{int}(I, E'') \ge 1-\delta$. (And vice-versa for completeness).

The thresholds $\epsilon, \delta \in [0,1]$ control the degree of allowed disagreement.

Expected theoretical results:

• Lattice structure: We will investigate whether the set of all $(\epsilon, \delta)$-approximate bonds for fixed thresholds forms a complete lattice, similar to exact bonds.
• Robustness property: We will formally prove that approximate bonds are more robust to data perturbations (e.g., flipping bits in the context matrices) than exact bonds.

Work plan

• Months 1-4: Formalize the definitions and explore different choices for the similarity operators.
• Months 5-8: Analyze the algebraic structure of the set of approximate bonds. Prove or disprove the lattice property.
• Months 9-11: Develop a use-case (e.g., a recommender system with noisy data) to demonstrate the practical advantage of approximate bonds over exact bonds.
• Month 12: Write the manuscript.

Potential target journals

1. International Journal of Approximate Reasoning (Q2): The perfect venue for a paper whose core contribution is a formal model of approximation.
2. Fuzzy Sets and Systems (Q1): A top choice, as the concepts of similarity and thresholds are central to fuzzy set theory.

MVA strategy

• Paper 1 (The MVA):
  • Scope: The full theoretical paper as described. It should introduce the formal definition, analyze its properties, and provide a compelling, illustrative example of its utility.
  • Goal: To introduce the foundational theory of approximate bonds to the FCA community.
  • Target venue: International Journal of Approximate Reasoning.