Non-Redundant Data Association

• Non-redundant data association is the process of extracting maximally informative, non-overlapping rules to enhance predictive insights.
• It employs criteria such as equivalence, subsumption, closure-based and statistical measures to filter out redundant associations.
• Algorithms like AGT and FCIL efficiently prune redundant rules, reducing rule clutter by up to 80% compared to traditional methods.

Non-redundant data association is the problem of extracting sets of associations or rules that are both logically precise and maximally informative, without the exponential pattern explosion caused by redundant or overlapping structures. The notion of non-redundancy is rigorously defined in both data-mining (association rules, itemset-based dependencies) and combinatorial data association (multi-object tracking, one-to-one correspondences), and is operationalized using equivalence, subsumption, closure-based, statistical, and algorithmic criteria. This article surveys the principal definitions and algorithms for non-redundant association, formal bases, and efficient rule-set extraction, as established in the research literature.

1. Formal Definitions of Non-Redundancy

Redundancy in data associations and rule mining is characterized by the presence of multiple rules or associations where at least one can be inferred from or is subsumed by another, offering no additional predictive power or information. Foundational formalizations include:

• Set-Inclusion Redundancy: Rule $R_2: A_2 \Rightarrow C_2$ is redundant w.r.t. $R_1: A_1 \Rightarrow C_1$ if $A_1 \subseteq A_2$, $C_1 = C_2$, and both meet the same confidence requirement. This captures the intuition that adding extra context without improving prediction or precision is superfluous (Sarker et al., 2018).
• Minimal Non-Redundant Association Rules (MNAR): A rule $G \Rightarrow C \setminus G$ (where $C$ is a frequent closed itemset and $G$ is a minimal generator) is non-redundant if no other rule with the same support and confidence is more general (i.e., has a strictly smaller antecedent and larger consequent) (Vo et al., 2011).
• Maximal Non-Redundant Rules: A rule is maximal non-redundant if it is neither redundant (in the sense above) nor subsumed by any rule that is at least as strong in both support and confidence, with more general antecedent and more specific consequent (Weidner et al., 2019).
• Redundancy in Attribute Associations: In multivariate attribute association mining, an association set $S$ is non-redundant if it demonstrates high statistical significance, and no proper subset of $S$ holds significant association—guaranteeing parsimony (Chanda et al., 2012).
• Speciousness: A stronger criterion, where a dependency is pruned if it disappears or reverses when conditioning on another variable (i.e., the marginal dependency is explained away), based on conditional leverages or information measures (Hämäläinen et al., 2017).
• One-to-One Correspondence (Discrete Assignment): In data association for matching, non-redundancy is enforced as a constraint that no entity in one set is matched to more than one entity in the other set, i.e., $\sum_q X_{pq} \le 1$ and $\sum_p X_{pq} \le 1$ for assignment matrix $X$ (Lusk et al., 2020).

2. Non-Redundant Association Rule Mining Algorithms

Several algorithmic paradigms are used to extract non-redundant association rules:

• Context-Precedence Trees (AGT): As proposed by Sarker and Salim, the Association Generation Tree (AGT) is built breadth-first with context splitting by information gain, assigning dominant behaviors to nodes. Redundant child nodes sharing both behavior and minimum confidence with their parents are marked and pruned, guaranteeing that no rule's antecedent is a proper superset of another predicting the same behavior at threshold (Sarker et al., 2018).
• Frequent Closed Itemset Lattices (FCIL): The CHARM-L algorithm is used to mine closed itemsets, building a lattice structure, after which MNARs are generated by enumerating minimal generators for each closed itemset, emitting rules only once per unique (support, confidence) pair, and ensuring all rules are minimal w.r.t. redundancy (Vo et al., 2011).
• Post-Mining Pruning (Maximality and Subsumption): Post-processing over all mined rules to remove any that are redundant (covered by confidence-1 rules with more general antecedents) or subsumed (by rules of greater or equal support/confidence and more general/specific sets) (Weidner et al., 2019).
• Attribute-Association Mining with Redundancy and Bound-Based Pruning: Two-stage approaches first enumerate all statistically significant non-redundant attribute combinations (“combinations of interest” and “special combinations of interest”), using redundancy filtering (e.g., negative normalized k-way interaction information), then traverse only the reduced set for interaction discovery (Chanda et al., 2012).
• Specious Rule Elimination: Pruning is based on testing for loss of dependency under conditioning, using exact hypergeometric or information-theoretic measures, and exploiting order-homomorphic properties to restrict redundancy comparison to stronger rules only (Hämäläinen et al., 2017).
• Graph-Theoretic Methods for Data Association: Formulating "densest consistent subgraph" or maximum-weight clique problems under one-to-one constraints, and relaxing to efficient continuous optimization (e.g., via penalized affinity matrices and projected gradient ascent) to recover non-redundant global associations (Lusk et al., 2020).

3. Theoretical Foundations: Redundancy, Closure, and Minimal Bases

The theory of association-rule redundancy is grounded in notions of logical entailment and closure operators:

• Plain and Standard Redundancy: Various definitions (standard, simple, strict, cover) are shown to be equivalent—redundancy reduces to coverage by more general antecedents and consequences (Balcazar, 2010).
• Deductive Calculi: Sound and complete deduction schemes are provided for both plain redundancy (using right-reduction, right-augmentation, and left-augmentation) and closure-based redundancy (incorporating datasets' implication bases), yielding formal characterization of rule redundancy and entailment (Balcazar, 2010).
• Minimal Bases: Complete, minimum-size bases for association rules are constructed by identifying, for each closed set $Y$, its valid minimal antecedents $X$, such that $X \rightarrow Y \setminus X$ forms the core non-redundant rule set. This is further refined with closure-based or double-support variants, ensuring absolute minimality (Balcazar, 2010).
• Multi-Premise Entailment: Non-trivial redundancy can arise from the conjunction of two strict partial rules; Balcázar provides the first full treatment and deductive scheme for two-premise redundancy, identifying when proper entailment (not inferable from any single rule) occurs (Balcazar, 2010).

4. Statistical and Information-Theoretic Criteria for Non-Redundancy

Rigorous data-driven criteria underpin non-redundancy in both attribute associations and rule mining:

• Total Correlation Information (TCI) and K-Way Interaction Information (KWII): Only attribute subsets whose joint TCI is highly significant and not replicated by any strict subset are kept. KWII quantifies synergy or redundancy invisible to lower-order marginals (Chanda et al., 2012).
• Bound and Sample-Size Pruning: Efficient algorithms exploit information-theoretic lower and upper bounds to preemptively eliminate candidate combinations or rulesets not meeting criteria, based on fast entropy/mutual information computations and sample adequacy (Chanda et al., 2012).
• Conditional Dependency Elimination: Specious association mining directly targets dependencies that arise purely from confounding, operationalized using conditional mutual information, signed conditional MI, or exact combinatorial tests. Only rules remaining significant under all proper conditionals are kept (Hämäläinen et al., 2017).
• Empirical Results: Across multiple datasets, specious-pruning can eliminate up to 90% of initially mined associations, revealing the true and concise dependency structures (Hämäläinen et al., 2017).

5. Computational Complexity and Empirical Performance

Non-redundant association mining algorithms are designed to combat combinatorial explosion:

• Tree- or Lattice-Based Approaches: AGT and FCIL methods drastically reduce the number of generated rules; AGT, for example, yields 60–80% fewer rules than classic Apriori for the same threshold (Sarker et al., 2018), and FCIL-based MNAR mining realizes 5–20x speedups over frequent itemset lattice approaches whenever the number of closed sets is much less than the set of all frequent itemsets (Vo et al., 2011).
• Post-Pruning Algorithms: Subquadratic complexity in the number of rules can be achieved by efficient set-inclusion checks using bit-vectors or sorted lists, and the practical impact is a drastic reduction in rule-set size to the subset most interpretable and actionable for end-users (Weidner et al., 2019).
• Continuous Relaxations in Assignment Problems: In one-to-one correspondence tasks with millions of putative associations, projected-gradient solvers enforce one-to-one structure and geometric consistency, achieving high-precision matches at $O(|E|)$ complexity (for sparse affinity matrices) with runtimes in the tens to hundreds of milliseconds, even under extreme noise and outlier regimes (Lusk et al., 2020).

6. Practical Applications and Extensions

Practical deployments of non-redundant data association span diverse domains:

• Context-Aware Mobile Applications: Concise behavioral rule-sets extracted using AGT or similar are directly usable for smartphone call-handling agents, notification predictors, or app-utilization profiling, avoiding unnecessary rule proliferation (Sarker et al., 2018).
• Spatio-Temporal Pattern Mining: Rule pruning by maximal non-redundancy/subsumption is immediately applicable to analyzing structured event sequences, e.g., sports analytics or surveillance (Weidner et al., 2019).
• Genomic Data and Complex Trait Modeling: Statistically non-redundant attribute association sets (COI, SCOI) and KWII mining expose genuine epistatic or multi-locus combinatorial effects, while pruning SNPs in strong linkage as functionally redundant (Chanda et al., 2012).
• Robust Data Association in Vision and Robotics: Non-redundant assignment solvers such as CLIPPER provide state-of-the-art precision/recall in object matching, registration, and tracking tasks, under severe data noise/outlier conditions (Lusk et al., 2020).

7. Limitations and Outlook

Principal limitations include:

• Combinatorial Scaling: While redundancy-pruning reduces output size dramatically, underlying enumeration (closed itemsets, attribute subsets, context splittings) can still incur exponential cost in dimension for dense data—a challenge partially mitigated by closure, information-theoretic bounding, or parallel/distributed computation (Vo et al., 2011, Sarker et al., 2018).
• Dependence on Quality of Underlying Structure: Non-redundancy often presumes that the closure operator or lattice structure is well-behaved; rare or noisy data, or "borderline" confidence thresholds, can degrade performance or informativeness (Balcazar, 2010).
• Attribute Redundancy and Domain Knowledge: Statistical redundancy does not always align with semantic redundancy; domain expertise may be needed to refine or reinterpret the final non-redundant rule-set (Chanda et al., 2012).
• Future Directions: Research avenues include incremental or streaming non-redundant rule mining, refined closure/operator theory for complex data types, user-guided or constraint-based redundancy models, and scalable multi-user or cross-domain deployments.

Non-redundant data association forms the foundation for interpretable, efficient, and robust data mining and matching frameworks across scientific and engineering disciplines, enabling reliable discovery and action in high-dimensional, noisy, or complex environments (Balcazar, 2010, Chanda et al., 2012, Vo et al., 2011, Sarker et al., 2018, Hämäläinen et al., 2017, Lusk et al., 2020, Weidner et al., 2019).