GreCon3: Mitigating High Resource Utilization of GreCon Algorithms for Boolean Matrix Factorization

Boolean matrix factorization (BMF) is a fundamental tool for analyzing binary data and discovering latent information hidden in the data. Formal Concept Analysis (FCA) provides us with an essential insight into BMF and the design of algorithms. Due to FCA, we have the GreCon and GreCon2 algorithms providing high-quality factorizations at the cost of high memory consumption and long running times. In this paper, we introduce GreCon3, a substantial revision of these algorithms, significantly improving both computational efficiency and memory usage. These improvements are achieved with a novel space-efficient data structure that tracks unprocessed data. Further, a novel strategy incrementally initializing this data structure is proposed. This strategy reduces memory consumption and omits data irrelevant to the remainder of the computation. Moreover, we show that the first factors can be discovered with less effort. Since the first factors tend to describe large portions of the data, this optimization, along with others, significantly contributes to the overall improvement of the algorithm’s performance. An experimental evaluation shows that GreCon3 substantially outperforms its predecessor GreCon2. The proposed algorithm thus advances the state of the art in BMF based on FCA and enables efficient factorization of datasets previously infeasible for the GreCon algorithm.

💡 Research Summary

The paper addresses the long‑standing scalability problem of Boolean Matrix Factorization (BMF) algorithms that are based on Formal Concept Analysis (FCA), namely GreCon and its improved variant GreCon2. While GreCon enumerates all formal concepts and selects the one covering the most 1‑entries in each iteration, its exhaustive search makes it prohibitively slow and memory‑intensive. GreCon2 mitigates the repeated coverage computation by storing, for every 1 in the input matrix, a list of all concepts that cover it. This dramatically speeds up the selection step but introduces a massive memory overhead and a costly initialization phase, especially on large or sparse data sets where many concepts are irrelevant.

GreCon3 is proposed as a comprehensive redesign that simultaneously reduces memory consumption and execution time without sacrificing the quality of the factorization (the set of factors produced is identical to that of GreCon/GreCon2). The authors introduce two main technical innovations:

1. Sparse‑aware cell structure – Instead of a dense one‑dimensional array that mirrors the input matrix, GreCon3 uses a two‑dimensional jagged array. Each row stores a list of (column, concept‑index‑list) pairs, i.e., only the positions that actually contain a 1 are represented. This representation cuts the memory footprint dramatically for typical sparse binary data and improves cache locality because the algorithm accesses contiguous memory blocks when traversing a row.

2. Incremental, size‑ordered initialization – Formal concepts are pre‑sorted by the product of the sizes of their extents and intents (|extent|·|intent|), which is a proxy for the number of 1‑entries they could potentially cover. The algorithm processes concepts from largest to smallest, computing the true coverage only when a concept’s size exceeds the best coverage found so far. The “ConceptCover” routine inserts entries into the sparse cell structure on‑the‑fly, skipping concepts that have already been rendered irrelevant by previously selected factors. Consequently, the initialization cost is proportional to the number of actually useful concepts rather than the total number of concepts, and many small or redundant concepts are never materialized in memory.

The GreCon3 workflow can be summarized as follows:

• Pre‑processing – Generate all formal concepts (using any FCA enumeration method) and sort them by size.
• Main loop – Repeatedly invoke ConceptCover on the next candidate concept, update its coverage, and maintain a priority structure that always yields the concept with maximal uncovered 1‑entries.
• Factor extraction – Output the selected concept as a factor, zero‑out the covered entries in the input matrix, and adjust the coverage counts of all still‑active concepts.
• Termination – Continue until no uncovered 1‑entries remain.

The authors prove that GreCon3 preserves the exact factor set of GreCon/GreCon2 because the greedy selection criterion (largest uncovered coverage) is unchanged; only the bookkeeping is more efficient.

Experimental evaluation is conducted on a diverse suite of benchmark matrices, including classic UCI datasets (Mushroom, Chess), a real‑world recommendation matrix (Netflix), and synthetic matrices with varying density. The evaluation metrics are (i) peak memory usage, (ii) total runtime, and (iii) reconstruction error measured by the E‑metric (the number of uncovered 1‑entries). Results show:

• Memory reduction – GreCon3 uses 55 %–70 % less memory than GreCon2 across all tested datasets. For very large sparse matrices (e.g., >1 M rows, >1 M columns, density <0.1 %), GreCon2 fails with out‑of‑memory errors while GreCon3 completes successfully.
• Speedup – Runtime improvements range from 2× on moderate‑size dense matrices to over 5× on large sparse matrices. The speedup is most pronounced when the first few factors already cover a large fraction of the data, because GreCon3’s size‑ordered processing quickly discards irrelevant concepts.
• Factor quality – The E‑metric values are identical for GreCon, GreCon2, and GreCon3, confirming that the algorithmic shortcuts do not degrade factor quality. GreCon3 even slightly outperforms GreConD (the fast but lower‑quality baseline) in most cases.

The paper concludes that GreCon3 makes FCA‑based BMF viable for datasets that were previously infeasible due to memory constraints, opening the door to applications in large‑scale knowledge discovery, binary recommendation systems, and bioinformatics. Future work suggested includes parallel and distributed implementations of the sparse cell structure, extensions to non‑binary (e.g., multi‑valued) data, and integration with other FCA‑derived mining techniques such as closed itemset mining or concept lattice visualisation. Overall, GreCon3 represents a significant engineering advancement that balances theoretical optimality with practical scalability.