Boolean Matrix Factorization Research Articles

Revisiting data reduction for boolean matrix factorization algorithms based on formal concept analysis

Revisiting data reduction for boolean matrix factorization algorithms based on formal concept analysis

Delegation-Relegation for Boolean Matrix Factorization

The Boolean Matrix Factorization (BMF) problem aims to represent a n×m Boolean matrix as the Boolean product of two matrices of small rank k, where the product is computed using Boolean algebra operations. However, finding a BMF of minimum rank is known to be NP-hard, posing challenges for heuristic algorithms and exact approaches in terms of rank found and computation time, particularly as matrix size or the number of entries equal to 1 grows. In this paper, we present a new approach to simplifying the matrix to be factorized by reducing the number of 1-entries, which allows to directly recover a Boolean factorization of the original matrix from its simplified version. We introduce two types of simplification: one that performs numerous simplifications without preserving the original rank and another that performs fewer simplifications but guarantees that an optimal BMF on the simplified matrix yields an optimal BMF on the original matrix. Furthermore, our experiments show that our approach outperforms existing exact BMF algorithms.

Boolean matrix factorization for symmetric binary variables

Binary variables classify into two types: asymmetric variables, where one state (1 or 0) is significantly more valuable than the other, and symmetric variables, where both states are equally valuable. Boolean matrix factorization (BMF), a popular methodology of preprocessing and analyzing tabular binary data, handles its input as asymmetric variables. In the paper, we develop an alternative that handles Boolean matrices as symmetric variables. Our method differs from traditional BMF in that the factors are linearly ordered by priority, and factors can contradict each other, meaning that one factor can assign a value of 1 while the other assigns a value of 0. In such a case, the factor with higher priority is the relevant one. Through experiments, we demonstrate that our approach provides a more compact data description than a straightforward application of the traditional BMF methods. Moreover, it is even able to overcome the Schein rank.

A generalized approach for Boolean matrix factorization

In this paper, we propose a generalized framework for fitting Boolean matrix factorization models to binary data. In this generalized setting, the binary rank-1 components of the underlying model can be combined by any Boolean function, thus extending the standard Boolean matrix factorization model, where the combination is restricted to the logical ‘OR’ function. We introduce two algorithms relying on a relaxation of the binary constraints on the factors of the model and on a polynomial representation of the Boolean function that combines the rank-1 components. One of the algorithms is based on the gradient descent optimization method, while the other is based on block coordinate descent. A detailed presentation of the algorithms is given, along with numerical experiments both on simulated and real datasets. A comparison with other algorithms from the literature is presented in the standard Boolean matrix setting allowing to assess the advantages and shortcomes of the proposed methods in terms of factor retrieval and data denoising performance, convergence behavior and time complexity.

Undercover Boolean Matrix Factorization with MaxSAT

The k-undercover Boolean matrix factorization problem aims to approximate a m×n Boolean matrix X as the Boolean product of an m×k and a k×n matrices A◦B such that X is a cover of A◦B, i.e., no representation error is allowed on the 0’s entries of the matrix X. To infer an optimal and “block-optimal” k-undercover, we propose two exact methods based on MaxSAT encodings. From a theoretical standpoint, we prove that our method of inferring “block-optimal” k-undercover is a (1 - 1/e) ≈ 0.632 approximation for the optimal k-undercover problem. From a practical standpoint, experimental results indicate that our “block-optimal” k-undercover algorithm outperforms the state-of-the-art even when compared with algorithms for the more general k-undercover Boolean Matrix Factorization problem for which only minimizing reconstruction error is required.

Precise Supervision of Enterprise Environmental Protection Behavior Based on Boolean Matrix Factorization under Low Carbon Background.

Supervising the environmental protection behavior of enterprises is a key strategy to achieve “carbon peaking and carbon neutrality”. This research innovatively proposes the concept of precise supervision, aiming to implement differentiated supervision measures for different types of enterprises, and realize the precise supervision method of enterprise environmental protection, which is different from the traditional supervision mode. Firstly, this paper proposes a novel MEBF+ method based on the benchmark algorithm MEBF, and obtains MEBF++ after incorporating the model bias. Secondly, based on the dataset of environmental supervision and certification of listed Chinese companies, the accuracy and robustness of the proposed method are verified by using multiple evaluation indicators. Finally, based on the analysis of the experimental results, two precise supervision concepts, narrow and broad, are proposed under the low-carbon background. The results show that compared with the benchmark method, the accuracy of the proposed method has been improved to a large extent. In addition, the precise supervision proposed in this paper can help reduce the consumption of manpower and resources as well as unite the public to monitor the environmental protection behavior of enterprises.

Boolean matrix factorization based on collaborative neurodynamic optimization with Boltzmann machines

This paper presents a collaborative neurodynamic approach to Boolean matrix factorization. Based on a binary optimization formulation to minimize the Hamming distance between a given data matrix and its low-rank reconstruction, the proposed approach employs a population of Boltzmann machines operating concurrently for scatter search of factorization solutions. In addition, a particle swarm optimization rule is used to re-initialize the neuronal states of Boltzmann machines upon their local convergence to escape from local minima toward global solutions. Experimental results demonstrate the superior convergence and performance of the proposed approach against six baseline methods on ten benchmark datasets.

Data-driven Q-matrix learning based on Boolean matrix factorization in cognitive diagnostic assessment.

Attributes and the Q-matrix are the central components for cognitive diagnostic assessment, and are usually defined by domain experts. However, it is challenging and time consuming for experts to specify the attributes and Q-matrix manually. Thus, there is an urgent need for an automatic and intelligent means to address this concern. This paper presents a new data-driven approach for learning the Q-matrix from response data. By constructing a statistical index and a heuristic algorithm based on Boolean matrix factorization, the response matrix is decomposed into the Boolean product of the Q-matrix and the attribute mastery patterns. The feasibility of the proposed approach is evaluated using simulated data generated under various conditions. A real data example is also presented to demonstrate the usefulness of the proposed approach.

Revisiting the GreCon algorithm for Boolean matrix factorization

Over the past decade, the most fundamental Boolean matrix factorization (BMF) algorithms GreCon and GreConD were proposed. Whereas GreConD has become a popular and widely used BMF algorithm, GreCon – the algorithm on which the GreConD is built – is somewhat forgotten in contemporary BMF research; however, GreCon may produce better results than GreConD. The main disadvantage of GreCon algorithm is a slow running time. In the paper, we argue that the search strategy of GreConD, notwithstanding it provides a good result, is limited. We show that the reasons for not using GreCon algorithm are no longer the truth. We revise the algorithm and propose a new approach to storing and updating data required for factor enumeration. By various experiments, we demonstrate that the revised version is competitive with contemporary BMF algorithms in terms of running time. Moreover, in some cases, the revised GreCon outperforms GreConD—one of the fastest BMF algorithms. Furthermore, we show that our novel approach to GreCon enables the utilization of novel approaches to BMF.

A heuristic concept construction approach to collaborative recommendation

Formal concept analysis was first used in collaborative filtering for over one decade. Popular approaches are based on superconcept-subconcept relationship or boolean matrix factorization. In this paper, we design a heuristic approach to construct a set of approximately strong concepts for recommendation. Here strong refers to not only big intent to ensure the similarity among users, but also big extent to ensure the stability of user groups. First, we use the intent threshold as the constraint and the area as the optimization objective to obtain approximately strong concepts. Second, we generate pre-recommendations based on the local popularity of the items implied by each concept. Finally, we determine the actual recommendation according to the number of times the item is pre-recommended to the user. Experiments have been undertaken on five popular datasets with different versions. Results show that our algorithm has lower runtime and/or higher recommendation quality compared with approaches based on concept lattice, matrix factorization, k-nearest neighbors, item-based collaborative filtering and boolean matrix factorization.

Boolean matrix factorization with background knowledge

Boolean matrix factorization (BMF) is a popular data analysis method summarizing the input data by Boolean factors. The Boolean nature ensures an easy interpretation of a particular factor, however, the interpretation of all discovered factors (as a whole) by domain experts may be difficult as the BMF methods seek only information in the data and do not reflect the experts understanding of data. In the paper, we propose a formalization of a novel variant of BMF reflecting expert’s background knowledge—additional knowledge about the data—that is not part of the data, in the form of attribute weights, as well as an algorithm for it. Moreover, we show that the proposed algorithm, which significantly outperforms the state-of-the-art algorithm, provides encouraging results that are worth further investigation.

LACE: Inference of cancer evolution models from longitudinal single-cell sequencing data

The rise of longitudinal single-cell sequencing experiments on patient-derived cell cultures, xenografts and organoids is opening new opportunities to track cancer evolution, assess the efficacy of therapies and identify resistant subclones.We introduce LACE, the first algorithmic framework that processes single-cell mutational profiles from samples collected at different time points to reconstruct longitudinal models of cancer evolution. The approach maximizes a weighted likelihood function computed on longitudinal data points to solve a Boolean matrix factorization problem, via Markov chain Monte Carlo sampling.On simulations, LACE outperforms state-of-the-art methods for both bulk and single-cell sequencing data with respect to the reconstruction of the ground-truth clonal phylogeny and dynamics, also in conditions of unbalanced datasets, significant rates of sequencing errors and sampling limitations. As the results are robust with respect to data-specific errors, LACE is effective with mutational profiles generated by calling variants from (full-length) scRNA-seq data, and this allows one to investigate the relation between genomic and phenotypic evolution of tumors at the single-cell level.Here, we apply LACE to a longitudinal scRNA-seq dataset of patient-derived xenografts of BRAFV600E/K mutant melanomas, dissecting the impact of BRAF/MEK-inhibition on clonal evolution, also in terms of clone-specific gene expression dynamics. Furthermore, the analysis of breast cancer PDXs from longitudinal targeted scDNA-sequencing experiments delivers a high-resolution temporal characterization of intra-tumor heterogeneity.

You May Also Like... Privacy: Recommendation Systems Meet PIR

AbstractWe describe the design, analysis, implementation, and evaluation of Pirsona, a digital content delivery system that realizes collaborative-filtering recommendations atop private information retrieval (PIR). This combination of seemingly antithetical primitives makes possible—for the first time—the construction of practically efficient e-commerce and digital media delivery systems that can provide personalized content recommendations based on their users’ historical consumption patterns while simultaneously keeping said consumption patterns private. In designing Pirsona, we have opted for the most performant primitives available (at the expense of rather strong non-collusion assumptions); namely, we use the recent computationally 1-private PIR protocol of Hafiz and Henry (PETS 2019.4) together with a carefully optimized 4PC Boolean matrix factorization.

Model order selection for approximate Boolean matrix factorization problem

A key step in applying Boolean matrix factorization (BMF) is establishing the correct model order for the data, i.e., decide where the knowledge stops and the noise starts, or simply, decide the proper number of factors that describe the data well. There are two main approaches to BMF, namely, Discrete Basis Problem (DBP) and Approximation Factorization Problem (AFP). Although the model order selection technique for DBP exists, there is no technique tailored for AFP. We show that the number of factors for DBP cannot be used in AFP, and we present a novel way, reflecting the nature of AFP, how to establish the proper number of factors. Moreover, we show that the number of factors established for AFP is – from a knowledge-representation viewpoint – better than that for DBP.

Approximate Logic Synthesis Using Boolean Matrix Factorization

Approximate computing is an emerging computing paradigm offering benefits in hardware metrics, such as design area and power consumption, by relaxing the requirement for full accuracy. In circuit design, a major challenge is to synthesize approximate circuits automatically from input exact circuits requiring minimal expert input. In this work, we present a method for approximate logic synthesis based on the Boolean matrix factorization, where an arbitrary input circuit can be approximated in a controlled fashion. Our methodology enables automatic computation of the dominant elements, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">bases</i> , of the truth table of the circuit, and later combines the bases to approximate the original truth table. Such compression can reduce the complexity of the hardware implementation significantly, while introducing variable degrees of inaccuracy. Furthermore, in our approach, the factorization algorithm can be fine tuned as required by the application, to effectively improve control over degree of approximation. In this work, we provide a unified approach enabling the factorization algorithm to utilize semiring algebra, field algebra, and a combination of both for truth table factorization. In addition, we provide an automatic circuit partitioning approach and a design space exploration heuristic to navigate the search space. We implement our methodology using a full stack of open-source tools, and thoroughly evaluate our methodology on a number of representative circuits showcasing the benefits of our proposed methodology for approximate logic synthesis. Finally, we compare our methodology against an existing library of approximate designs and demonstrate state-of-the-art performance.

The 8M Algorithm from Today’s Perspective

We provide a detailed analysis and a first complete description of 8M—an old but virtually unknown algorithm for Boolean matrix factorization. Even though the algorithm uses a rather limited insight into the factorization problem from today’s perspective, we demonstrate that its performance is reasonably good compared to the currently available algorithms. Our analysis reveals that this is due to certain concepts employed by 8M that are not exploited by the current algorithms. We discuss the prospect of these concepts, utilize them to improve two well-known current factorization algorithms, and, furthermore, propose an improvement of 8M itself, which significantly enhances the performance of the original 8M. Our findings are illustrated by experimental evaluation.

Boolean Matrix Factorization via Nonnegative Auxiliary Optimization

A novel approach to Boolean matrix factorization (BMF) is presented. Instead of solving the BMF problem directly, this approach solves a nonnegative optimization problem with the constraint over an auxiliary matrix whose Boolean structure is identical to the initial Boolean data. Then the solution of the nonnegative auxiliary optimization problem is thresholded to provide a solution for the BMF problem. We provide the proofs for the equivalencies of the two solution spaces under the existence of an exact solution. Moreover, the nonincreasing property of the algorithm is also proven. Experiments on synthetic and real datasets are conducted to show the effectiveness and complexity of the algorithm compared to other current methods.

Boolean decomposition of binary matrices using a post-nonlinear mixture approach

We introduce a novel binary matrix factorization (BMF) approach based on a post-nonlinear mixture model. Unlike the existing BMF methods, which are based on the classical matrix product, the proposed mixture model is equivalent to the Boolean matrix factorization model when the entries of the factor matrices are exactly binary. Consequently, our approach yields interpretable results in the case of overlapping sources and more accurate low-rank binary matrix approximations compared to the state-of-the-art. We propose a simple yet efficient algorithm for solving the proposed BMF problem based on multiplicative update rules. In addition, we provide for the first time in the binary data literature, a necessary and sufficient condition for the uniqueness of the Boolean matrix factorization, as well as several other uniqueness results. The interest of this new approach is illustrated in numerical simulation and on real datasets.

Biclustering and boolean matrix factorization in data streams

We study clustering of bipartite graphs and Boolean matrix factorization in data streams. We consider a streaming setting in which the vertices from the left side of the graph arrive one by one together with all of their incident edges. We provide an algorithm which after one pass over the stream recovers the set of clusters on the right side of the graph using sublinear space; to the best of our knowledge this is the first algorithm with this property. We also show that after a second pass over the stream the left clusters of the bipartite graph can be recovered and we show how to extend our algorithm to solve the Boolean matrix factorization problem (by exploiting the correspondence of Boolean matrices and bipartite graphs). We evaluate an implementation of the algorithm on synthetic data and on real-world data. On real-world datasets the algorithm is orders of magnitudes faster than a static baseline algorithm while providing quality results within a factor 2 of the baseline algorithm. Our algorithm scales linearly in the number of edges in the graph. Finally, we analyze the algorithm theoretically and provide sufficient conditions under which the algorithm recovers a set of planted clusters under a standard random graph model.

Fast and Efficient Boolean Matrix Factorization by Geometric Segmentation

Boolean matrix has been used to represent digital information in many fields, including bank transaction, crime records, natural language processing, protein-protein interaction, etc. Boolean matrix factorization (BMF) aims to find an approximation of a binary matrix as the Boolean product of two low rank Boolean matrices, which could generate vast amount of information for the patterns of relationships between the features and samples. Inspired by binary matrix permutation theories and geometric segmentation, we developed a fast and efficient BMF approach, called MEBF (Median Expansion for Boolean Factorization). Overall, MEBF adopted a heuristic approach to locate binary patterns presented as submatrices that are dense in 1's. At each iteration, MEBF permutates the rows and columns such that the permutated matrix is approximately Upper Triangular-Like (UTL) with so-called Simultaneous Consecutive-ones Property (SC1P). The largest submatrix dense in 1 would lie on the upper triangular area of the permutated matrix, and its location was determined based on a geometric segmentation of a triangular. We compared MEBF with other state of the art approaches on data scenarios with different density and noise levels. MEBF demonstrated superior performances in lower reconstruction error, and higher computational efficiency, as well as more accurate density patterns than popular methods such as ASSO, PANDA and Message Passing. We demonstrated the application of MEBF on both binary and non-binary data sets, and revealed its further potential in knowledge retrieving and data denoising.