Factorization of Binary Matrices: Rank Relations, Uniqueness and Model Selection of Boolean Decomposition

DeSantis, Derek; Skau, Erik; Truong, Duc P.; Alexandrov, Boian S.

doi:10.48550/arxiv.2012.10496

Cited by 3 publications

(6 citation statements)

References 8 publications

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“…Now we need to show that (Y, W, H) is an exact solution for the optimization problem (2). First, by construction, the optimized value is already zero.…”

Section: Nonnegative Auxiliary Optimization For Bmfmentioning

confidence: 99%

“…The NMF algorithm searches for a best nonnegative rank approximation to X = W H. With BANMF being algorithmically similar to NMF, we are naturally interested in the performance when there is a gap between the Boolean rank and the nonnegative rank. Theoretically, it is known that the nonnegative rank is greater than or equal to the Boolean rank [2]. To empirically investigate the effect of a gap between the ranks, we generate a suite of Boolean matrices where a gap between rank + (X) and rank B (X) exists.…”

Section: Banmf and Nmf On Different-rank Datamentioning

confidence: 99%

“…This model is a medical application for diagnosing lung cancer. 2 . The dimensions of the data are 2000 × 143.…”

Section: Lung Cancer With Probes Dataset (Lucap0)mentioning

confidence: 99%

“…The Boolean rank of X is the smallest k for which such an exact representation, X = W ⊗ B H, exists. Interestingly, in contrast to the non-negative rank, the Boolean rank can be not only bigger or equal, but also smaller than the real rank [2,10].…”

Section: Introductionmentioning

confidence: 99%

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Boolean Matrix Factorization via Nonnegative Auxiliary Optimization

Truong,

Skau,

Desantis

et al. 2021

Preprint

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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.

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“…Now we need to show that (Y, W, H) is an exact solution for the optimization problem (2). First, by construction, the optimized value is already zero.…”

Section: Nonnegative Auxiliary Optimization For Bmfmentioning

confidence: 99%

Section: Banmf and Nmf On Different-rank Datamentioning

confidence: 99%

“…This model is a medical application for diagnosing lung cancer. 2 . The dimensions of the data are 2000 × 143.…”

Section: Lung Cancer With Probes Dataset (Lucap0)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Boolean Matrix Factorization via Nonnegative Auxiliary Optimization

Truong,

Skau,

Desantis

et al. 2021

Preprint

Self Cite

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show abstract

“…The Boolean rank of X is the smallest k for which such an exact representation, X = AB, exists. Interestingly, in contrast to the non-negative rank, the Boolean rank can be not only bigger or equal, but also much smaller than the logarithm of the real rank (Monson et al, 1995;DeSantis et al, 2020). Hence, low-rank factorization for Boolean matrices is of particular interest.…”

Section: Introductionmentioning

confidence: 99%

Boolean Hierarchical Tucker Networks on Quantum Annealers

Pelofske,

Hahn,

O'Malley

et al. 2021

Preprint

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Quantum annealing is an emerging technology with the potential to solve some of the computational challenges that remain unresolved as we approach an era beyond Moore's Law. In this work, we investigate the capabilities of the quantum annealers of D-Wave Systems, Inc., for computing a certain type of Boolean tensor decomposition called Boolean Hierarchical Tucker Network (BHTN). Boolean tensor decomposition problems ask for finding a decomposition of a high-dimensional tensor with categorical, [true, false], values, as a product of smaller Boolean core tensors. As the BHTN decompositions are usually not exact, we aim to approximate an input high-dimensional tensor by a product of lower-dimensional tensors such that the difference between both is minimized in some norm. We show that BHTN can be calculated as a sequence of optimization problems suitable for the D-Wave 2000Q quantum annealer. Although current technology is still fairly restricted in the problems they can address, we show that a complex problem such as BHTN can be solved efficiently and accurately.

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Distributed non-negative RESCAL with Automatic Model Selection for Exascale Data

Bhattarai¹,

Kharat²,

Skau³

et al. 2022

Preprint

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With the boom in the development of computer hardware and software, social media, IoT platforms, and communications, there has been an exponential growth in the volume of data produced around the world. Among these data, relational datasets are growing in popularity as they provide unique insights regarding the evolution of communities and their interactions. Relational datasets are naturally non-negative, sparse, and extra-large. Relational data usually contain triples, (subject, relation, object), and are represented as graphs/multigraphs, called knowledge graphs, which need to be embedded into a low-dimensional dense vector space. Among various embedding models, RESCAL allows learning of relational data to extract the posterior distributions over the latent variables and to make predictions of missing relations. However, RESCAL is computationally demanding and requires a fast and distributed implementation to analyze extra-large real-world datasets. Here we introduce a distributed non-negative RESCAL algorithm for heterogeneous CPU/GPU architectures with automatic selection of the number of latent communities (model selection), called pyDRESCALk. We demonstrate the correctness of pyDRESCALk with real-world and large synthetic tensors, and the efficacy showing near-linear scaling that concurs with the theoretical complexities. Finally, pyDRESCALk determines the number of latent communities in an 11-terabyte dense and 9-exabyte sparse synthetic tensor.

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Factorization of Binary Matrices: Rank Relations, Uniqueness and Model Selection of Boolean Decomposition

Cited by 3 publications

References 8 publications

Boolean Matrix Factorization via Nonnegative Auxiliary Optimization

Boolean Matrix Factorization via Nonnegative Auxiliary Optimization

Boolean Hierarchical Tucker Networks on Quantum Annealers

Distributed non-negative RESCAL with Automatic Model Selection for Exascale Data

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