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Ternary Matrix Factorization: problem definitions and algorithms

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Abstract

Can we learn from the unknown? Logical data sets of the ternary kind are often found in information systems. They contain unknown as well as true/false values. An unknown value may represent a missing entry (lost or indeterminable) or have meaning, like a Don’t Know response in a questionnaire. In this paper, we introduce algorithms for reducing the dimensionality of logical data (categorical data in general) in the context of a new data mining challenge: Ternary Matrix Factorization (TMF). For a ternary data matrix, TMF exploits ternary logic to produce a basis matrix (which holds the major patterns in the data) and a usage matrix (which maps patterns to original observations). Both matrices are interpretable, and their ternary matrix product approximates the original matrix. TMF has applications in (1) finding targeted structure in ternary data, (2) imputing values through pattern discovery in highly incomplete categorical data sets, and (3) solving instances of its encapsulated Binary Matrix Factorization problem. Our elegant algorithm FasTer (FASt TERnary Matrix Factorization) has linear run-time complexity with respect to the dimensions of the data set and is parameter-robust. A variant of FasTer that exploits useful results from combinatorics provides accuracy bounds for a core part of the algorithm in certain situations. Experiments on synthetic and real-world data sets show that our algorithms are able to outperform state-of-the-art techniques in all three TMF applications with respect to run-time and effectiveness. Finally, convincing speedup and efficiency results on a parallel version of FasTer demonstrate its suitability for weak- and strong-scaling scenarios.

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Notes

  1. Data from the UCI Machine Learning Repository.

  2. Other mapping choices yield analogous results.

  3. Using \(\tau =0.5\) for the Asso rounding threshold gives the most sensible results here.

  4. dropbox.com/s/znut1tsxutyjfvu/tmf.zip.

  5. For imputation problems, values of \(\mathfrak {u}\) are replaced with \(\mathfrak {f}\) in this selection to ensure that only binary values are present in the initial basis vector.

  6. We focus on the TUP rather than the TBP because, in the general case, the TBP involves ternary logic and cannot be easily compared with or reduced to combinatorial problems.

  7. dropbox.com/s/znut1tsxutyjfvu/tmf.zip.

  8. Aside from \(\rho _\mathfrak {u}\) which is not applicable in BMF.

  9. The publicly available implementation was used.

  10. We thank Radim Belohlavek for sharing an up-to-date implementation.

  11. We thank Pauli Miettinen for sharing an up-to-date implementation.

  12. We use the author’s original implementation with YALMIP and SDPT3 as the SPD solver.

  13. api.stackexchange.com.

  14. dropbox.com/s/znut1tsxutyjfvu/tmf.zip.

  15. “Search and research” is the first point of advice on SO’s How to Ask page.

  16. One of the highest-rated questions on the site, for example, is simply “What is a plain English explanation of Big-O?”

  17. dropbox.com/s/znut1tsxutyjfvu/tmf.zip.

  18. A nonparametric test that does not assume the population to be normally distributed.

  19. dropbox.com/s/znut1tsxutyjfvu/tmf.zip.

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Acknowledgments

This work is supported by the Helmholtz Young Investigators Groups programme.

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Authors and Affiliations

  1. Helmholtz Zentrum München, Technische Universität München, 85764, Neuherberg, Germany

    Samuel Maurus & Claudia Plant

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  1. Samuel Maurus
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  2. Claudia Plant
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Correspondence to Samuel Maurus.

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Maurus, S., Plant, C. Ternary Matrix Factorization: problem definitions and algorithms. Knowl Inf Syst 46, 1–31 (2016). https://doi.org/10.1007/s10115-015-0838-3

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