Binary Matrix Factorization and Consensus Algorithms

Fu, Yinghua; Nian-ping, Jiang; Sun, Hong

doi:10.1109/icece.2010.1455

Cited by 3 publications

(7 citation statements)

References 7 publications

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“…gives us a matrix with real values, we have rounded the matrix values to the nearest integer and called it the output matrix and then calculated error. We can see clearly from Table 3 and Figure 3 that PLRMF (7) which works only on finite fields is performing far better than NMF. It is important to note here that the input rank used in both the algorithms PLRMF (7) and NMF in Table 3 varies from 20 to 100.…”

Section: Experimental Results On Realworld Datamentioning

confidence: 89%

“…We can see clearly from Table 3 and Figure 3 that PLRMF (7) which works only on finite fields is performing far better than NMF. It is important to note here that the input rank used in both the algorithms PLRMF (7) and NMF in Table 3 varies from 20 to 100. However, since the output of NMF algorithm is two real-valued factor matrices U and V, and to get the output image the real values in UV were rounded up to the nearest integer.…”

Section: Experimental Results On Realworld Datamentioning

confidence: 89%

“…This image is obtained by changing values between 0−255 to 7 distinct values of an image from [34]. For running PLRMF (7) on this image, we mapped those 7 shades to the numbers 0, 1, . .…”

Section: Experimental Results On Realworld Datamentioning

confidence: 99%

“…Gillis et al [28] proved that F 2 -MF is NP-hard already for r = 1. Since the problems over finite fields are computationally much more challenging, it is not surprising that most of the practical approaches for handling these problems are heuristics [7][8][9][10][11].…”

Section: Could the Ideas Behind The Theoretical Advances On Bmf Be Us...mentioning

confidence: 99%

“…The large number of applications requiring Boolean or binary matrix factorization has given raise to many interesting heuristic algorithms for solving these computationally hard problems [7][8][9][10][11][12]. In the theory community, also several algorithms for such problems were developed, including efficient polynomial-time approximation schemes (EPTAS) [13,14].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Boolean and $\mathbb{F}_p$-Matrix Factorization: From Theory to Practice

Fomin¹,

Panolan²,

Patil³

et al. 2022

Preprint

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Boolean Matrix Factorization (BMF) aims to find an approximation of a given binary matrix as the Boolean product of two low-rank binary matrices. Binary data is ubiquitous in many fields, and representing data by binary matrices is common in medicine, natural language processing, bioinformatics, computer graphics, among many others. Factorizing a matrix into low-rank matrices is used to gain more information about the data, like discovering relationships between the features and samples, roles and users, topics and articles, etc. In many applications, the binary nature of the factor matrices could enormously increase the interpretability of the data.Unfortunately, BMF is computationally hard and heuristic algorithms are used to compute Boolean factorizations. Very recently, the theoretical breakthrough was obtained independently by two research groups. Ban et al. (SODA 2019) and Fomin et al. (Trans. Algorithms 2020) show that BMF admits an efficient polynomial-time approximation scheme (EPTAS). However, despite the theoretical importance, the high double-exponential dependence of the running times from the rank makes these algorithms unimplementable in practice. The primary research question motivating our work is whether the theoretical advances on BMF could lead to practical algorithms.The main conceptional contribution of our work is the following. While EPTAS for BMF is a purely theoretical advance, the general approach behind these algorithms could serve as the basis in designing better heuristics. We also use this strategy to develop new algorithms for related F p -Matrix Factorization. Here, given a matrix A over a finite field GF(p) where p is a prime, and an integer r, our objective is to find a matrix B over the same field with GF(p)-rank at most r minimizing some norm of A − B. Our empirical research on synthetic and real-world data demonstrates the advantage of the new algorithms over previous * The research received funding from the Research Council of Norway via the project BWCA (grant no. 314528) and IIT Hyderabad via Seed grant (SG/IITH/F224/2020-21/SG-79). The work is conducted while Anurag Patil and Adil Tanveer were students at IIT Hyderabad.

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Section: Experimental Results On Realworld Datamentioning

confidence: 89%

Section: Experimental Results On Realworld Datamentioning

confidence: 89%

Section: Experimental Results On Realworld Datamentioning

confidence: 99%

Section: Could the Ideas Behind The Theoretical Advances On Bmf Be Us...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Boolean and $\mathbb{F}_p$-Matrix Factorization: From Theory to Practice

Fomin¹,

Panolan²,

Patil³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

Boolean and $\mathbb{F}_{p}$-Matrix Factorization: From Theory to Practice

Fomin

Panolan

Patil³

et al. 2022

2022 International Joint Conference on Neural Networks (IJCNN)

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Boolean Matrix Factorization (BMF) aims to find an approximation of a given binary matrix as the Boolean product of two low-rank binary matrices. Binary data is ubiquitous in many fields, and representing data by binary matrices is common in medicine, natural language processing, bioinformatics, computer graphics, among many others. Factorizing a matrix into lowrank matrices is used to gain more information about the data, like discovering relationships between the features and samples, roles and users, topics and articles, etc. In many applications, the binary nature of the factor matrices could enormously increase the interpretability of the data.Unfortunately, BMF is computationally hard and heuristic algorithms are used to compute Boolean factorizations. Very recently, the theoretical breakthrough was obtained independently by two research groups. Ban et al. (SODA 2019) and Fomin et al. (Trans. Algorithms 2020) show that BMF admits an efficient polynomial-time approximation scheme (EPTAS). However, despite the theoretical importance, the high double-exponential dependence of the running times from the rank makes these algorithms unimplementable in practice. The primary research question motivating our work is whether the theoretical advances on BMF could lead to practical algorithms.The main conceptional contribution of our work is the following. While EPTAS for BMF is a purely theoretical advance, the general approach behind these algorithms could serve as the basis in designing better heuristics. We also use this strategy to develop new algorithms for related Fp-Matrix Factorization. Here, given a matrix A over a finite field GF(p) where p is a prime, and an integer r, our objective is to find a matrix B over the same field with GF(p)-rank at most r minimizing some norm of A − B. Our empirical research on synthetic and realworld data demonstrates the advantage of the new algorithms over previous works on BMF and Fp-Matrix Factorization.

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Approximation Schemes for Low-rank Binary Matrix Approximation Problems

Fomin

Golovach

Lokshtanov

et al. 2019

ACM Trans. Algorithms

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We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constrains. The new constrained clustering problem encompasses a number of problems and by solving it, we obtain the first linear timeapproximation schemes for a number of well-studied fundamental problems concerning clustering of binary vectors and low-rank approximation of binary matrices. Among the problems solvable by our approach are Low GF(2)-Rank Approximation, Low Boolean-Rank Approximation, and various versions of Binary Clustering. For example, for Low GF(2)-Rank Approximation problem, where for an m × n binary matrix A and integer r > 0, we seek for a binary matrix B of GF(2) rank at most r such that 0 norm of matrix A − B is minimum, our algorithm, for any > 0 in time f (r, ) · n · m, where f is some computable function, outputs a (1+ )-approximate solution with probability at least (1− 1 e ). Our approximation algorithms substantially improve the running times and approximation factors of previous works. We also give (deterministic) PTASes for these problems running in time n f (r) 1 2 log 1 , where f is some function depending on the problem. Our algorithm for the constrained clustering problem is based on a novel sampling lemma, which is interesting in its own. * The research leading to these results have been supported by the Research Council of Norway via the projects "CLASSIS" and "MULTIVAL".

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Binary Matrix Factorization and Consensus Algorithms

Cited by 3 publications

References 7 publications

Boolean and $\mathbb{F}_p$-Matrix Factorization: From Theory to Practice

Boolean and $\mathbb{F}_p$-Matrix Factorization: From Theory to Practice

Boolean and $\mathbb{F}_{p}$-Matrix Factorization: From Theory to Practice

Approximation Schemes for Low-rank Binary Matrix Approximation Problems

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