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.
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.
Introduction of locking compression plate (LCP) has revolutionised the management of supracondylar fracture femur. Material & Methods: Prospective observational study carried out JMF ACPM Medical College on the patients having fracture distal femur between November 2015 to December 2017. Study done on 36 patients, (NEER'S Classification) age range of 18 to 70 years (irrespective of their sex), were subjected to fixation by locking compression plate after obtaining thorough written informed consent.Result: Functional evaluation of LCP by Neers criteria was excellent in 61.5% patient's upto 50 years age and in 40% more than 50 years old. There was statistically no significant (p>0.05) difference of functional status of patients in between the age group of patients with distal femur fracture, which were stabilized by LCP. Locking compression plate is safe procedure for supracondylar femur fracture with excellent functional outcome and early clinical and radiological union with minimum complications.
Introduction: Traumatic dislocation of hip associated with ipsilateral femur neck and acetabulum fracture is a rare and devastating injury as a result of high-energy trauma, with potential complications of femur head aseptic necrosis and hip joint degenerative arthritis. Patients, postoperatively, may encounter decreased functional outcomes and serious challenge, during activities of daily living. Management guidelines lack clarity, with possible options including fracture fragments fixation or hip joint replacement. Case Report: A 45-year-old patient was diagnosed with posterior dislocation of hip with ipsilateral femur neck and acetabulum fracture. Plan for emergent hip reduction and primary surgical fixation of fracture fragments was made. Intraoperatively, posterior wall acetabulum fracture was found to be excessively comminuted and non-amenable for fixation and femoral head was found to be avascularized. Subsequently, plan was changed to total hip replacement with acetabular defect reconstruction. At 1-year follow-up, no aseptic loosening was observed on radiographic views of pelvis, with total Harris Hip Score of 91. Conclusion: Early operative management for such complex injuries is a necessity. Definitive management in the form of primary fracture fixation/arthroplasty should be decided intraoperatively. Keywords: Femur neck, hip dislocation, acetabulum fracture, ipsilateral, case report.
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