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High precision analytical approximation is proposed for variance-covariance based risk allocation in a portfolio of risky assets. A general case of a single-period multi-factor Merton-type model with stochastic recovery is considered. The…

Risk Management · Quantitative Finance 2009-09-28 Mikhail Voropaev

This paper provides a theoretical explanation on the clustering aspect of nonnegative matrix factorization (NMF). We prove that even without imposing orthogonality nor sparsity constraint on the basis and/or coefficient matrix, NMF still…

Machine Learning · Computer Science 2010-06-15 Andri Mirzal , Masashi Furukawa

We apply the vectorized Non-negative Matrix Factorization (NMF) method to post-processing of direct imaging data for exoplanetary systems such as circumstellar disks. NMF is an iterative approach, which first creates a non-orthogonal and…

Instrumentation and Methods for Astrophysics · Physics 2018-03-20 Bīn Rén , Laurent Pueyo , Guangtun Ben Zhu , John Debes , Gaspard Duchêne

In this article we propose a method to refine the clustering results obtained with the nonnegative matrix factorization (NMF) technique, imposing consistency constraints on the final labeling of the data. The research community focused its…

Computer Vision and Pattern Recognition · Computer Science 2016-09-16 Rocco Tripodi , Sebastiano Vascon , Marcello Pelillo

Bayesian Non-negative Matrix Factorization (NMF) is a promising approach for understanding uncertainty and structure in matrix data. However, a large volume of applied work optimizes traditional non-Bayesian NMF objectives that fail to…

Machine Learning · Statistics 2018-03-19 M. Arjumand Masood , Finale Doshi-Velez

Non-negative matrix factorization (NMF) is one of the most popular decomposition techniques for multivariate data. NMF is a core method for many machine-learning related computational problems, such as data compression, feature extraction,…

Numerical Analysis · Computer Science 2017-12-07 Gabriele Torre , Michael Graber

Nonnegative matrix factorization (NMF) seeks a low-rank approximation $X \approx UV^T$ with nonnegative factors and is commonly solved using interior methods that enforce feasibility throughout optimization. We show that such…

Machine Learning · Computer Science 2026-05-20 Qiujing Lu , Tonmoy Monsoor , Ehsan Ebrahimzadeh , Kartik Sharma , Vwani Roychowdhury

In this paper we consider the Nonnegative Matrix Factorization (NMF) problem: given an (elementwise) nonnegative matrix $V \in \R_+^{m\times n}$ find, for assigned $k$, nonnegative matrices $W\in\R_+^{m\times k}$ and $H\in\R_+^{k\times n}$…

Optimization and Control · Mathematics 2007-05-23 Lorenzo Finesso , Peter Spreij

Symmetric Nonnegative Matrix Factorization (SNMF) models arise naturally as simple reformulations of many standard clustering algorithms including the popular spectral clustering method. Recent work has demonstrated that an elementary…

Computer Vision and Pattern Recognition · Computer Science 2016-09-20 Reza Borhani , Jeremy Watt , Aggelos Katsaggelos

Portfolio optimization methods have evolved significantly since Markowitz introduced the mean-variance framework in 1952. While the theoretical appeal of this approach is undeniable, its practical implementation poses important challenges,…

Portfolio Management · Quantitative Finance 2024-05-28 Adil Rengim Cetingoz , Olivier Guéant

Non-negative Matrix Factorization (NMF) is a useful method to extract features from multivariate data, but an important and sometimes neglected concern is that NMF can result in non-unique solutions. Often, there exist a Set of Feasible…

Applications · Statistics 2021-01-20 Ragnhild Laursen , Asger Hobolth

There has been a recent critical need to study fairness and bias in machine learning (ML) algorithms. Since there is clearly no one-size-fits-all solution to fairness, ML methods should be developed alongside bias mitigation strategies that…

Machine Learning · Computer Science 2026-03-06 Lara Kassab , Erin George , Deanna Needell , Haowen Geng , Nika Jafar Nia , Aoxi Li

Non-Negative Matrix Factorization (NMF) is a widely used dimension reduction method that factorizes a non-negative data matrix into two lower dimensional non-negative matrices: One is the basis or feature matrix which consists of the…

Applications · Statistics 2022-11-03 Yun Cai , Hong Gu , Toby Kenney

Non-negative matrix factorisation (NMF) is a widely used tool for unsupervised learning and feature extraction, with applications ranging from genomics to text analysis and signal processing. Standard formulations of NMF are typically…

Machine Learning · Computer Science 2026-03-11 Elisabeth Sommer James , Asger Hobolth , Marta Pelizzola

Non-negative matrix factorization (NMF) has become a popular method for representing meaningful data by extracting a non-negative basis feature from an observed non-negative data matrix. Some of the unique features of this method in…

Optimization and Control · Mathematics 2022-11-15 Sajad Fathi Hafshejani , Zahra Moaberfard

In this paper, we introduce and provide a short overview of nonnegative matrix factorization (NMF). Several aspects of NMF are discussed, namely, the application in hyperspectral imaging, geometry and uniqueness of NMF solutions,…

Numerical Analysis · Computer Science 2017-03-03 Nicolas Gillis

Symmetric Nonnegative Matrix Factorization (SymNMF) is a technique in data analysis and machine learning that approximates a symmetric matrix with a product of a nonnegative, low-rank matrix and its transpose. To design faster and more…

Machine Learning · Computer Science 2024-12-02 Koby Hayashi , Sinan G. Aksoy , Grey Ballard , Haesun Park

A fractal approach to the long-short portfolio optimization is proposed. The algorithmic system based on the composition of market-neutral spreads into a single entity was considered. The core of the optimization scheme is a fractal walk…

Portfolio Management · Quantitative Finance 2016-12-20 Sergey Kamenshchikov , Ilia Drozdov

The model described in this paper belongs to the family of non-negative matrix factorization methods designed for data representation and dimension reduction. In addition to preserving the data positivity property, it aims also to preserve…

Machine Learning · Computer Science 2022-09-23 Rachid Hedjam , Abdelhamid Abdesselam , Abderrahmane Rahiche , Mohamed Cheriet

Symmetric nonnegative matrix factorization (SymNMF) has important applications in data analytics problems such as document clustering, community detection and image segmentation. In this paper, we propose a novel nonconvex variable…

Optimization and Control · Mathematics 2017-03-27 Songtao Lu , Mingyi Hong , Zhengdao Wang