International Statistical Review

A Unifying Framework and Comparison of Algorithms for Non‐negative Matrix Factorisation

Early View

Summary Non‐negative matrix factorisation (NMF) is an increasingly popular unsupervised learning method. However, parameter estimation in the NMF model is a difficult high‐dimensional optimisation problem. We consider algorithms of the alternating least squares type. Solutions to the least squares problem fall in two categories. The first category is iterative algorithms, which include algorithms such as the majorise–minimise (MM) algorithm, coordinate descent, gradient descent and the Févotte‐Cemgil expectation–maximisation (FC‐EM) algorithm. We introduce a new family of iterative updates based on a generalisation of the FC‐EM algorithm. The coordinate descent, gradient descent and FC‐EM algorithms are special cases of this new EM family of iterative procedures. Curiously, we show that the MM algorithm is never a member of our general EM algorithm. The second category is based on cone projection. We describe and prove a cone projection algorithm tailored to the non‐negative least square problem. We compare the algorithms on a test case and on the problem of identifying mutational signatures in human cancer. We generally find that cone projection is an attractive choice. Furthermore, in the cancer application, we find that a mix‐and‐match strategy performs better than running each algorithm in isolation.

Related Topics

Related Publications

Related Content

Site Footer

Address:

This website is provided by John Wiley & Sons Limited, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ (Company No: 00641132, VAT No: 376766987)

Published features on StatisticsViews.com are checked for statistical accuracy by a panel from the European Network for Business and Industrial Statistics (ENBIS)   to whom Wiley and StatisticsViews.com express their gratitude. This panel are: Ron Kenett, David Steinberg, Shirley Coleman, Irena Ograjenšek, Fabrizio Ruggeri, Rainer Göb, Philippe Castagliola, Xavier Tort-Martorell, Bart De Ketelaere, Antonio Pievatolo, Martina Vandebroek, Lance Mitchell, Gilbert Saporta, Helmut Waldl and Stelios Psarakis.